Answers to Holt Chapter 9

CHAPTER 

9 

Solutions Key 

Quadratic Functions and Equations 

ARE YOU READY? PAGE 587 

1. A 2. D 

3. C 4. B 

5. x y = -2x + 8 (x, y) 

-4 y = -2(-4) + 8 = 16 (-4, 16) 

-2 y = -2(-2) + 8 = 12 (-2, 12) 

0 y = -2(0) + 8 = 8 (0, 8) 

2 y = -2(2) + 8 = 4 (2, 4) 

4 y = -2(4) + 8 = 0 (4, 0) 

 

 

8. x y = 2 x 2 (x, y) 

-2 y = 2 (-2) 2 = 8 (-2, 8) 

-1 y = 2 (-1) 2 = 2 (-1, 2) 

0 y = 2 (0) 2 = 0 (0, 0) 

1 y = 2 (1) 2 = 2 (1, 2) 

2 y = 2 (2) 2 = 8 (2, 8) 

Draw a smooth curve through all the points to show 

all ordered pairs that satisfy the function. Draw 

arrowheads on both “ends” of the curve. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6. x y = (x + 1) 2 (x, y) 

-3 y = (-3 + 1) 2 = 4 (-3, 4) 

-2 y = (-2 + 1) 2 = 1 (-2, 1) 

-1 y = (-1 + 1) 2 = 0 (-1, 0) 

0 y = (0 + 1) 2 = 1 (0, 1) 

1 y = (1 + 1) 2 = 4 (1, 4) 

 

 

 

 

 

7. x y = x 2 + 3 (x, y) 

-2 y = (-2) 2 + 3 = 7 (-2, 7) 

-1 y = (-1) 2 + 3 = 4 (-1, 4) 

0 y = (0) 2 + 3 = 3 (0, 3) 

1 y = (1) 2 + 3 = 4 (1, 4) 

2 y = (2) 2 + 3 = 7 (2, 7) 

 

 

9. (m + 2)(m + 5) 

= m(m) + m(5) + 2(m) + 2(5) 

= m 2 + 5m + 2m + 10 

= m 2 + 7m + 10 

10. (y - 7)(y + 2) 

= y(y) + y(2) - 7(y) - 7(2) 

= y 2 + 2y - 7y - 14 

= y 2 - 5y - 14 

11. (2a + 4)(5a + 6) 

= 2a(5a) + 2a(6) + 4(5a) + 4(6) 

= 10 a 2 + 12a + 20a + 24 

= 10 a 2 + 32a + 24 

12. (x + 1)(x + 1) = (x) 2 + 2(x)(1) + (1) 2 

= x 2 + 2x + 1 

13. (t + 5)(t + 5) = (t) 2 + 2(t)(5) + (5) 2 

= t 2 + 10t + 25 

14. (3n - 8)(3n - 8) = (3n) 2 - 2(3n)(8) + (8) 2 

= 9 n 2 - 48n + 64 

16. Factors of -2 Sum 

15. Factors of 1 Sum 

= (x - 1) 2 

1 · 1 2 ✗ 

2 · -1 1 ✗ 

(-1) · (-1) -2 ✓ 

-2 · 1 -1 ✓ 

(x - 1)(x - 1) 

(x - 2)(x + 1) 

17. Factors of 5 Sum 

18. Factors of -12 Sum 

 

 

 

 

 

5 · 1 6 ✗ 

(-5) · (-1) -6 ✓ 

(x - 5)(x - 1) 

(-12) · 1 -11 ✗ 

(-6) · 2 -4 ✗ 

(-4) · 3 -1 ✓ 

(x - 4)(x + 3) 

Copyright © by Holt, Rinehart and Winston. 333 Holt Algebra 1 

All rights reserved.

19. Factors of 18 Sum 20. Factors of -18 Sum 

(-18) · (-1) -19 ✗ (-18) · 1 -17 ✗ 

(-9) · (-2) -11 ✗ (-9) · 2 -7 ✓ 

(-6) · (-3) -9 ✓ (x - 9)(x + 2) 

(x - 6)(x - 3) 

21. √ 36 = √ 6 2 = 6 22. √ 121 = √ 11 2 = 11 

23. - √ 64 = - √ 8 2 = -8 

24. √ 16 √ 81 = √ 4 2 √ 9 2 = (4)(9) = 36 

___ 

( 3__ 

5) 2 = 3__ 

5 

25. √ 9 

25 = √ 

26. - √ 6(24) = - √ 144 = - √ 12 2 = -12 

27. 3m + 5 = 11 

______ - 5 -5 ___ 

3m = 6 

___ 3m 

3 = 6__ 

3 

m = 2 

29. 5n + 13 = 28 

_______ - 13 ____ -13 

5n = 15 

___ 5n 

5 = ___ 15 

5 

n = 3 

31. 10 = r__ 

3 + 8 

___ -8 _____ - 8 

2 = r__ 

3 

3(2) = 3 ( r__ 3) 

6 = r 

28. 3t + 4 = 10 

_____ - 4 ___ -4 

3t = 6 

__ 3t 

3 = 6__ 

3 

t = 2 

30. 2(k - 4) + k = 7 

2(k) + 2(-4) + k = 7 

2k - 8 + k = 7 

3k - 8 = 7 

______ + 8 ___ +8 

3k = 15 

___ 3k 

3 = ___ 15 

3 

k = 5 

32. 2(y - 6) = 8.6 

2(y) + 2(-6) = 8.6 

2y - 12 = 8.6 

_______ + 12 ______ +12 

2y = 20.6 

___ 2y 

2 = ____ 20.6 

2 

y = 10.3 

2a. 

b. 

x y = x 2 + 2 

 

-2 6 

-1 3 

0 2 

1 3 

 

2 6 

x y = -3 x 2 + 1 

-2 -11 

-1 -2 

0 1 

1 -2 

2 -11 

3a. f(x) = -4 x 2 - x + 1 

a = -4 

Since a < 0, the parabola opens downward. 

b. y - 5 x 2 = 2x - 6 

______ + 5 x 2 ______ +5 x 2 

y = 5 x 2 + 2x - 6 

a = 5 

Since a > 0, the parabola opens upward. 

4a. The vertex is (-2, 5) and the maximum is 5. 

b. The vertex is (3, -1) and the minimum is -1. 

5a. The vertex is (-2, -4), so the minimum is -4. 

Domain: all real numbers; Range: y ≥ -4 

b. The vertex is (2, 3), so the maximum is 3. 

Domain: all real numbers; Range: y ≤ 3 

 

 

THINK AND DISCUSS, PAGE 593 

1. If the second differences are constant, the function 

is quadratic. If the function can be written in the form 

y = a x 2 + bx + c, it is quadratic. 

2. 

 

 

 

 

 

 

 

9-1 IDENTIFYING QUADRATIC 

FUNCTIONS, PAGES 590–597 

 

 

 

CHECK IT OUT! PAGES 591–593 

1a. This function is quadratic. The second differences 

are constant. 

b. y + x = 2 x 2 

_____ - x ___ -x 

y = 2 x 2 - x 

This is a quadratic function because it can be written 

in the form y = a x 2 + bx + c, where a = 2, b = -1, 

and c = 0. 

 

 

EXERCISES, PAGES 594–597 

GUIDED PRACTICE, PAGE 594 

1. minimum 

2. y + 6x = -14 

_____ - 6x ____ -6x 

y = -6x - 14 

This is not a quadratic function because the value of 

a is 0, and the function cannot be written in the form 

y = a x 2 + bx + c. 



3. 2 x 2 + y = 3x - 1 

________ -2 x 2 ______ -2 x 2 

y = -2 x 2 + 3x - 1 


in the form y = a x 2 + bx + c, where a = -2, b = 3, 

and c = -1. 

4. This function is quadratic. The second differences 

are constant. 

5. This function is not quadratic. The second 

differences are not constant. 

6. 

x y = 4 x 2 

 

-2 16 

-1 4 

 

0 0 

 

1 4 

2 16 

7. 

8. 

9. 

x 

y = 1__ 

-2 2 

-1 1__ 

2 

0 0 

1 1__ 

2 

2 2 

x y = - x 2 + 1 

-2 -3 

-1 0 

0 1 

1 0 

2 -3 

2 x 2 

 

 

 

 

 

x y = -5 x 2 

 

 

-2 -20 

 

-1 -5 

0 0 

1 -5 

2 -20 

10. y = -3 x 2 + 4x 

a = -3 


11. y = 1 - 2x + 6 x 2 

a = 6 


12. y + x 2 = -x - 2 

_____ - x 2 _______ - x 2 

y = - x 2 - x - 2 

a = -1 


 

 

 

 

 

 

 

13. y + 2 = x 2 

____ - 2 ___ -2 

y = x 2 - 2 


14. y - 2 x 2 = -3 

______ + 2 x 2 _____ +2 x 2 

y = 2 x 2 - 3 


15. y + 2 + 3 x 2 = 1 

__________ - 3 x 2 _____ -3 x 2 

y + 2 = -3 x 2 + 1 

____ - 2 ________ - 2 

y = -3 x 2 - 1 


16. The vertex is (-1, 3) and the maximum is 3. 

17. The vertex is (-3, -4) and the minimum is -4. 

18. The vertex is (-1, -4), so the minimum is -4. 


19. The vertex is (3, 4), so the maximum is 4. 




21. The vertex is (4, -4), so the minimum is -4. 


PRACTICE AND PROBLEM SOLVING, PAGES 595–597 

22. This function is not quadratic. The second 

differences are not constant. 

23. -3 x 2 + x = y - 11 

________ + 11 ______ + 11 

-3 x 2 + x + 11 = y 



and c = 11. 


are constant. 

25. y = 2__ 

3 x - 4__ 

9 + 1__ 

6 x 2 

y = 1__ 

6 x 2 + 2__ 

3 x - 4__ 

9 


26. 

in the form y = a x 2 + bx + c, where a = 1__ 

and c = - 

4__ 

9 . 

x y = x 2 - 5 

-2 -1 

-1 -4 

0 -5 

1 -4 

2 -1 

 

 

 

 

 

 

 

6 , b = 2__ 

3 , 



27. 

28. 

29. 

x 

1__ y = - 

2 x 2 

 

 

 

-2 -2 

-1 - 

1__ 

2 

0 0 

1 - 

1__ 

2 

2 -2 

x y = -2 x 2 + 2 

-2 -6 

-1 0 

0 2 

1 0 

2 -6 

x y = 3 x 2 - 2 

-2 10 

-1 1 

0 -2 

1 1 

2 10 

30. y = 7 x 2 - 4x 

a = 7 


31. x - 3 x 2 + y = 5 

___________ 

-x ___ -x 

-3 x 2 + y = -x + 5 

________ +3 x 2 _______ +3 x 2 

y = 3 x 2 - x + 5 

a = 3 


32. y = - 

2__ 

3 x 2 

a = - 

2__ 

3 


33. The vertex is (0, -5) and the minimum is -5. 

34. The vertex is (1, -3) and the maximum is -3. 



36. The vertex is (-4, 2), so the minimum is 2. 

Domain: all real numbers; Range: y ≥ 2 

37. The vertex is (-2, -2), so the minimum is -2. 



Domain: all real anumbers; Range: y ≤ 4 

 

 

 

 

 

 

 

 

39. never 40. never 

41. always 42. sometimes 

43. sometimes 44. always 

45. This is not a quadratic function because the value of 

a is 0. 

 

 

 

 

 

 

46. y = 2 x 2 - 5 + 3x 

y = 2 x 2 + 3x - 5 


in the form y = a x 2 + bx + c, where a = 2, b = 3, 

and c = -5. 

47. y = (x + 1) 2 

y = (x) 2 + 2(x)(1) + (1) 2 

y = x 2 + 2x + 1 



and c = 1. 

48. y = 5 - (x - 1) 2 

y = 5 - ( (x) 2 - 2(x)(1) + (1) 2 ) 

y = 5 - ( x 2 - 2x + 1) 

y = 5 - x 2 - (-2x) - (1) 

y = 5 - x 2 + 2x - 1 

y = - x 2 + 2x + 4 



and c = 4. 

49. This is a quadratic function because it can be written 


and c = -9. 

50. This is not a quadratic function because it contains 

a power greater than 2. 

51a. about 0.375 s b. about 2.25 m 

c. The independent variable x represents the time 

since the volleyball is served, so this makes sense 

only for nonnegative numbers. 

52a. 

b. x ≥ 0 

y = 16 x 2 

0 0 

1 32 

 

2 32 

 

3 0 

 

x 48x - 

 

 

 

 

 

 

c. No; the maximum y-value on the parabola is less 

than 50. 

53. y = 1__ 

2 x - x 2 

y = - x 2 + 1__ 

2 x 


in the form y = a x 2 + bx + c, where a = -1, b = 1__ 

2 , 

and c = 0. 

54. This is a linear function because it can be written in 

the form y = mx + b, where m = 1__ and b = -3. 

2 

 



55. y + 3 = - x 2 

____ - 3 ___ -3 

y = - x 2 - 3 



and c = -3. 

56. y - 2 x 2 = 0 

______ + 2 x 2 _____ +2 x 2 

y = 2 x 2 



and c = 0. 

57. y = 1__ 

2 x ( x 2 ) 

y = 1__ 

2 x 3 

This is not a quadratic or linear function because the 

power of x is greater than 2. 

58. This is not a quadratic or linear function because the 

x appears in the denominator of a fraction. 

59. This is a linear function because it can be written in 

the form y = mx + b, where m = 3__ and b = 0. 

2 



and c = 1. 

61a. 

x 

y = -16 t 2 + 32t 

0 0 

1 16 

2 0 

 

 

 

b. t ≥ 0 c. 16 ft 

d. 2 s 

 

 

 

 

 

 

62a. In a linear function, the first differences will be 

constant for a constant change in x. In a quadratic 

function, the second differences will be constant for 

a constant change in x. 

b. A linear function can be written in the form 

y = mx + b. A quadratic function can be written in 

the form y = a x 2 + bx + c, where a, b, and c are 

real numbers and a is not equal to 0. 

c. The graph of a linear function is a line. The graph 

of a quadratic function is a parabola. 

 

63a. 

b. 

 

Time (s) Height (m) 

0 0 

1 34.3 

2 58.8 

3 73.5 

4 78.4 

5 73.5 

6 58.8 

7 34.3 

8 0 

 

 

 

 

 

 

 

+ 34.3 

+ 24.5 

+ 14.7 

+ 4.9 

- 4.9 

- 14.7 

- 24.5 

- 34.3 

c. The acceleration is the absolute value of the 

second differences. 

- 9.8 

- 9.8 

- 9.8 

- 9.8 

- 9.8 

- 9.8 

- 9.8 

64. After the equation is solved for y, x 2 will be negative. 

This means the parabola opens downward and has 

a maximum. 

TEST PREP, PAGE 597 

65. C; since graph C is a parabola, it represents a 

quadratic function. 

66. J; since y + 3 x 2 = 9 is the same as 

y = -3 x 2 + 9, the value of a is less than 0. 

Therefore, the parabola opens downward and the 

function has a maximum. 

67. f(x) = 5 - 2 x 2 + 3x 

f(x) = -2 x 2 + 3x + 5 

x y = 5 - 2 x 2 + 3x 

 

-2 -9 

-1 0 

0 5 

1 6 

 

2 3 

Yes; the function can be written in the form 

y = a x 2 + bx + c, and the graph forms a parabola. 

CHALLENGE AND EXTEND, PAGE 597 

68. A = lw 

= (2x + 10)(2x + 6) 

= 2x(2x) + 2x(6) + 10(2x) + 10(6) 

= 4 x 2 + 12x + 20x + 60 

= 4 x 2 + 32x + 60 

69. f(x) = x 2 - 4 


f(x) = - (x + 2) 2 


 

 



SPIRAL REVIEW, PAGE 597 

70. 10,000 = 10 · 10 · 10 · 10 = 10 4 

71. 16 = (-2)(-2)(-2)(-2) = (-2) 4 

72. ___ 8 

27 = 2__ 

3) 3 

3 · 2__ 

3 · 2__ 

3 = ( 2__ 

73. Let x represent the acutal distance. 

14 

1__ 

1__ 

3 = ____ 4 

x 

x = 3 ( 14 1__ 4) 

x = 42 3__ 

4 

The actual distance from Lin’s home to the park is 

42 3__ 

4 mi. 

74. r(c) = 25c + 400, where r is the total registration 

money and c is the number of additional campers. 

The additional number of campers can only be 

whole numbers up to 3. A reasonable domain is 

{0, 1, 2, 3} 

c 0 1 2 3 

r(c) = 25c + 400 400 425 450 475 

So a reasonable range is {400, 425, 450, 475}. 

75. p(h) = 9h, where p is Sal’s pay and h is the number 

of hours worked. 

Sal works between 30 and 35 hours. A reasonable 

domain is {31, 32, 33, 34}. 

h 31 32 33 34 

p(h) = 9h 279 288 297 306 

So a reasonable range is {279, 288, 297, 306}. 

ALGEBRA LAB: EXPLORE THE AXIS OF 

SYMMETRY, PAGE 598 

ACTIVITY, PAGE 598 

1. a 1 -2 -1 

b -2 -8 4 

b__ 

a 

Axis of Symmetry 

(from Graph) 

2. - 

1__ 

2 

TRY THIS, PAGE 598 

-2 4 -4 

x = 1 x = -2 x = 2 

3. x = - 

1__ 

2( b__ 

1. x = -3 2. x = -1 

3. x = 2 4. x = 3__ 

2 

5. x = 0 6. x = - 

1__ 

6 

___ 

= 

a) -b 

2a 

9-2 CHARACTERISTICS OF QUADRATIC 



1a. The graph does not cross the x-axis, so there are no 

zeros of this function. 

b. The only zero appears to be 3. 

2a. The axis of symmetry is x = -3. 

b. _______ -2 + 4 = 2__ 

2 2 = 1 

The axis of symmetry is x = 1. 

3. a = 2, b = 1 

b 

x = -___ 

2a = - ____ 1 

2(2) = - 1__ 

4 

The axis of symmetry is x = - 

1__ 

4 . 

4. a = 1, b = -4 

b 

x = -___ 

2a = - ____ -4 

2(1) = 4__ 

2 = 2 

y = x 2 - 4x - 10 

= (2) 2 - 4(2) - 10 

= 4 - 8 - 10 = -14 

The vertex is (2, -14). 

5. a = -0.07, b = 0.42 

b 

x = -___ 

2a = - ________ 0.42 

2(-0.07) = - ______ 0.42 

-0.14 = 3 

f(x) = -0.07 x 2 + 0.42x + 6.37 

f(3) = -0.07 (3) 2 + 0.42(3) + 6.37 

= -0.07(9) + 1.26 + 6.37 

= -0.63 + 1.26 + 6.37 = 7 

The height of the rise is 7 ft. 


1. Possible answer: Find the point(s) where the graph 

intersects the x-axis. The values of the 

x-coordinates are the zeros. 

2. Possible answer: Using the quadratic function, 

y = a x 2 + bx + c, find the values of a, b, and c. 

b 

Then use the formula x = -___ 

to find x, the axis of 

2a 

symmetry. 

3. 

 

 

 

 

 

 

 

 

 

 

 

 

 


GUIDED PRACTICE, PAGES 603–604 

 

 

 

 

 

 

 

 

 

1. An x-intercept is a value of x where f(x) = 0. 

2. The axis of symmetry is the vertical line passing 

through the vertex that divides the parabola into 

2 symmetric halves. 

3. The only zero appears to be -1. 

 



4. The zeros appear to be -3 and 3. 

5. The graph does not cross the x-axis, so there are no 


6. _______ -4 + 1 = ___ -3 3__ 

2 2 = - 2 


3__ 

2 . 

7. _____ 0 + 4 = 4__ 

2 2 = 2 


8. The axis of symmetry is x = -4. 

9. a = 1, b = 4 

b 

x = -___ 

2a = - ____ 4 

2(1) = - 4__ 

2 = -2 

The axis of symmetry is x = -2. 

10. a = 3, b = -18 

b 

x = -___ 

2a = - ____ -18 

2(3) = ___ 18 

6 = 3 


11. a = 2, b = 3 

b 

x = -___ 

2a = - ____ 3 3__ 

2(2) = - 4 


3__ 

4 . 

12. a = -3, b = 1 

b 

x = -___ 

2a = - _____ 1 

2(-3) = - ___ 1 

-6 = 1__ 

6 

The axis of symmetry is x = 1__ 

6 . 

13. a = -5, b = 10 

b 

x = -___ 

2a = - _____ 10 

2(-5) = ____ -10 

-10 = 1 

y = -5 x 2 + 10x + 3 

= -5 (1) 2 + 10(1) + 3 

= -5(1) + 10 + 3 

= -5 + 10 + 3 = 8 

The vertex is (1, 8). 

14. a = 1, b = 4 

b 

x = -___ 

2a = - ____ 4 

2(1) = - 4__ 

2 = -2 

y = x 2 + 4x - 7 

= (-2) 2 + 4(-2) - 7 

= 4 - 8 - 7 = -11 

The vertex is (-2, -11). 

15. a = 1__ 

2 , b = 2 

b 

x = -___ 

2a = - ____ 2 

2 ( 2) 1__ 

= - 

2__ 

1 = -2 

y = 1__ 

2 x 2 + 2x 

= 1__ 

2 (-2) 2 + 2(-2) 

= 1__ 

2 (4) - 4 

= 2 - 4 = -2 


16. a = -1, b = 6 

b 

x = -___ 

2a = - _____ 6 

2(-1) = - ___ 6 

-2 = 3 

y = - x 2 + 6x + 1 

= - (3) 2 + 6(3) + 1 

= -9 + 18 + 1 = 10 


17. The zeros are -5 and 1. 

x = _______ -5 + 1 = ___ -4 

2 2 = -2 

y = x 2 + 4x - 5 

= (-2) 2 + 4(-2) - 5 

= 4 - 8 - 5 = -9 


18. a = -16, b = 63 

b 

t = -___ 

2a = - ______ 63 

2(-16) = - ____ 63 

-32 = ___ 63 

32 

y = -16 t 2 + 63t + 4 

___ 

___ 

= -16 ( 32) 63 2 + 63 ( 32) 63 + 4 

= -16 ( _____ 

1024) 3969 + _____ 3969 

32 + 4 

3969 

= -_____ 

64 + _____ 7938 

64 + ____ 256 

64 

= _____ 4225 

64 ≈ 66 

Since the maximum height of the arrow is about 

66 ft, the arrow cannot pass over a tree that is 68 ft 

tall. 




20. The only zero appears to be 0. 

21. The zeros appear to be -8 and -2. 

_______ 

___ 

22. -3 + 1 = -2 

2 2 = -1 


23. The axis of symmetry is x = 6. 

24. _______ -3 + 0 = ___ -3 3__ 

2 2 = - 2 


3__ 

2 . 

25. a = 1, b = 1 

b 

x = -___ 

2a = - ____ 1 

2(1) = - 1__ 

2 


1__ 

2 . 

26. a = 3, b = -2 

b 

x = -___ 

2a = - ____ -2 

3(2) = 2__ 

6 = 1__ 

3 

The axis of symmetry is x = 1__ 

3 . 

27. a = 1__ 

2 , b = -5 

b 

x = -___ 

2a = - -5 

____ 

2 ( 2) 1__ 

5__ = 

1 = 5 




28. a = -2, b = 1__ 

3 

1__ 1__ 

b 

x = -___ 

2a = - _____ 3 

2(-2) = - ___ 3 

-4 = ___ 1 

12 

The axis of symmetry is x = ___ 1 

12 . 

29. a = 1, b = 7 

b 

x = -___ 

2a = - ____ 7 

2(1) = - 7__ 

2 = -3.5 

y = x 2 + 7x 

= (-3.5) 2 + 7(-3.5) 

= 12.25 - 24.5 = -12.25 

The vertex is (-3.5, -12.25). 

30. a = -1, b = 8 

b 

x = -___ 

2a = - _____ 8 

2(-1) = - ___ 8 

-2 = 4 

y = - x 2 + 8x + 16 

= - (4) 2 + 8(4) + 16 

= -16 + 32 + 16 = 32 


31. a = -2, b = -8 

b 

x = -___ 

2a = - _____ -8 

2(-2) = - ___ -8 

-4 = -2 

y = -2 x 2 - 8x - 3 

= -2(-2) 2 - 8(-2) - 3 

= -2(4) + 16 - 3 

= -8 + 16 - 3 = 5 

The vertex is (-2, 5). 

32. a = -1, b = 1__ 

2 

1__ 1__ 

b 

x = -___ 

2a = - _____ 2 

2(-1) = - ___ 2 

-2 = 1__ 

4 

y = - x 2 + 1__ 

2 x + 2 

= - ( 4) 1__ 2 + 1__ 2( 1__ 

4) + 2 

= -___ 

1 

16 + 1__ 

8 + 2 = 2 ___ 1 

16 

4 , 2 ___ 

16) 1 . 

The vertex is ( 1__ 


x = _______ -2 + 4 = 2__ 

2 2 = 1 

y = - 

1__ 

2 x 2 + x + 4 

= - 

1__ 

2 (1) 2 + 1 + 4 

= - 

1__ 

2 + 1 + 4 = 4.5 

The vertex is (1, 4.5). 

34. a = -6.28, b = 4.5 

b 

x = -___ 

2a = - ________ 4.5 

2(-6.28) = - _______ 4.5 

-12.56 ≈ 0.36 

y = -6.28 x 2 + 4.5x 

≈ -6.28 (0.36) 2 + 4.5(0.36) 

≈ -6.28(0.1296) + 1.62 

≈ -0.814 + 1.62 = 0.806 

Yes; the highest point of the arch support is about 8 

feet above the creek. 

35. The axis of symmetry is the y-axis or x = 0. 

36a. about (4, 77) 

b. the time and height at the moment when the rocket 

was at its highest point 

c. 0 and 8; the starting and ending times. 

d. x = 4; it is the midpoint between the two zeros and 

the x-coordinate of the vertex. 

37. 0 38. 1 

39. 2 

40. The two zeros are on either side of the axis of 

symmetry and are equidistant from it. 


41. B; since (-2) 2 - 2(-2) - 8 = 4 + 4 - 8 = 0 and 

(4) 2 - 2(4) - 8 = 16 - 8 - 8 = 0. 

42. J; Rearranging equation J into standard form gives 

y = 2 x 2 b 

+ 2x - 5. Since -___ 

2a = - ____ 2 

2(2) = - 1__ 

2 , J is 

correct. 

43. 2 

a = -5, b = 20 

b 

x = -___ 

2a = - _____ 20 

2(-5) = - ____ 20 

-10 = 2 


44. The domain is all real numbers and the range is 

y ≤ 0. 

45. Graph f(x) = -0.01 x 2 + 20 and y = -5. The height 

is the distance between f(x) and y = -5 on the line 

x = 0; 25 ft. The width is the distance between the 

two points where f(x) = -5; 100 ft. 


46. ___ -4 

2 = __ y 6 

2y = -24 

___ 2y 

2 = ____ -24 

2 

y = -12 

48. 4y = 12x - 8 

___ 4y 

4 = _______ 12x - 8 

4 

y = 3x - 2 

47. 2x + y = 3 

_______ -2x ____ -2x 

y = -2x + 3 

49. 10 - 5y = 20x 

________ 

-10 ____ -10 

-5y = 20x - 10 

____ -5y 

= ________ 20x - 10 

-5 -5 

y = -4x + 2 


a is 0. 



51. x 2 - 5x = 2 + y 

_______ - 2 ______ -2 

x 2 - 5x - 2 = y 


in the form y = a x 2 + bx + c, where a = 1, b = -5, 

and c = -2. 


in the form y = a x 2 + bx + c, where a = -1, 

b = -6, and c = 0. 

9-3 GRAPHING QUADRATIC FUNCTIONS, 

PAGES 606–611 


1a. a = 2, b = 6 

b 

x = -___ 

2a = - ____ 6 6__ 3__ 

2(2) = - 4 = - 2 

y = 2 x 2 + 6x + 2 

= 2 ( - 

3__ 

= 2 ( 9__ 

= 9__ 

2) 2 + 6 ( - 

4) - 9 + 2 

3__ 

2) + 2 

5__ 

2 - 9 + 2 = - 2 

The vertex is ( - 

3__ 5__ 

2 , - 2) . 

y = 2 x 2 + 6x + 2 → y = 2 x 2 + 6x + (2) 

The y-intercept is 2; the graph passes through 

(0, 2). 

Let x = -1. 

y = 2 x 2 + 6x + 2 

= 2 (-1) 2 + 6(-1) + 2 

= 2(1) - 6 + 2 

= 2 - 6 + 2 = -2 

Let x = 1. 

y = 2 x 2 + 6x + 2 

= 2 (1) 2 + 6(1) + 2 

= 2(1) + 6 + 2 

= 2 + 6 + 2 = 10 

Two other points are (-1, -2) and (1, 10) 

Graph the axis of symmetry, and all points. Reflect 

the points across the axis of symmetry. Connect the 

points with a smooth curve. 

 

 

 

 

 

 

b. y + 6x = x 2 + 9 

_____ - 6x ______ -6x 

y = x 2 - 6x + 9 

a = 1, b = -6 

b 

x = -___ 

2a = - ____ -6 

2(1) = 6__ 

2 = 3 

y = x 2 - 6x + 9 

= (3) 2 - 6(3) + 9 

= 9 - 18 + 9 = 0 


y = x 2 - 6x + 9 → y = x 2 - 6x + (9) 


(0, 9). 

Let x = 1. 

y = x 2 - 6x + 9 

= (1) 2 - 6(1) + 9 

= 1 - 6 + 9 = 4 

Let x = 2. 

y = x 2 - 6x + 9 

= (2) 2 - 6(2) + 9 

= 4 - 12 + 9 = 1 

Two other points are (1, 4) and (2, 1). 




 

 

2. a = -16, b = 24 

b 

x = -___ 

2a = - ______ 24 

2(-16) = - ____ 24 

-32 = 0.75 

f(x) = -16 x 2 + 24x 

f(0.75) = -16 (0.75) 2 + 24(0.75) 

= -16(0.5625) + 18 

= -9 + 18 = 9 

The vertex is (0.75, 9). 

 

f(x) = -16 x 2 + 24x → f(x) = -16 x 2 + 24x + 0 


(0, 0). 

Let x = 0.5. 

f(x) = -16 x 2 + 24x 

f(0.5) = -16 (0.5) 2 + 24(0.5) 

= -16(0.25) + 12 

= -4 + 12 = 8 

Another point is (0.5, 8). 




The vertex is (0.75, 9). So at 0.75 s, Molly reached 

her maximum height of 9 ft. The graph shows the 

zeros of the function are 0 and 1.5. At 0 s, Molly has 

not jumped and at 1.5 s she reaches the pool. Molly 

takes 1.5 s to reach the pool. 

 

 




1. Possible answer: First subtract a x 2 from both sides 

of the equation. Then divide both sides by -1. The 

resulting value of c is the y-intercept. 

2. Possible answer: Find the axis of symmetry, 

the vertex, and any zeros of the graph. Draw a 

curved line that passes through the points and is 

symmetrical about the axis of symmetry. 

3. Possible answer: The vertex is the maximum height 

of Molly’s dive at the midpoint of the time she was in 

the air. The zeros are the start time of the dive and 

the time when she reaches the pool. 

4. 

 

 

 

 

 

 

 

 

 



1. a = 1, b = -2 

b 

x = -___ 

2a = - ____ -2 

2(1) = 2__ 

2 = 1 

y = x 2 - 2x - 3 

= (1) 2 - 2(1) - 3 

= 1 - 2 - 3 = -4 


 

 

 

 

 

 

y = x 2 - 2x - 3 → y = x 2 - 2x + (-3) 

The y-intercept is -3; the graph passes through 

(0, -3). 

Let x = -1. 

y = x 2 - 2x - 3 

= (-1) 2 - 2(-1) - 3 

= 1 + 2 - 3 = 0 

Let x = -2. 

y = x 2 - 2x - 3 

= (-2) 2 - 2(-2) - 3 

= 4 - 4 - 3 = -3 

Two other points are (-1, 0) and (-2, -3). 




 

 

 

 

 

 

2. -y - 3 x 2 = -3 

________ + 3 x 2 _____ +3 x 2 

-y = 3 x 2 - 3 

-1(-y) = -1( 3 x 

2 - 3 ) 

y = -3 x 2 + 3 

a = -3, b = 0 

b 

x = -___ 

2a = - _____ 0 

2(-3) = 0 

y = -3 x 2 + 3 

= -3 (0) 2 + 3 

= -3(0) + 3 = 3 

The vertex is (0, 3). This is the y-intercept. 

Let x = 1. 

y = -3 x 2 + 3 

= -3(1) 2 + 3 

= -3(1) + 3 

= -3 + 3 = 0 

Let x = 2. 

y = -3 x 2 + 3 

= -3 (2) 2 + 3 

= -3(4) + 3 

= -12 + 3 = -9 

Two other points are (1, 0) and (2, -9). 




 

 

 

 

 

 

 

 



3. a = 2, b = 2 

b 

x = -___ 

2a = - ____ 2 

2(2) = - 2__ 

4 = - 1__ 

2 

y = 2 x 2 + 2x - 4 

= 2 ( - 

1__ 

= 2 ( 1__ 

= 1__ 

2 

2) 2 + 2 ( - 

4) - 1 - 4 

1__ 

2) - 4 

- 1 - 4 = -4 

1__ 

2 

, -4 

1__ 


1__ 

2 

2) . 

y = 2 x 2 + 2x - 4 → y = 2 x 2 + 2x + (-4) 


(0, -4). 

Let x = 1. 

y = 2 x 2 + 2x - 4 

= 2 (1) 2 + 2(1) - 4 

= 2(1) + 2 - 4 

= 2 + 2 - 4 = 0 


Let x = 2. 

y = 2 x 2 + 2x - 4 

= 2 (2) 2 + 2(2) - 4 

= 2(4) + 4 - 4 

= 8 + 4 - 4 = 8 




 

 

 

 

4. a = 1, b = 4 

b 

x = -___ 

2a = - ____ 4 

2(1) = - 4__ 

2 = -2 

y = x 2 + 4x - 8 

= (-2) 2 + 4(-2) - 8 

= 4 - 8 - 8 = -12 


y = x 2 + 4x - 8 → y = x 2 + 4x + (-8) 


(0, -8). 

Let x = -1. 

y = x 2 + 4x - 8 

= (-1) 2 + 4(-1) - 8 

= 1 - 4 - 8 = -11 

 

 

 

Let x = 1. 

y = x 2 + 4x - 8 

= (1) 2 + 4(1) - 8 

= 1 + 4 - 8 = -3 

Two other points are (-1, -11) and (1, -3). 




 

 

 

 

 

 

5. y + x 2 + 5x + 2 = 0 

_______________ 

-y ___ -y 

x 2 + 5x + 2 = -y 

-1 ( x 2 + 5x + 2) = -1(-y) 

- x 2 - 5x - 2 = y 

a = -1, b = -5 

b 

x = -___ 

2a = - _____ -5 

2(-1) = ___ 5 5__ 

-2 = - 2 

y = - x 2 - 5x - 2 

= - ( - 

5__ 

5__ 

2) 2 - 5 ( - 

2) - 2 

25 

= -___ 

4 + ___ 25 

2 - 2 = ___ 17 

4 

2 , ___ 17 

4 

) . 


5__ 

y = - x 2 - 5x - 2 → y = - x 2 - 5x + (-2) 


(0, -2). 

Let x = -2. 

y = - x 2 - 5x - 2 

= - (-2) 2 - 5(-2) - 2 

= -4 + 10 - 2 = 4 

Let x = -1. 

y = - x 2 - 5x - 2 

= - (-1) 2 - 5(-1) - 2 

= -1 + 5 - 2 = 2 

Two other points are (-2, 4) and (-1, 2). 




 

 

 

 

 

 

 

 



6. a = 4, b = 0 

b 

x = -___ 

2a = - ____ 0 

2(4) = 0 

y = 4 x 2 + 2 

= 4 (0) 2 + 2 

= 4(0) + 2 = 2 


Let x = 1. 

y = 4 x 2 + 2 

= 4 (1) 2 + 2 

= 4(1) + 2 

= 4 + 2 = 6 

Let x = 2. 

y = 4 x 2 + 2 

= 4 (2) 2 + 2 

= 4(4) + 2 

= 16 + 2 = 18 





 

7. a = -16, b = 96 

b 

x = -___ 

2a = - ______ 96 

2(-16) = - ____ 96 

-32 = 3 

f(x) = -16 x 2 + 96x 

f(3) = -16 (3) 2 + 96(3) 

= -16(9) + 288 

= -144 + 288 = 144 


 

 

f(x) = -16 x 2 + 96x → f(x) = -16 x 2 + 96x + (0) 


(0, 0). 

Let x = 1. 

f(x) = -16 x 2 + 96x 

f(1) = -16(1) 2 + 96(1) 

= -16(1) + 96 

= -16 + 96 = 80 

 

 

 

Let x = 2. 

f(x) = -16 x 2 + 96x 

f(2) = -16 (2) 2 + 96(2) 

= -16(4) + 192 

= -64 + 192 = 128 





The vertex is (3, 144). So at 3 s, the golf ball 

reached its maximum height of 144 ft. The graph 

shows the zeros of the function are 0 and 6. 

At 0 s the golf ball had not left the ground, and at 6 s 

it reaches the ground. The golf ball was in the air for 

6 s. 

180 

120 

y 

(3, 144) 


8. a = -4, b = 12 

b 

x = -___ 

2a = - _____ 12 

2(-4) = - ___ 12 

-8 = 3__ 

2 

y = -4 x 2 + 12x - 5 

= -4 ( 3__ 

2) 2 + 12 ( 2) 3__ - 5 

4) + 18 - 5 

= -9 + 18 - 5 = 4 


2 , 4 ) . 

= -4 ( 9__ 

y = -4 x 2 + 12x - 5 → y = -4 x 2 + 12x + (-5) 


(0, -5). 

Let x = 1. 

y = -4 x 2 + 12x - 5 

= -4 (1) 2 + 12(1) - 5 

= -4(1) + 12 - 5 

= -4 + 12 - 5 = 3 

Let x = -1. 

y = -4 x 2 + 12x - 5 

= -4 (-1) 2 + 12(-1) - 5 

= -4(1) - 12 - 5 

= -4 - 12 - 5 = -21 

Two other points are (1, 3) and (-1, -21). 




 

 

 

 

 

 

 

 

60 

(6, 0) 

(0, 0) 

x 

0 2 4 6 



9. a = 3, b = 12 

b 

x = -___ 

2a = - ____ 12 

2(3) = - ___ 12 

6 = -2 

y = 3 x 2 + 12x + 9 

= 3 (-2) 2 + 12(-2) + 9 

= 3(4) - 24 + 9 

= 12 - 24 + 9 = -3 


y = 3 x 2 + 12x + 9 → y = 3 x 2 + 12x + (9) 


(0, 9). 

Let x = -1. 

y = 3 x 2 + 12x + 9 

= 3 (-1) 2 + 12(-1) + 9 

= 3(1) - 12 + 9 

= 3 - 12 + 9 = 0 

Two other points are (-1, 0) and (1, 24). 

Let x = 1. 

y = 3 x 2 + 12x + 9 

= 3 (1) 2 + 12(1) + 9 

= 3(1) + 12 + 9 

= 3 + 12 + 9 = 24 




 

 

 

 

 

 

10. y - 7 x 2 - 14x = 3 

____________ + 14x _____ +14x 

y - 7 x 2 = 14x + 3 

______ + 7 x 2 ________ +7 x 2 

y = 7 x 2 + 14x + 3 

a = 7, b = 14 

b 

x = -___ 

2a = - ____ 14 

2(7) = - ___ 14 

14 = -1 

y = 7 x 2 + 14x + 3 

= 7 (-1) 2 + 14(-1) + 3 

= 7(1) - 14 + 3 

= 7 - 14 + 3 = -4 


y = 7 x 2 + 14x + 3 → y = 7 x 2 + 14x + (3) 


(0, 3). 

Let x = 1 

y = 7 x 2 + 14x + 3 

= 7 (1) 2 + 14(1) + 3 

= 7(1) + 14 + 3 

= 7 + 14 + 3 = 24 

Let x = 2. 

y = 7 x 2 + 14x + 3 

= 7 (2) 2 + 14(2) + 3 

= 7(4) + 28 + 3 

= 28 + 28 + 3 = 59 





 

 

 

 

 

 

11. a = -1, b = 2 

b 

x = -___ 

2a = - _____ 2 

2(-1) = ___ -2 

-2 = 1 

y = - x 2 + 2x 

= - (1) 2 + 2(1) 

= -1 + 2 = 1 

The vertex is at (1, 1). 

y = - x 2 + 2x → y = - x 2 + 2x + (0) 


(0, 0). 

Let x = -1. 

y = - x 2 + 2x 

= - (-1) 2 + 2(-1) 

= -1 - 2 = -3 

Let x = -2. 

y = - x 2 + 2x 

= - (-2) 2 + 2(-2) 

= -4 - 4 = -8 

Two other points are (-1, -3) and (-2, -8). 




 

 

 

 

 

 



12. y - 1 = 4 x 2 + 8x 

____ + 1 ________ +1 

y = 4 x 2 + 8x + 1 

a = 4, b = 8 

b 

x = -___ 

2a = - ____ 8 8__ 

2(4) = - 8 = -1 

y = 4 x 2 + 8x + 1 

= 4 (-1) 2 + 8(-1) + 1 

= 4(1) - 8 + 1 

= 4 - 8 + 1 = -3 


y = 4 x 2 + 8x + 1 → y = 4 x 2 + 8x + (1) 


(0, 1). 

Let x = 1. 

y = 4 x 2 + 8x + 1 

= 4 (1) 2 + 8(1) + 1 

= 4(1) + 8 + 1 

= 4 + 8 + 1 = 13 

Two other points at (1, 13) and (2, 33). 

Let x = 2. 

y = 4 x 2 + 8x + 1 

= 4 (2) 2 + 8(2) + 1 

= 4(4) + 16 + 1 

= 16 + 16 + 1 = 33 




 

 

 

 

 

13. a = -2, b = -3 

b 

x = -___ 

2a = - _____ -3 

2(-2) = ___ 3 3__ 

-4 = - 4 

y = -2 x 2 - 3x + 4 

= -2 ( - 

3__ 

4) 2 - 3 ( - 

4) + 4 

= -2 ( ___ 

16) 9 + 9__ 

4 + 4 

= - 

9__ 

8 + 9__ + 4= 5 

1__ 

4 8 

3__ 

8) . 

y = -2 x 2 - 3x + 4 → y = -2 x 2 - 3x + (4) 


(0, 4) 

The vertex is at ( - 

3__ 

4 , 5 1__ 

Let x = 1. 

y = -2 x 2 - 3x + 4 

= -2(1) 2 - 3(1) + 4 

= -2(1) - 3 + 4 

= -2 - 3 + 4 = -1 

Let x = 2. 

y = -2 x 2 - 3x + 4 

= -2 (2) 2 - 3(2) + 4 

= -2(4) - 6 + 4 

= -8 - 6 + 4 = -10 

Two other points are (1, -1) and (2, -10). 




 

 

 

 

 

 



14. a = -16, b = 16 

b 

x = -___ 

2a = - ______ 16 

2(-16) = - ____ 16 

-32 = 0.5 

f(x) = -16 x 2 + 16x 

f(0.5) = -16 (0.5) 2 + 16(0.5) 

= -16(0.25) + 8 

= -4 + 8 = 4 

The vertex is (0.5, 4). 

f(x) = -16 x 2 + 16x → f(x) = -16 x 2 + 16x + (0) 


(0, 0). 

Let x = 0.25. 

f(x) = -16 x 2 + 16x 

f(0.25) = -16 (0.25) 2 + 16(0.25) 

= -16(0.625) + 4 

= -1 + 4 = 3 

Another point is (0.25, 3). 




The vertex is (0.5, 4). So at 0.5 s, the ring reached 

its maximum height of 8 ft. The graph shows the 

zeros of the function are 0 and 1. At 0 s the ring had 

not left the juggler’s hand, and at 1 s it returns to the 

juggler. The ring was in the air for 1 s. 

4 

2 

(0, 0) 

-1 

-2 

y 

(0.5, 8) 

(1, 0) 

0 2 

15. a = 1, b = -8 

b 

x = -___ 

2a = - ____ -8 

2(1) = 8__ 

2 = 4 

x 

y = x 2 - 8x 

= (4) 2 - 8(4) 

= 16 - 32 = -16 

The axis of symmetry is x = 4 and the vertex is 

(4, -16). 

16. a = -1, b = 6 

b 

x = -___ 

2a = - _____ 6 

2(-1) = - ___ 6 

-2 = 3 

y = - x 2 + 6x - 4 

= - (3) 2 + 6(3) - 4 

= -9 + 18 - 4 = 5 

The axis of symmetry is x = 3, and the vertex is 

(3, 5). 

17. a = -3, b = 0 

b 

x = -___ 

2a = - _____ 0 

2(-3) = 0 

y = 4 - 3 x 2 

= 4 - 3 (0) 2 

= 4 - 3(0) = 4 


(0, 4). 

18. a = -2, b = 0 

b 

x = -___ 

2a = - _____ 0 

2(-2) = 0 

y = -2 x 2 - 4 

= -2 (0) 2 - 4 

= -2(0) - 4 = -4 


(0, -4). 

19. a = -1, b = -1 

b 

x = -___ 

2a = - _____ -1 

2(-1) = ___ 1 

-2 = - 1__ 

2 

y = - x 2 - x - 4 

= - ( - 

1__ 

= - 

1__ 

4 + 1__ 

1__ 

2) 2 - ( - 

2) - 4 

2 - 4 = - ___ 15 

4 


1__ 

, and the vertex is 

2 

( - 1__ 

2 , - ___ 15 

4 ) . 

20. a = 1, b = 8 

b 

x = -___ 

2a = - ____ 8 8__ 

2(1) = - 2 = -4 

y = x 2 + 8x + 16 

= (-4) 2 + 8(-4) + 16 

= 16 - 32 + 16 = 0 

The axis of symmetry is x = -4, and the vertex is 

(-4, 0). 

21. a = -1, b = 0 

b 

x = -___ 

2a = - _____ 0 

2(-1) = 0 

y = - x 2 

= -(0) 2 

= -(0) = 0 


Let x = 1. 

Let x = 2. 

y = - x 2 

y = - x 2 

= -(1) 2 

= -(2) 2 

= -(1) = -1 

= -(4) = -4 





 

 

 

 

 

 

 

 



22. a = -1, b = 4 

b 

x = -___ 

2a = - _____ 4 

2(-1) = - ___ 4 

-2 = 2 

y = - x 2 + 4x 

= - (2) 2 + 4(2) 

= -4 + 8 = 4 


y = - x 2 + 4x → y = - x 2 + 4x + (0) 


(0, 0). 

Let x = 1. 

y = - x 2 + 4x 

= -(1) 2 + 4(1) 

= -1 + 4 = 3 

Let x = -1. 

y = - x 2 + 4x 

= - (-1) 2 + 4(-1) 

= -1 - 4 = -5 

Two other points are (1, 3) and (-1, -5). 




 

 

 

 

 

 

 

 

24. a = 1, b = -1 

b 

x = -___ 

2a = - ____ -1 

2(1) = 1__ 

2 

y = x 2 - x 

= ( 2) 1__ 2 - 1__ 

2 

= 1__ 

4 - 1__ 

2 = - 1__ 

4 

The vertex is at ( 1__ 

2 , - 1__ 

4) . 

y = x 2 - x → y = x 2 - x + (0) 


(0, 0). 

Let x = -1. 

y = x 2 - x 

= (-1) 2 - (-1) 

= 1 + 1 = 2 

Let x = -2. 

y = x 2 - x 

= (-2) 2 - (-2) 

= 4 + 2 = 6 





 

 

 

 

23. a = 1, b = -6 

b 

x = -___ 

2a = - ____ -6 

2(1) = 6__ 

2 = 3 

y = x 2 - 6x + 4 

= (3) 2 - 6(3) + 4 

= 9 - 18 + 4 = -5 


y = x 2 - 6x + 4 → y = x 2 - 6x + (4) 


(0, 4). 

Let x = 1. 

y = x 2 - 6x + 4 

= (1) 2 - 6(1) + 4 

= 1 - 6 + 4 = -1 

Let x = 2. 

y = x 2 - 6x + 4 

= (2) 2 - 6(2) + 4 

= 4 - 12 + 4 = -4 





 

 

 

 

 

 

 

 

25. a = 3, b = 0 

x = ___ -b 

2a = - ____ 0 

2(3) = 0 

 

 

 

 

y = 3 x 2 - 4 

= 3 (0) 2 - 4 

= 3(0) - 4 = -4 

The vertex is (0, -4). This is the y-intercept. 

Let x = 1. 

y = 3 x 2 - 4 

= 3 (1) 2 - 4 

= 3(1) - 4 

= 3 - 4 = -1 

Let x = 2. 

y = 3 x 2 - 4 

= 3 (2) 2 - 4 

= 3(4) - 4 

= 12 - 4 = 8 

Two other points are (1, -1) and (2, 8). 




 

 

 

 

 

 

 

 

 

 



26. a = -2, b = -16 

b 

x = -___ 

2a = - _____ -16 

2(-2) = ___ 16 

-4 = -4 

27a. 

y = -2 x 2 - 16x - 25 

= -2 (-4) 2 - 16(-4) - 25 

= -2(16) + 64 - 25 

= -32 + 64 - 25 = 7 


y = -2 x 2 - 16x - 25 → y = -2 x 2 - 16x + (-25) 


(0, -25). 

Let x = -3. 

y = -2 x 2 - 16x - 25 

= -2 (-3) 2 - 16(-3) - 25 

= -2(9) + 48 - 25 

= -18 + 48 - 25 = 5 

Let x = -2. 

y = -2 x 2 - 16x - 25 

= -2 (-2) 2 - 16(-2) - 25 

= -2(4) + 32 - 25 

= -8 + 32 - 25 = -1 

Two other points are (-3, 5) and (-2, -1). 




 

 

 

 

 

 

 

 

 

 

 

 

 

 

b. Domain: 0 ≤ x ≤ 3.16; Range: 0 ≤ y ≤ 50 

c. 3.16 s 

28. Student A is incorrect; the student incorrectly 

identified the values of a and b. 

29. (-1, 4); reflect the given point across the axis of 

symmetry. 

 

30. x = 3 

The radius of the pipe is 3 cm when the velocity is 

7 cm/s. 

31. y = 12 

The velocity of the pipe is 12 cm/s when the radius 

is 2 cm. 

 

 

 

 

 

 

32. Domain: x ≥ 0; a negative radius does not make 

sense. 

33. (-1, 6); the axis of symmetry is a vertical line 

through the vertex. So its equation is x = 0. Reflect 

the point (1, 6) across the axis of symmetry. 

34. The x-coordinate of the vertex gives the axis of 

symmetry. If the range includes values greater than 

the y-coordinate of the vertex, then the parabola 

opens upward. If the range includes values less 

than the y-coordinate of the vertex, then the 

parabola opens downward. 

35a. h = - 

1__ 

2 a t 2 + v i 

t + h i 

h = - 

1__ 

2 (32) t 2 + (45)t + 50 

h = -16 t 2 + 45t + 50 

b. a = -16, b = 45 

b 

x = -___ 

2a = - ______ 45 

2(-16) = - ____ 45 

-32 ≈ 1.4 

h = -16 t 2 + 45t + 50 

= -16 (1.4) 2 + 45(1.4) + 50 

= -16(1.96) + 63 + 50 

= -31.36 + 63 + 50 ≈ 81.6 

The vertex is about (1.4, 81.6). 

c. h = - t 2 + 45t + 50 → h = - t 2 + 45t + (50) 


(0, 50). 

Let t = 1. 

h = -16 t 2 + 45t + 50 

= -16 (1) 2 + 45(1) + 50 

= -16(1) + 45 + 50 

= -16 + 45 + 50 = 79 

Another point is (1, 79). 


the points across the axis of symmetry. Connect 

the points with a smooth curve. 

 

 

 

 

 

 

 

 

 

 

 

d. The vertex represents the time, 1.4 s, that the 

water bottle rocket has spent in the air when it 

reaches its highest point, 81.6 ft. 



36. Graph Axis of 

Domain and 

Vertex Zeros 

Opens Symmetry Range 

D: all real 

Upward x = 0 (0, 4) none numbers 

R: y ≥ 4 

Downward x = 0 (0, 4) -2, 2 

D: all real 

numbers 

R: y ≤ 4 

D: all real 

Upward x = 1 (1, -9) -2, 4 numbers 

R: y ≥ -9 


b 

37. A; The axis of symmetry is x = -___, where a = 

1__ 

2a 2 , 

and b = -5. Evaluating gives x = 5. 

38. J 

a = 1, b = -5 

b 

x = -___ 

2a = - ____ -5 

2(1) = 5__ 

2 

f(x) = x 2 - 5x + 6 

2) = ( 2) 5__ 2 - 5 ( 5__ 

2) + 6 

= ___ 25 

4 - ___ 25 

2 + 6 = - 1__ 

4 

f ( 5__ 

4) . 


2 , - 1__ 

39. D; Since for equation D, b = 6 and a = -3, 

b 

-___ 

= 1 and since 

2a 

-3(1) 2 + 6(1) + 5 = -3 + 6 + 5 = 8, D is correct. 

40. a = 1, b = 3 

b 

x = -___ 

2a = - ____ 3 3__ 

2(1) = - 2 

y = x 2 + 3x + 2 

= ( - 

3__ 

= 9__ 

4 - 9__ 

3__ 

2) 2 + 3 ( - 

2) + 2 

2 + 2 = - 1__ 

4 


3__ 

2 , - 1__ 

4) . 

y = x 2 + 3x + 2 → y = x 2 + 3x + (2) 


(0, 2). 

Let x = -1. 

y = x 2 + 3x + 2 

= (-1) 2 + 3(-1) + 2 

= 1 - 3 + 2 = 0 

Let x = 1. 

y = x 2 + 3x + 2 

= (1) 2 + 3(1) + 2 

= 1 + 3 + 2 = 6 





 

 

 

 

 

 

The zeros are -2 and -1. The axis of symmetry is 

x = - 

3__ 

2 . The vertex is ( - 3__ 

2 , - 1__ 

 

 

4) . 


41. -1; the axis of symmetry is x = 1. The given zero 

is 2 units right from the axis of symmetry. The other 

zero is 2 units left from the axis of symmetry. 

42. (0, 6); maximum, the parabola opens downward 

because the zeros are below the vertex, so the 

vertex is a maximum. 


43. The y-intercept is 6. The x-intercept is 3. 

44. The y-intercept is -4. The x-intercept is 8. 

45. The y-intercept is 3. There is no x-intercept. 



46. 3x - y = 2 

__________________ 

+ (-3)(x + 4y = 18) 

3x - y = 2 

__________________ 

+ (-3x - 12y = -54) 

0 - 13y = -52 

-13y = -52 

_____ -13y 

-13 = ____ -52 

-13 

y = 4 

x + 4y = 18 

x + 4(4) = 18 

x + 16 = 18 

______ - 16 ____ -16 

x = 2 

(2, 4) 

48. 3(-2x + 3y = 12) 

_______________ 

+ 6x + y = 4 

-6x + 9y = 36 

_____________ 

+ 6x + y = 4 

0 + 10y = 40 

10y = 40 

____ 10y 

10 = ___ 40 

10 

y = 4 

6x + y = 4 

6x + 4 = 4 

_____ - 4 ___ -4 

6x = 0 

___ 6x 

6 = 0__ 

6 

x = 0 

(0, 4) 

49. a = 1, b = 2 

b 

x = -___ 

2a = - ____ 2 

2(1) = - 2__ 

2 = -1 

y = x 2 + 2x - 15 

= (-1) 2 + 2(-1) - 15 

= 1 - 2 - 15 = -16 


50. a = -3, b = 12 

b 

x = -___ 

2a = - _____ 12 

2(-3) = - ___ 12 

-6 = 2 

y = -3 x 2 + 12x - 4 

= -3 (2) 2 + 12(2) - 4 

= -3(4) + 24 - 4 

= -12 + 24 - 4 = 8 


51. a = -1, b = -2 

b 

x = -___ 

2a = - _____ -2 

2(-1) = ___ 2 

-2 = -1 

y = -2x - x 2 + 3 

= -2(-1) - (-1) 2 + 3 

= 2 - 1 + 3 = 4 


47. 2x + 3y = 3 

_______________ 

+ 3(4x - y = 13) 

2x + 3y = 3 

______________ 

+ 12x - 3y = 39 

14x + 0 = 42 

14x = 42 

____ 14x 

14 = ___ 42 

14 

x = 3 

2x + 3y = 3 

2(3) + 3y = 3 

6 + 3y = 3 

_______ -6 ___ -6 

3y = -3 

___ 3y 

3 = ___ -3 

3 

y = -1 

(3, -1) 

TECHNOLOGY LAB: THE FAMILY OF 

QUADRATIC FUNCTIONS, PAGE 612 


1. much narrower 2. much wider 

3. The closer the value of a to 0, the wider the 

parabola. The greater the value of a, the narrower 

the parabola. 

4. All the graphs open downward. The graph of 

y = - 

1__ 

2 x 2 is the widest. The graphs of y = -2 x 2 and 

y = -3 x 2 are narrower than the parent function. 

5. much narrower 

6. When a is negative, the parabolas open downward. 

Otherwise the conjecture is the same. 

7. The graph of y = x 2 - 1 is lower than the parent 

function. The graphs of y = x 2 + 2 and y = x 2 + 4 

are higher. 

8. lower, with a vertex at (0, -7) 

9. The value of c causes the parabola to move up if 

c > 0 and down if c < 0. 

9-4 TRANSFORMING QUADRATIC 



1a. ⎪-1⎥ = 1 ⎪ 2__ 3⎥ = 2__ 

3 

f(x) = - x 2 , g(x) = 2__ 

3 x 2 

b. ⎪-4⎥ = 4 ⎪6⎥ = 6 ⎪0.2⎥ = 0.2 

g(x) = 6 x 2 , f(x) = -4 x 2 , h(x) = 0.2 x 2 

2a. The graph of g(x) is the same width, has the same 

axis of symmetry, opens downward, and its vertex is 

translated 4 units down to (0, -4). 

b. The graph of g(x) is narrower, has the same axis of 

symmetry, and also opens upward, and its vertex is 

translated 9 units up to (0, 9). 

c. The graph of g(x) is wider, has the same axis of 



3a. h 1 

(t) = -16 t 2 + 16; h 2 

(t) = -16 t 2 + 100 

The graph of the ball that is dropped from a height 

of 100 ft is a vertical translation of the graph of the 

ball that is dropped from a height of 16 ft. The 

y-intercept of the graph of the ball that is dropped 

from 100 ft is 84 units higher. 

b. The ball that is dropped from 16 ft reaches the 

ground in 1 s. The ball that is dropped from 100 ft 

reaches the ground in 2.5 s. 




1. When c is positive, the graph of y = x 2 + c is a 

translation c units up of the graph of y = x 2 . When c 

is negative, the graph of y = x 2 + c is a translation 

c units down of the graph of y = x 2 . 

2. The graph of g(x) = a x 2 is narrower than the graph 

of f(x) = x 2 if ⎪a⎥ > 1, and wider if ⎪a⎥ < 1. 

3. 

 

 

 

 

 

 

 

 

 



1. ⎪3⎥ = 3 (2) = 2 

f(x) = 3 x 2 , g(x) = 2 x 2 

 

 

 

 

 

 

2. ⎪5⎥ = 5 ⎪-5⎥ = 5 

Since the absolute values are equal, the graphs 

have the same width. 

3. ⎪ 3__ 

4⎥ = 3__ 

4 

⎪-2⎥ = 2 ⎪-8⎥ = 8 

h(x) = -8 x 2 , g(x) = -2 x 2 , f(x) = 3__ 

4 x 2 

4. ⎪1⎥ = 1 ⎪ - 4__ 

5⎥ = 4__ 

5 

⎪3⎥ = 3 

h(x) = 3 x 2 , f(x) = x 2 , g(x) = - 

4__ 

5 x 2 

5. The graph of g(x) is the same width, has the same 

axis of symmetry, also opens upward, and its vertex 

is translated 6 units up to (0, 6). 

6. The graph of g(x) is narrower, has the same axis 

of symmetry, opens downward, and its vertex is 


7. The graph of g(x) is wider, has the same axis of 

symmetry, also opens upward, and has the same 

vertex. 

8. The graph of g(x) is wider, has the same axis 



9a. h 1 

(t) = -16 t 2 + 16; h 2 

(t) = -16 t 2 + 256 

The graph of h 2 

is a vertical translation of the graph 

of h 1 

. The y-intercept of the graph h 2 

is 240 units 

higher. 

b. The baseball that is dropped from 256 ft reaches the 

ground in 4 s. The baseball that is dropped from 16 

ft reaches the ground in 1 s. 


10. ⎪1⎥ = 1 ⎪4⎥ = 4 

g(x) = 4 x 2 , f(x) = x 2 11. ⎪-2⎥ = 2 ⎪ 1__ 2⎥ = 1__ 

2 

f(x) = -2 x 2 , g(x) = 1__ 

12. ⎪-1⎥ = 1 

5__ 

⎪ - 8⎥ = 5__ 

8 ⎪ 1__ 2⎥ = 1__ 

2 

f(x) = - x 2 , g(x) = - 

5__ 

8 x 2 , h(x) = 1__ 

2 x 2 

13. ⎪-5⎥ = 5 

3__ 

⎪ - 8⎥ = 3__ ⎪3⎥ = 3 

8 

f(x) = -5 x 2 , h(x) = 3 x 2 , g(x) = - 

3__ 

8 x 2 

19. always 20. never 

23a. Check students’ work. 

26. Possible answer: f(x) = x 2 

27. Possible answer: f(x) = -10 x 2 + 2 

28. Possible answer: f(x) = 1__ 

2 x 2 + 1 

29. B 30. D 

31. A 32. C 

35a. Possible answer: f(x) = x 2 + 2 

b. Possible answer: f(x) = - x 2 + 2 

c. Possible answer: f(x) = 5 x 2 + 2 

2 x 2 


symmetry, also opens upward, and its vertex is 






symmetry, also opens upward, and its vertex is 


17. The graph of g(x) is wider, has the same axis 


translated 1 unit up to (0, 1). 

18a. h 1 

(t) = -16 t 2 + 10,000; h 2 

(t) = -16 t 2 + 14,400 

The graph of h 2 

is a vertical translation of the 

graph of h 1 

. The y-intercept of h 2 

is 4400 units 

higher. 

b. The raindrop falling from 10,000 ft reaches the 

ground in 25 s. The raindrop falling from 14,400 ft 

reaches the ground in 30 s. 

21. never 22. sometimes 

b. Possible answer: air resistance or wind may have 

affected the fall of the object. 

24. f(x) = x 2 25. f(x) = 3 x 2 - 6 

33. c = 0 and a ≠ 0 34a. 50 ft and 75 ft 

b. h(t) = -16 t 2 + 75 c. about 1.8 s and 2.2 s 

36. c is the y-intercept of y = x 2 + c, and b is the 

y-intercept of y = x + b. 



37. The graph of f(x) = ___ 1 

10 x 2 - 5 will look like the 

graph of f(x) = x 2 , except that it will be wider 

( because ___ 1 

10 < 1 ) and it will be translated 5 units 

down (because c = -5). 

38a. 

y = (x - 3) 2 and y = x 2 - 3 have the same width 

as y = x 2 , and all of these parabolas open upward. 

y = (x - 3) 2 is translated 3 units right of y = x 2 , 

has an axis of symmetry of x = 3, and has 

1 zero (like y = x 2 ) . y = x 2 - 3 has 2 zeros and is 

translated 3 units down from y = x 2 . 

b. 64 ft; 4 s 

c. The x-coordinate of the vertex is the amount 

subtracted from x in the quantity that is squared, 

and the y-coordinate is the constant term in the 

equation. On the graph, find the greatest value of 

y, and then find the x-coordinate that corresponds 

to this y-coordinate. 


39. D; To shift 3 units down, subtract 3 from f(x) to get 

g(x) = - x 2 - 7. 

40. G; Since both function have vertex (0, 4), G is 

correct. 

41. 0 


42. The graph of f(x) = (x - h) 2 is the result of 

translating the graph of f(x) = x 2 by h units along the 

x-axis. 

43a. f(x) = x 2 - 7 b. f(x) = - x 2 + 2 

c. f(x) = 1__ 

2 x 2 + 1 


44a. Distributive Property b. Commutative Property 

c. Associative Property d. Combine like terms 

45. no correlation 46. positive correlation 

47. a = 2, b = 0 

b 

x = -___ 

2a = - ____ 0 

2(2) = 0 

y = 2 x 2 - 1 

= 2 (0) 2 - 1 

= 2(0) - 1 = -1 


Let x = 1. 

y = 2 x 2 - 1 

= 2 (1) 2 - 1 

= 2(1) - 1 

= 2 - 1 = 1 

Let x = 2. 

y = 2 x 2 - 1 

= 2 (2) 2 - 1 

= 2(4) - 1 

= 8 - 1 = 7 





 

 

 

48. a = 1, b = -2 

b 

x = -___ 

2a = - ____ -2 

2(1) = 2__ 

2 = 1 

y = x 2 - 2x - 2 

= (1) 2 - 2(1) - 2 

= 1 - 2 - 2 = -3 


y = x 2 - 2x - 2 → y = x 2 - 2x + (-2) 


(0, -2). 

Let x = -1. 

y = x 2 - 2x - 2 

= (-1) 2 - 2(-1) - 2 

= 1 + 2 - 2 = 1 

 

 

 

Let x = -2 

y = x 2 - 2x - 2 

= (-2) 2 - 2(-2) - 2 

= 4 + 4 - 2 = 6 





 

 

 

 

 

 



49. a = -3, b = -1 

b 

x = -___ 

2a = - _____ -1 

2(-3) = ___ 1 

-6 = - 1__ 

6 

y = -3 x 2 - x + 6 

= -3 ( - 

1__ 

1__ 

6) 2 - ( - 

6) + 6 

= -3 ( ___ 

36) 1 + 1__ 

6 + 6 

= -___ 

1 

12 + 1__ 

6 + 6 = 6 ___ 1 

12 

6 , 6 ___ 

12) 1 . 


1__ 

y = -3 x 2 - x + 6 → y = -3 x 2 - x + (6) 


(0, 6). 

Let x = 1. 

y = -3 x 2 - x + 6 

= -3(1) 2 - 1 + 6 

= -3(1) - 1 + 6 

= -3 - 1 + 6 = 2 


Let x = 2. 

y = -3 x 2 - x + 6 

= -3 (2) 2 - 2 + 6 

= -3(4) - 2 + 6 

= -12 - 2 + 6 = -8 




 

MULTI-STEP TEST PREP, PAGE 620 

 


are constant. 

2. 

 

 

 

 

 

 

 

 

3. 0 and 6; x = 3; (3, 144) 

4. The x-coordinate represents the time when the 

water bottle rocket reaches it maximum height, 

which is represented by the y-coordinate. 

5. 1.775 s; for y = 110 the corresponding x-coordinate 

is 1.775. 

 

 

 

READY TO GO ON? PAGE 621 

1. y + 2 x 2 = 3x 

______ - 2 x 2 _____ -2 x 2 

y = -2 x 2 + 3x 


in the form y = a x 2 + bx + c where a = -2, b = 3, 

and c = 0. 

2. x 2 + y = 4 + x 2 

_______ - x 2 ______ - x 2 

y = 4 


a is 0. 


are constant. 

4. Since a < 0, the parabola opens downward. The 

parabola has a maximum. 

5. y - 2 x 2 = 4x + 3 

______ + 2 x 2 _______ +2 x 2 

y = 2 x 2 + 4x + 3 

Since a > 0, the parabola opens upward. The 

parabola has a minimum. 

6. Since a < 0, the parabola opens downward. The 

parabola has a maximum. 

7. 

 

x 

 

y = 1__ 

2 x 2 - 2 

-2 0 

-1 - 

3__ 

2 

0 -2 

1 - 

3__ 

2 

2 0 

 

 

 

 



x = _______ -1 + 3 = 2__ 

2 2 = 1 


9. The only zero appears to be -2. 




The vertex is at (1, 1), so the axis of symmetry is 

x = 1. 

11. a = 1, b = 6 

b 

x = -___ 

2a = - ____ 6 6__ 

2(1) = - 2 = -3 

y = x 2 + 6x + 2 

= (-3) 2 + 6(-3) + 2 

= 9 - 18 + 2 = -7 


 

 

 

 



12. a = -2, b = 4 

b 

x = -___ 

2a = - _____ 4 

2(-2) = - ___ 4 

-4 = 1 

y = 3 + 4x - 2 x 2 

= 3 + 4(1) - 2 (1) 2 

= 3 + 4 - 2(1) 

= 3 + 4 - 2 = 5 


13. a = 3, b = 12 

b 

x = -___ 

2a = - ____ 12 

2(3) = - ___ 12 

6 = -2 

y = 3 x 2 + 12x - 12 

= 3 (-2) 2 + 12(-2) - 12 

= 3(4) -24 -12 

= 12 - 24 - 12 = -24 


14. a = -0.02, b = 1.6 

b 

x = -___ 

2a = - ________ 1.6 

2(-0.02) = - ______ 1.6 

-0.04 = 40 

y = -0.02 x 2 + 1.6x 

= -0.02 (40) 2 + 1.6(40) 

= -0.02(1600) + 64 

= -32 + 64 = 32 

The hangar is 32 ft tall. 

15. a = 1, b = 3 

b 

x = -___ 

2a = - ____ 3 3__ 

2(1) = - 2 

y = x 2 + 3x + 9 

= ( - 

3__ 

= 9__ 

4 - 9__ 

2) 2 + 3 ( - 

3__ 

2) + 9 

2 + 9 = ___ 27 

4 

2 , ___ 27 

4 

) . 


3__ 

y = x 2 + 3x + 9 → y = x 2 + 3x + (9) 


(0, 9). 

Let x = -1. 

y = x 2 + 3x + 9 

= (-1) 2 + 3(-1) + 9 

= 1 - 3 + 9 = 7 

Let x = 1. 

y = x 2 + 3x + 9 

= (1) 2 + 3(1) + 9 

= 1 + 3 + 9 = 13 





 

 

16. a = 1, b = -2 

b 

x = -___ 

2a = - ____ -2 

2(1) = 2__ 

2 = 1 

y = x 2 - 2x - 15 

= (1) 2 - 2(1) -15 

= 1 - 2 - 15 = -16 


y = x 2 - 2x - 15 → y = x 2 - 2x + (-15) 


(0, -15). 

Let x = -1. 

y = x 2 - 2x - 15 

= (-1) 2 - 2(-1) - 15 

= 1 + 2 - 15 = -12 

Let x = -2. 

y = x 2 - 2x - 15 

= (-2) 2 - 2(-2) - 15 

= 4 + 4 - 15 = -7 





 

 

17. a = 1, b = -2 

b 

x = -___ 

2a = - ____ -2 

2(1) = 2__ 

2 = 1 

y = x 2 - 2x - 8 

= (1) 2 - 2(1) - 8 

= 1 - 2 - 8 = -9 


 

y = x 2 - 2x - 8 → y = x 2 - 2x + (-8) 


(0, -8). 

Let x = -1. 

y = x 2 - 2x - 8 

= (-1) 2 - 2(-1) - 8 

= 1 + 2 - 8 = -5 

 

 

 

Let x = -2. 

y = x 2 - 2x - 8 

= (-2) 2 - 2(-2) - 8 

= 4 + 4 - 8 = 0 

Two other points are (-1, -5) and (-2, 0). 




 

 

 

 

 

 

 

 

 

 

 

 

 



18. a = 2, b = 0 

b 

x = -___ 

2a = - ____ 0 

2(2) = 0 

y = 2 x 2 - 6 

= 2 (0) 2 - 6 

= 2(0) - 6 = -6 


Let x = 1. 

y = 2 x 2 - 6 

= 2 (1) 2 - 6 

= 2(1) - 6 

= 2 - 6 = -4 

Let x = 2. 

y = 2 x 2 - 6 

= 2 (2) 2 - 6 

= 2(4) - 6 

= 8 - 6 = 2 

Two other points are (1, -4) and (2, 2). 




 

 

 

19. a = 4, b = 8 

b 

x = -___ 

2a = - ____ 8 8__ 

2(4) = - 8 = -1 

y = 4 x 2 + 8x - 2 

= 4 (-1) 2 + 8(-1) - 2 

= 4(1) - 8 - 2 

= 4 - 8 - 2 = -6 


 

y = 4 x 2 + 8x - 2 → y = 4 x 2 + 8x + (-2) 


(0, -2). 

Let x = 1. 

y = 4 x 2 + 8x - 2 

= 4 (1) 2 + 8(1) - 2 

= 4(1) + 8 - 2 

= 4 + 8 - 2 = 10 

 

 

 

Let x = 2. 

y = 4 x 2 + 8x - 2 

= 4 (2) 2 + 8(2) - 2 

= 4(4) + 16 - 2 

= 16 + 16 - 2 = 30 





 

 

 

 

 

 

 

 

20. a = 2, b = 10 

b 

x = -___ 

2a = - ____ 10 

2(2) = - ___ 10 5__ 

4 = - 2 

y = 2 x 2 + 10x + 1 

= 2 ( - 

5__ 

5__ 

2) 2 + 10 ( - 

2) + 1 

= 2 ( ___ 25 

4 ) - ___ 50 

2 + 1 

___ 

___ 

___ 

= 25 

2 - 50 

2 + 1 = - 23 

2 

2 , - ___ 23 

2 ) . 


5__ 

y = 2 x 2 + 10x + 1 → y = 2 x 2 + 10x + (1) 


(0, 1). 

Let x = -2. 

y = 2 x 2 + 10x + 1 

= 2 (-2) 2 + 10(-2) + 1 

= 2(4) - 20 + 1 

= 8 - 20 + 1 = -11 

Let x = -1. 

y = 2 x 2 + 10x + 1 

= 2 (-1) 2 + 10(-1) + 1 

= 2(1) - 10 + 1 

= 2 - 10 + 1 = -7 





 

 

 

 

 

21. The graph of g(x) is the same width, has the same 

axis of symmetry, also opens upward, and its vertex 

is translated 2 units down to (0, -2). 


symmetry, also opens upward, and its vertex is the 

same. 


of symmetry, also opens upward, and its vertex is 


24. The graph of g(x) has the same width, has the same 



25a. h 1 

(t) = -16 t 2 + 196; h 2 

(t) = -16 t 2 + 676 

b. 3.5 s; 6.5 s 



9-5 SOLVING QUADRATIC EQUATIONS 

BY GRAPHING, PAGES 622–627 


1a. x 2 - 8x - 16 = 2 x 2 

______________ 

-2 x 2 _____ -2 x 2 

- x 2 - 8x - 16 = 0 

Graph y = - x 2 - 8x - 16. The axis of symmetry is 

x = -4. The vertex is (-4, 0). The y-intercept is 

-16. Another point is (-1, -9). Graph the points 

and reflect them across the axis of symmetry. 

y 

x 

-6 -4 

0 

-4 

The only zero appears to be 

-4. 

b. 6x + 10 = - x 2 

_______ + x 2 ____ + x 2 

x 2 + 6x + 10 = 0 

Graph y = x 2 + 6x + 10. The axis of symmetry is 


10. Another point is (-1, 5). Graph the points and 

reflect them across the axis of symmetry. 

y 

2. Graph y = -16 x 2 + 32x. 

height 

16 

12 

8 

4 

0 

l 

1 2 

time (s) 

The zeros appear to be 0 and 2. 

The dolphin leaves the water at 0 s and reenters the 

water at 2 s. 

The dolphin is out of the water for about 2 s. 


1. The vertex is on the x-axis. 

2. Either the vertex is above the x-axis and the graph 

opens upward, or the vertex is below the x-axis and 

the graph opens downward. 

3. Either the vertex is above the x-axis and the graph 

opens downward, or the vertex is below the x-axis 

and the graph opens upward. 

4. 

 

 

t 

-6 

-4 

-2 

4 

2 

0 

x 

The function appears to have 

no zeros. 

c. Graph y = - x 2 + 4. The axis of symmetry is 

x = 0. The vertex is (0, 4). Another point is (1, 3). 

Graph the points and reflect them across the axis of 

symmetry. 

8 

y 

 

 

 

 



1. zeros, x-intercepts 

 

2. Graph y = x 2 - 4. The axis of symmetry is 

x = 0. The vertex is (0, -4). Another point is (1, -3). 


symmetry. 

y 

8 

-4 

-4 

-8 

0 

2 4 

x 

The zeros appear to be 

-2 and 2. 

-4 

4 

-2 

0 

-8 

2 4 

x 


-2 and 2. 



3. x 2 = 16 

____ - 16 -16 ____ 

x 2 - 16 = 0 

Graph y = x 2 - 16. The axis of symmetry is x = 0. 

The vertex is (0, -16). Another point is (1, -15). 


symmetry. 

y 

-8 

x 

-8 -4 

0 

4 8 

-16 

-24 


-4 and 4. 

4. Graph y = -2 x 2 - 6. The axis of symmetry is 



symmetry. 

y 

x 

-4 -2 

0 

2 4 

-4 

-12 


no zeros. 

5. Graph y = - x 2 + 12x - 36. The axis of symmetry is 

x = 6. The vertex is (6, 0). The y-intercept is 

-36. Another point is (3, -9). Graph the points and 


-8 

-16 

-24 

0 

y 

6. - x 2 = -9 

6 

12 

x 

The only zero appears to be 6. 

____ + x 2 + ____ x 2 

0 = x 2 - 9 

Graph y = x 2 - 9. The axis of symmetry is 



symmetry. 

y 

-4 

4 

-2 

0 

-4 

2 4 

x 


-3 and 3. 

7. 2 x 2 = 3 x 2 - 2x - 8 

_____ -2 x 2 _____________ 

-2 x 2 

0 = x 2 - 2x - 8 

Graph y = x 2 - 2x - 8. The axis of symmetry is 

x = 1. The vertex is (1, -9). The y-intercept is -8. 

Another point is (-1, -5). Graph the points and 


y 

-2 

4 

-8 

0 

2 4 

x 


-2 and 4. 

8. Graph y = x 2 - 6x + 9. The axis of symmetry is 

x = 3. The vertex is (3, 0). The y-intercept is 9. 

Another point is (1, 4). Graph the points and reflect 

them across the axis of symmetry. 

12 

8 

4 

0 

y 

2 4 6 

x 


9. 8x = -4 x 2 - 4 

____ -8x ________ 

-8x 

0 = -4 x 2 - 8x - 4 

Graph y = -4 x 2 - 8x - 4. The axis of symmetry 

is x = -1. The vertex is (-1, 0). The y-intercept is 



-4 

-2 

-4 

-8 

-12 

y 

0 

2 

x 


-1. 

10. Graph y = x 2 + 5x + 4. The axis of symmetry is 

x = -2.5. The vertex is (-2.5, -2.25). The 

y-intercept is 4. Another point is (1, 10). Graph the 

points and reflect them across the axis of symmetry. 

y 

4 

-6 

-4 

2 

-2 

-2 

-4 

0 

x 


-4 and -1. 



11. Graph y = x 2 + 2. The axis of symmetry is x = 0. 

The vertex is (0, 2). Another point is (1, 3). Graph 

the points and reflect them across the axis of 

symmetry. 

-4 

-2 

6 

4 

2 

0 

y 

2 4 

x 


no zeros. 

12. x 2 - 6x = 7 

_______ - 7 ___ -7 

x 2 - 6x - 7 = 0 

Graph y = x 2 - 6x - 7. The axis of symmetry is 

x = 3. The vertex is (3, -16). The y-intercept is 



-4 

-8 

-12 

y 

0 

4 8 12 

x 


-1 and 7. 

13. x 2 + 5x = -8 

_______ + 8 ___ +8 

x 2 + 5x + 8 = 0 

Graph y = x 2 + 5x + 8. The axis of symmetry is 

x = -2.5. The vertex is (-2.5, 1.75). The y-intercept 

is 8. Another point is (-1, 4). Graph the points and 


-6 

-4 

-2 

12 

8 

4 

0 

y 

x 


no zeros. 

14. Graph y = -16 x 2 + 80x. 


The baseball leaves the machine at 0 s and hits the 

ground at 5 s. The baseball was in the air for 5 s. 


15. Graph y = - x 2 + 16. The axis of symmetry is 



symmetry. 

-8 

8 

-8 

0 

y 

8 

x 


-4 and 4. 

16. 3 x 2 = -7 

___ + 7 ___ +7 

3 x 2 + 7 = 0 

Graph y = 3 x 2 + 7. The axis of symmetry is 



symmetry. 

4 

x 

-4 -2 

0 

2 4 

y 

17. 5 x 2 - 12x + 10 = x 2 + 10x 

______________ 

- 10x ________ - 10x 

5 x 2 - 22x + 10 = x 2 


no zeros. 

______________ 

- x 2 ____ - x 2 

4 x 2 - 22x + 10 = 0 

Graph y = 4 x 2 - 22x + 10. The axis of symmetry 

is x = 2.75. The vertex is (2.75, -20.25). The 

y-intercept is 10. Another point is (1, -8). Graph the 


-8 

-16 

-24 

0 

y 

2 4 6 

x 


1__ 

, and 5. 

2 

18. Graph y = x 2 + 10x + 25. The axis of symmetry 




-12 

-8 

-4 

24 

16 

8 

0 

y 

x 


-5. 

19. -4 x 2 - 24x = 36 

__________ - 36 ____ -36 

-4 x 2 - 24x - 36 = 0 

Graph y = -4 x 2 - 24x - 36. The axis of symmetry 


-36. Another point is (-2, -4). Graph the points 


-6 

-8 

0 

y 

x 


-3. 



20. -9 x 2 + 10x - 9 = -8x 

_____________ +8x ____ +8x 

-9 x 2 + 18x - 9 = 0 

Graph y = -9 x 2 + 18x - 9. The axis of symmetry 

is x = 1. The vertex is (1, 0). The y-intercept is -9. 

Another point is (-1, -36). Graph the points and 


-2 

0 

-8 

-16 

y 

2 

x 

4 6 


21. Graph y = - x 2 - 1. The axis of symmetry is 



symmetry. 

-4 

-2 

0 

y 

x 

2 4 

24. Graph h(t) = -16 t 2 + 1450. 

The zeros appear to be -9.5 and 9.5. 

The water droplet begins to fall at 0 s and reaches 

the bottom at 9.5 s. 

The droplet is in the air for about 9.5 s. 

-6 

800 

400 

y 

0 6 



29. never 

30a. 

x 

-4 

-6 


no zeros. 

22. Graph y = 3 x 2 - 27. The axis of symmetry is 

x = 0. The vertex is (0, -27). Another point is 

(2, -15). Graph the points and reflect them across 

the axis of symmetry. 

-4 

-2 

0 

-8 

-16 

y 

x 

2 4 


-3 and 3. 

23. 4 x 2 - 4x + 5 = 2 x 2 

_____________ 

-2 x 2 _____ -2 x 2 

2 x 2 - 4x + 5 = 0 

Graph y = 2 x 2 - 4x + 5. The axis of symmetry 

is x = 1. The vertex is (1, 3). The y-intercept is 5. 

Another point is (-1, 13). Graph the points and 


6 

y 

b. 784 ft c. 14 s 

31a. 4 s b. 10 ft 

c. According to the graph, x = 4 is a solution of the 

related equation, but x = 4 is not a solution of 

0 = -5 x 2 + 10x. 

32. -3, 4 

33. Graph y = -5 x 2 + 7x. 

The zeros appear to be 0 and 1.4. 

The kangaroo jumps at 0 s and returns to the 

ground at 1.4 s. 

The kangaroo was in the air for 1.4 s. 

34. -1 35. -1, 1 

36. no real solutions 37a. x - 4, x - 8 

b. c 2 = a 2 + b 2 

(x) 2 = (x - 4) 2 + (x - 8) 2 

x 2 = (x) 2 - 2(x)(4) + (4) 2 + (x) 2 - 2(x)(8) + (8) 2 

x 2 = x 2 - 8x + 16 + x 2 - 16x + 64 

x 2 = 2 x 2 - 24x + 80 

____ - x 2 ______________ 

- x 2 

0 = x 2 - 24x + 80 

4 

-2 

2 

0 

2 4 6 

x 


no zeros. 



c. Graph y = x 2 - 24x + 80. The axis of symmetry 

is x = 12. The vertex is (12, -64). The y-intercept 

is 80. Another point is (11, -63). Graph the points 


y 

x 

0 6 12 18 

-20 

-40 

-60 


d. Two possible lengths for the hypotenuse; no; if 

x = 4, then the length of one leg would be 0 and 

the other would be negative, which is impossible. 

38. Look for places where the output values are 0. The 

corresponding input values are the solutions. 

If there are no output values equal to 0, then look 

for sign changes. The solution is between the input 

values that correspond to the y-value with the sign 

change. 

39. The graph of y = a x 2 - c, where a > 0 and 

c > 0, will always have two solutions because it 

is a parabola that opens upward whose vertex is 

below the origin. So it will always cross the x-axis in 

two places. By contrast, the graph of y = a x 2 + c, 

where a > 0 and c > 0, opens upward but its vertex 

is above the origin. Therefore, it will never cross the 

x-axis. 

40a. Graph h = -16 t 2 + 80t. 

height (ft) 

80 

60 

40 

20 

0 

h 

2 4 6 8 

time (s) 


The golf ball leaves the ground at 0 s and hits the 

ground at 5 s. The golf ball is in the air for 5 s. 

b. 100 ft c. 2.5 s 

d. 84 ft; yes; at 1.5 s; 3.5 is 1 unit right of the axis of 

symmetry (x = 2.5). The ball will have the same 

height at the time represented by the point 1 unit 

left of the axis of symmetry. 


41. C; the graph of y = -2 x 2 + 2 cross the x-axis twice, 

so it has 2 zeros. Therefore, the equation has two 

solutions. 

42. G; to find the solutions of x 2 = -4x + 12, use 

y = x 2 + 4x + 12, which is shown in G. 

t 

43. Graph y = 2 x 2 + x - 1. The axis of symmetry is x = 

-0.25. The vertex is (-0.25, -0.625). The 

y-intercept is -1. Another point is (1, 2). Graph the 


 

 

 

 

 

The zeros appear to be -1 and 1__ 

2 . 

The x-intercepts are the solutions of the equation. 


44. x ≈ -2.3 or x ≈ 1 45. no real solutions 

46. x ≈ -1 or x ≈ 0.75 47. x ≈ -1.6 or x ≈ 0.86 


48. y - y 1 

= m(x - x 1 

) 

y - 3 = 1__ 

49. y - y 1 

= m(x - x 1 

) 

(x - 2) 

y - 4 = -3(x - (-2)) 

2 

y = 1__ 

y - 4 = -3(x + 2) 

2 x + 2 y = -3x - 2 

50. y - y 1 

= m(x - x 1 

) 

y - 1 = 0(x - 2) 

y = 1 

52. 

______ 5 2 · 2 4 

= 5 2 - 1 · 2 4 - 2 

5 · 2 2 

54. 

__ 

( 2) x 3 -3 

y 

= 5 1 · 2 2 

= 5 · 4 = 20 

= 

__ 

( y 2 

3) 3 

x 

( 

= 

_____ y 2 ) 3 

( x 3 ) 3 

__ 

= y 6 

x 9 

55. 

____ 

( a 2 b 3 

) 3 

= ( a 2 - 1 · b 3 - 2 ) 3 

a b 2 

56. ( ___ 4s 

= ( a 1 · b 1 ) 3 

= (ab) 3 

= a 3 b 3 

___ 

3t ) -2 = ( 3t 

4s) 2 

____ 

= (3t) 2 

(4s) 2 

____ 

= 3 2 t 2 

4 2 s 2 

____ 

= 9 t 2 

16 s 2 

 

51. 

__ 3 4 

3 = 3 4 - 1 

( x 4 ) 5 

53. 

_____ 

= 3 3 = 27 

( x 3 ) = ___ x 20 

3 

x 9 

= x 20 - 9 

= x 11 



57. ( 2__ 

__ 

3) -3 · ( a 3 

58. 

____ 

( - k 2 

) -3 

5 k 3 

b 

) 

-2 

= ( 3__ 

2) 3 · 

( b__ 

= 27 

8 · b 2 

a 3) 2 

___ _____ 

( a 3 ) 2 

____ 

= 27 b 2 

8 a 6 

= 

____ 

( 5 k 3 

2) 3 

- k 

= ( -5 · k 

3 - 2) 3 

= ( -5 k 

1) 3 

= (-5) 3 k 3 

= -125 k 3 

59. The parabola is narrower, and has the same vertex 

and the same zero. 

60. The parabola is the same width, has a vertex 

translated 8 units down from f(x), and has 2 zeros. 

61. The parabola is wider than f(x), has a vertex 

translated 2 units up from f(x), and has no zeros. 

TECHNOLOGY LAB: EXPLORE ROOTS, 

ZEROS, AND X-INTERCEPTS, 

PAGES 628–629 

TRY THIS, PAGES 628–629 

1. -1, 5 2. -2, 3 

3. -1, 0.5 4. -0.8, 2 

5. Possible answer: Change Tblstart to 1.2 and △Tbl 

to 0.01 and scroll until I found zero. 

6. Possible answer: The function has no zeros and the 

equation has no solutions. 

7. -1, 1.5 8. -2, -0.6 

9. -0.5, 0.8 10. -0.4, 2.4 

11. Possible answer: Set the window dimensions wider 

to see both zeros, and then trace to find the other 

zero. 


BY FACTORING, PAGES 630–635 


1a. (x)(x + 4) = 0 

x = 0 or x + 4 = 0 

x = -4 

The solutions are 0 and -4. 

b. (x + 4)(x - 3) = 0 

x + 4 = 0 or x - 3 = 0 

x = -4 or x = 3 

The solutions are -4 and 3. 

2a. x 2 - 6x + 9 = 0 

(x - 3)(x - 3) = 0 

x - 3 = 0 

x = 3 

The only solution is 3. 

b. x 2 + 4x = 5 

_______ - 5 ___ -5 

x 2 + 4x - 5 = 0 

(x + 5)(x - 1) = 0 

x + 5 = 0 or x - 1 = 0 

x = -5 or x = 1 


c. 30x = -9 x 2 - 25 

_____ -30x _________ 

-30x 

0 = -9 x 2 - 30x - 25 

0 = -1 (9 x 2 + 30x + 25) 

0 = -1(3x + 5)(3x + 5) 

-1 ≠ 0 or 3x + 5 = 0 

3x = -5 

x = - 

5__ 

3 

The only solution is - 

5__ 

3 . 

d. 3 x 2 - 4x + 1 = 0 

(3x - 1)(x - 1) = 0 

3x - 1 = 0 or x - 1 = 0 

3x = 1 or x = 1 

x = 1__ 

3 

The solutions are 1__ 

3. h = -16 t 2 + 8t + 24 

0 = -16 t 2 + 8t + 24 

0 = -8 ( 2 t 

2 - t - 3 ) 

0 = -8(2t - 3)(t + 1) 

and 1. 

3 

-8 ≠ 0, 2t - 3 = 0 or t + 1 = 0 

2t = 3 or t = -1 ✗ 

t = 3__ 

2 

It takes the diver 1.5 s to reach the water. 


1. Possible answer: One method would be to factor 

and set each factor equal to 0. Another method 

would be to graph and find the x-intercepts of 

the graph. In both methods, 0 is important. In the 

factoring method, one side of the equation must be 

0. In the graphing method, the solutions are where 

f(x) = 0. 

2. -2, 6 

3. 

 

 

 

 

 

 



1. (x + 2)(x - 8) = 0 

x + 2 = 0 or x - 8 = 0 

x = -2 or x = 8 


 

 



2. (x - 6)(x - 5) = 0 

x - 6 = 0 or x - 5 = 0 

x = 6 or x = 5 

The solutions are 6 and 5. 

3. (x + 7)(x + 9) = 0 

x + 7 = 0 or x + 9 = 0 

x = -7 or x = -9 

The solutions are -7 and -9. 

4. (x)(x - 1) = 0 

x = 0 or x - 1 = 0 

x = 1 


5. (x)(x + 11) = 0 

x = 0 or x + 11 = 0 

x = -11 


6. (3x + 2)(4x - 1) = 0 

3x + 2 = 0 or 4x - 1 = 0 

3x = -2 or 4x = 1 

x = - 

2__ 

3 

or x = 

1__ 

4 

The solutions are - 

2__ 

3 

and 

1__ 

4 . 

7. x 2 + 4x - 12 = 0 

(x + 6)(x - 2) = 0 

x + 6 = 0 or x - 2 = 0 

x = -6 or x = 2 


8. x 2 - 8x - 9 = 0 

(x + 1)(x - 9) = 0 

x + 1 = 0 or x - 9 = 0 

x = -1 or x = 9 


9. x 2 - 5x + 6 = 0 

(x - 2)(x - 3) = 0 

x - 2 = 0 or x - 3 = 0 

x = 2 or x = 3 


10. x 2 - 3x = 10 

_______ - 10 ____ -10 

x 2 - 3x - 10 = 0 

(x + 2)(x - 5) = 0 

x + 2 = 0 or x - 5 = 0 

x = -2 or x = 5 


11. x 2 + 10x = -16 

________ + 16 ____ +16 

x 2 + 10x + 16 = 0 

(x + 8)(x + 2) = 0 

x + 8 = 0 or x + 2 = 0 

x = -8 or x = -2 


12. x 2 + 2x = 15 

_______ - 15 ____ -15 

x 2 + 2x - 15 = 0 

(x + 5)(x - 3) = 0 

x + 5 = 0 or x - 3 = 0 

x = -5 or x = 3 


13. x 2 - 8x + 16 = 0 

(x - 4)(x - 4) = 0 

x - 4 = 0 

x = 4 


14. -3 x 2 = 18x + 27 

_____ +3 x 2 _________ 

+3 x 2 

0 = 3 x 2 + 18x + 27 

0 = 3 ( x 2 + 6x + 9) 

0 = 3(x + 3)(x + 3) 

3 ≠ 0 or x + 3 = 0 

x = -3 

The only solution is -3. 

15. x 2 + 36 = 12x 

_______ -12x _____ -12x 

x 2 - 12x +36 = 0 

(x - 6)(x - 6) = 0 

x - 6 = 0 

x = 6 


16. x 2 + 14x + 49 = 0 

(x + 7)(x + 7) = 0 

x + 7 = 0 

x = -7 


17. x 2 - 16x + 64 = 0 

(x - 8)(x - 8) = 0 

x - 8 = 0 

x = 8 


18. 2 x 2 + 6x = -18 

________ + 18 ____ +18 

2 x 2 + 6x + 18 = 0 

2 ( x 2 + 3x + 9) = 0 

Since 2 ≠ 0 and since y = x 2 + 3x + 9 opens 

upward and has its vertex at (-1.5, 6.75), it does 

not intersect the x-axis. So there are no real 

solutions. 

19. h = -16 t 2 + 14t + 2 

0 = -16 t 2 + 14t + 2 

0 = -2 ( 8 t 

2 - 7t - 1 ) 

0 = -2(8t + 1)(t - 1) 

-2 ≠ 0, 8t + 1 = 0 or t - 1 = 0 

8t = -1 or t = 1 

t = - 

1__ 

8 ✗ 

The beanbag takes 1 s to reach the ground. 




20. (x - 8)(x + 6) = 0 

x - 8 = 0 or x + 6 = 0 

x = 8 or x = -6 


21. (x + 4)(x + 7) = 0 

x + 4 = 0 or x + 7 = 0 

x = -4 or x = -7 


22. (x - 2)(x - 5) = 0 

x - 2 = 0 or x - 5 = 0 

x = 2 or x = 5 


23. (x - 9)(x) = 0 

x - 9 = 0 or x = 0 

x = 9 


24. (x)(x + 25) = 0 

x = 0 or x + 25 = 0 

x = -25 


25. (2x + 1)(3x - 1) = 0 

2x + 1 = 0 or 3x - 1 = 0 

2x = -1 or 3x = 1 

or x = 

1__ 

x = - 

1__ 

2 


1__ 

2 

3 

and 

1__ 

3 . 

26. x 2 + 8x + 15 = 0 

(x + 5)(x + 3) = 0 

x + 5 = 0 or x + 3 = 0 

x = -5 or x = -3 


27. x 2 - 2x - 8 = 0 

(x + 2)(x - 4) = 0 

x + 2 = 0 or x - 4 = 0 

x = -2 or x = 4 


28. x 2 - 4x + 3 = 0 

(x - 1)(x - 3) = 0 

x - 1 = 0 or x - 3 = 0 

x = 1 or x = 3 


29. x 2 + 10x + 25 = 0 

(x + 5)(x + 5) = 0 

x + 5 = 0 

x = -5 


30. x 2 - x = 12 

______ - 12 ____ -12 

x 2 - x - 12 = 0 

(x + 3)(x - 4) = 0 

x + 3 = 0 or x - 4 = 0 

x = -3 or x = 4 


31. - x 2 = 4x + 4 

____ + x 2 _______ + x 2 

0 = x 2 + 4x + 4 

0 = (x + 2)(x + 2) 

x + 2 = 0 

x = -2 

The only solution appears to be -2. 

32. h = -16 t 2 + 95t + 6 

0 = -16 t 2 + 95t + 6 

0 = -1 ( 16 t 

2 - 95t - 6 ) 

0 = -1(16t + 1)(t - 6) 

-1 ≠ 0, 16t + 1 = 0 or t - 6 = 0 

16t = -1 or t = 6 

t = -___ 

1 

16 ✘ 

It takes the flare 6 s to reach the ground. 

33. (x + 8)(x + 8) = 0 

x + 8 = 0 

x = -8 

There is 1 solution. 

34. (x - 3)(x + 3) = 0 

x - 3 = 0 or x + 3 = 0 

x = 3 or x = -3 

There are 2 solutions. 

35. (x + 7) 2 = 0 

x + 7 = 0 

x = -7 


36. 3 x 2 + 12x + 9 = 0 

x 2 + 4x + 3 = 0 

(x + 3)(x + 1) = 0 

x + 3 = 0 or x + 1 = 0 

x = -3 or x = -1 


37. x 2 + 12x + 40 = 4 

____________ -4 ___ -4 

x 2 + 12x + 36 = 0 

(x + 6)(x + 6) = 0 

x + 6 = 0 

x = -6 


38. (x - 2) 2 = 9 

(x) 2 - 2(x)(2) + (2) 2 = 9 

x 2 - 4x + 4 = 9 

__________ - 9 ___ -9 

x 2 - 4x - 5 = 0 

(x + 1)(x - 5) = 0 

x + 1 = 0 or x - 5 = 0 

x = -1 or x = 5 


39. B; after factoring the polynomial, you should solve 

the equations x - 1 = 0 and x + 2 = 0. The correct 

solutions are 1 and -2. 



40. (x)(x + 2) = 24 

x(x) + x(2) = 24 

x 2 + 2x = 24 

______ - 24 ____ -24 

x 2 + 2x - 24 = 0 

(x + 6)(x - 4) = 0 

x + 6 = 0 or x - 4 = 0 

x = -6 or x = 4 

If x = -6, x + 2 = -4 and if x = 4, x + 2 = 6. The 

two possible sets of integers are -6 and -4, and 4 

and 6. 

41. A = 1__ 

2 bh 

15 = 1__ (h - 1)h 

2 

h(h) + 

1__ 

15 = 1__ 

2 

15 = 1__ 

2 h 

____ -15 ________ 

-15 

2 h 2 - 1__ 

0 = 1__ 

2 h 2 - 1__ 

2 h(-1) 

2 h - 15 

0 = h 2 - h - 30 

0 = (h + 5)(h - 6) 

h + 5 = 0 or h - 6 = 0 

h = -5 ✗ or h = 6 

The height of the triangle is 6 m. 

42. Let l represent the length and w represent the 

width. 

A = lw 

310 = (3w - 1)w 

310 = 3w(w) - 1(w) 

310 = 3 w 2 - w 

_____ -310 ________ 

-310 

0 = 3 w 2 - w - 310 

0 = (3w - 31)(w + 10) 

3w - 31 = 0 or w + 10 = 0 

3w = 31 or w = -10 ✗ 

w = 10 1__ 

3 

l = 3w - 1 

l = 3 ( 10 1__ 3) - 1 

l = 31 - 1 

l = 30 

The dimensions of the rectangle are 30 ft by 10 1__ 

3 ft. 

43. h = -5 t 2 + 30t 

0 = -5 t 2 + 30t 

0 = -5t(t - 6) 

-5t = 0 or t - 6 = 0 

t = 0 or t = 6 

It takes the rocket 6 s to reach the ground. 

44. A = 1__ 

2 h ( b 1 

+ b 2 ) 

48 = 1__ x(x + (x + 4)) 

2 

48 = 1__ x(2x + 4) 

2 

48 = 1__ x(2x) + 

1__ 

2 2 x(4) 

48 = x 2 + 2x 

____ -48 _______ -48 

0 = x 2 + 2x - 48 

0 = (x + 8)(x - 6) 

x + 8 = 0 or x - 6 = 0 

x = -8 ✗ or x = 6 

The length of the shorter base is 6 cm. 

45. No; the solutions of x - 2 = 5 and x + 3 = 5 

are 7 and 2; 7 and 2 are not solutions of 

(x - 2)(x + 3) = 5. 

46. The Zero Product Property states that if the product 

of two numbers is 0, then at least one of the 

numbers must be 0. When you factor a quadratic, 

the product of the factors is 0, so one of the factors 

must be 0. 

47a. 0 = -16 t 2 + 32t + 48 

0 = -16 ( t 2 - 2 - 3) 

0 = -16(t - 3)(t + 1) 

t - 3 = 0 or t + 1 = 0 

t = 3 or t = -1 ✗ 

The golf ball is in the air for 3 s. 

b. h = -16 t 2 + 32t + 48 

h = -16(1) 2 + 32(1) + 48 

h = -16 + 32 + 48 

h = 64 

The height of the golf ball is 64 ft after 1 s. 

c. Yes; the vertex is (1, 64), which represents the 

ball’s maximum height of 64 ft after 1 s in the air. 


48. C; The solutions of (x - 1)(2x + 5) = 0 are the 

same as the solutions of x - 1 = 0 and 2x + 5 = 0. 

Since the solutions of those equations are x = 1 and 

x = - 

5__ 

, C is correct. 

2 

49. F; The graph of y = x 2 - 5x + 6 could be used to 

solve the equation. 


50. 6 x 2 + 11x = 10 

________ - 10 ____ -10 

6 x 2 + 11x - 10 = 0 

(2x + 5)(3x - 2) = 0 

2x + 5 = 0 or 3x - 2 = 0 

2x = -5 or 3x = 2 

or x = 

2__ 

x = - 

5__ 

2 


5__ 

2 

3 

and 

2__ 

3 . 



51. 

52. 

0.2 x 2 + 1 = -1.2x 

________ +1.2x _____ +1.2x 

0.2 x 2 + 1.2x + 1 = 0 

0.2 ( x 2 + 6x + 5) = 0 

0.2(x + 5)(x + 1) = 0 

0.2 ≠ 0, x + 5 = 0 or x + 1 = 0 

x = -5 or x = -1 


1__ 

3 x 2 = 2x - 3 

- 

1__ 

- 

1__ 

_____ 3 x 2 ______ 3 x 2 

0 = - 

1__ 

3 x 2 + 2x - 3 

0 = - 

1__ 

3 ( x 2 - 6x + 9) 

0 = - 

1__ 

(x - 3)(x - 3) 

3 

- 

1__ 

3 ≠ 0 or x - 3 = 0 

x = 3 


53. 75x - 45 = -30 x 2 

________ +30 x 2 ______ +30 x 2 

30 x 2 + 75x - 45 = 0 

15 (2 x 2 + 5x - 3) = 0 

15(2x - 1)(x + 3) = 0 

15 ≠ 0, 2x - 1 = 0 or x + 3 = 0 

2x = 1 or x = -3 

x = 1__ 

2 

The solutions are 1__ and -3. 

2 

54. x 2 = -4(2x + 3) 

x 2 = -4(2x) - 4(3) 

x 2 = -8x - 12 

____ - x 2 _________ 

- x 2 

0 = - x 2 - 8x - 12 

0 = -1( x 2 + 8x + 12) 

0 = -1(x + 6)(x + 2) 

-1 ≠ 0, x + 6 = 0 or x + 2 = 0 

x = -6 or x = -2 


_______ 

x(x - 3) 

55. 

= 5 

2 

x(x - 3) 

2 (_______ 

2 

) = 2(5) 

x(x - 3) = 10 

x(x) + x(-3) = 10 

x 2 - 3x = 10 

______ - 10 ____ -10 

x 2 - 3x - 10 = 0 

(x + 2)(x - 5) = 0 

x + 2 = 0 or x - 5 = 0 

x = -2 or x = 5 


56. A = lw 

A = (x + 3)(x - 4) 

A = x(x) + x(-4) + 3(x) + 3(-4) 

A = x 2 - 4x + 3x - 12 

A = x 2 - x - 12 

The area is represented by x 2 - x - 12. 

57. A = lw 

A = (x)(3) 

A = 3x 

The area is represented by 3x. 

58. A = ( x 2 - x - 12) - (3x) 

A = x 2 - 4x - 12 

The area of the shaded region is represented by 

x 2 - 4x - 12. 

A = x 2 - 4x - 12 

48 = x 2 - 4x - 12 

____ -48 ___________ - 48 

0 = x 2 - 4x - 60 

0 = (x - 10)(x + 6) 

x - 10 = 0 or x + 6 = 0 

x = 10 or x = -6 

An x-value of 10 gives the desired area. 


59. f - 4 60. 5m 

61. √ 121 = √ 11 2 = 11 62. - √ 64 = - √ 8 2 = -8 

63. - √ 100 = - √ 10 2 = -10 

64. √ 225 = √ 15 2 = 15 

65. Graph y = x 2 - 49. The axis of symmetry is x = 0. 



symmetry. 

-8 

-4 

0 

-16 

-32 

y 

x 

4 8 

The solutions appear to be 

-7 and 7. 

66. x 2 = x + 12 

____ - x 2 ______ - x 2 

0 = - x 2 + x + 12 

Graph y = - x 2 + x + 12. The axis of symmetry is 

x = 0.5. The vertex is (0.5, 12.25). The y-intercept 

is 12. Another point is (1, 12). Graph the points and 


-2 

12 

8 

4 

0 

y 

2 4 

x 


4 and -3. 



67. - x 2 + 8x = 15 

________ - 15 ____ -15 

- x 2 + 8x - 15 = 0 

Graph y = - x 2 + 8x - 15. The axis of symmetry is 

x = 4. The vertex is (4, 1). The y-intercept is 



y 

2 

-2 

-4 

0 

2 4 6 

x 


3 and 5. 


BY USING SQUARE ROOTS, 

PAGES 636–641 


1a. x 2 = 121 

x = ± √ 121 

x = ±11 


b. x 2 = 0 

x = ± √ 0 

x = 0 

2a. 100 x 2 + 49 = 0 

_________ - 49 ____ -49 

_____ 100 x 2 

100 = ____ -49 

100 

x 2 49 

= -____ 

100 

x ≠ ± 

√ ____ 49 

100 

There are no real solutions. 

b. 36 x 2 = 1 

____ 36 x 2 

36 = ___ 1 

36 

x 2 = ___ 1 

36 

x = ± √ ___ 1 

36 

x = ± 

1__ 

6 


6 and - 1__ 

6 . 

3a. 0 = 90 - x 2 

c. x 2 = -16 

x ≠ ± √ 16 

There are no real 

solutions. 

____ + x 2 _______ + x 2 

x 2 = 90 

x = ± √ 90 

x ≈ ±9.49 

The approximate solutions are 9.49 and -9.49. 

b. 2 x 2 - 64 = 0 

_______ + 64 ____ +64 

___ 2 x 2 

2 = ___ 64 

2 

x 2 = 32 

x = ± √ 32 

x ≈ ±5.66 


c. x 2 + 45 = 0 

______ - 45 ____ -45 

x 2 = -45 

x ≠ ± √ 45 


4. A = 1__ 

2 h ( b 1 

+ b 2 ) 

6000 = 1__ (2x)(2x + x) 

2 

6000 = x(3x) 

6000 = 3 x 2 

_____ 

___ 

6000 

= 3 x 2 

3 3 

2000 = x 2 

± √ 2000 = x 

x ≈ 45 

Negative numbers are not reasonable for length, 

so x ≈ 45 is the only solution that makes sense. 

Therefore, the width of the front yard is 45 ft. 


1. Possible answer: There is no real number whose 

square is negative. 

2. Possible answer: The equation is equivalent to 

x 2 = 20, so x = ± √ 20 . √ 20 is between 4 and 5, so 

x is approximately ±4.5. 

3. 

 

 

 

 

 

 

 

 

 

EXERCISES, PAGE 639–641 


1. x 2 = 225 

x = ± √ 225 

x = ±15 


2. x 2 = 49 

x = ± √ 49 

x = ±7 


3. x 2 = -100 

x ≠ ± √ 100 




4. x 2 = 400 

11. 0 = 4 x 2 - 16 

____ 16 x 2 

16 = 121 

____ 170 

16 

x 2 = 121 

3 = ____ 3 w 2 

3 

____ 170 

16 

x = ± √ 3 = w 2 

16 

± √ 3 

x = ± 

___ 

11 

4 

The solutions are 11 

4 and - ___ 11 

4 . 

x = ± √ 400 

____ +16 ________ + 16 

x = ±20 

___ 16 


4 = ___ 4 x 2 

4 

5. -25 = x 2 

4 = x 2 

± √ 4 = x 

± √ 25 ≠ x 

±2 = x 



6. 36 = x 2 

12. 100 x 2 + 26 = 10 

± √ 36 = x 

_________ - 26 ____ -26 

±6 = x 

_____ 100 x 2 


100 = -16 

100 

7. 3 x 2 - 75 = 0 

x 2 = -____ 

16 

_______ + 75 +75 ____ 

100 

___ 3 x 2 

3 = 75 

x ≠ ± √ 100 

3 

x 2 There are no real solutions. 

= 25 

x = ± √ 25 

13. 3 x 2 = 81 

x = ±5 

___ 3 x 2 


3 = 81 

3 

8. 0 = 81 x 2 x 2 = 27 

- 25 

x = ± √ 27 

____ +25 _________ + 25 

___ 25 

81 = ____ 81 x 2 

81 

___ 25 

81 = x 2 

14. 0 = x 2 - 60 

± √ ____ +60 _______ + 60 

81 = x 

60 = x 2 

± 

5__ 

9 = x 

± √ 60 = x 

The solutions are 5__ 5__ 

9 and - 9 . 

9. 49 x 2 15. 100 - 5 x 2 = 0 

+ 64 = 0 

_________ + 5 x 2 _____ +5 x 2 

________ - 64 ____ -64 

____ 49 x 2 

49 = ____ 

-64 

100 

5 = ___ 5 x 2 

5 

49 

x 2 64 

= -___ 

20 = x 2 

49 

x ≠ ± √ ± √ 20 = x 

64 

49 


10. 16 x 2 + 10 = 131 

________ - 10 ____ -10 

16. A = lw 

170 = (3w)w 

170 = 3 w 2 

x ≈ ±5.20 


x ≈ 7.75 


x ≈ ±4.47 


w ≈ ±7.5 


so w ≈ 7.5 is the only solution that makes sense. 

Therefore, the width of the rectangle is 7.5 m. 


17. x 2 = 169 

x = ± √ 169 

x = ±13 




18. x 2 = 25 

26. 9 x 2 + 9 = 25 

x = ± √ 25 

______ - 9 ___ -9 

x = ±5 

___ 9 x 2 


9 = ___ 16 

9 

19. x 2 = -36 

x 2 = ___ 16 

9 

x ≠ ± √ 36 


x = ± √ ___ 16 

9 

20. x 2 = 10,000 

x = ± 

4__ 

3 

x = ± √ 10,000 


x = ±100 

3 and - 4__ 

3 . 


27. 49 x 2 + 1 = 170 

21. -121 = x 2 

_______ - 1 ___ -1 

± √ 121 ≠ x 

____ 49 x 2 


49 = ____ 169 

49 

22. 625 = x 2 

x 2 = ____ 169 

49 

± √ 625 = x 

±25 = x 

x = ± 

√ ____ 169 

49 


x = ± 

___ 13 

23. 4 - 81 x 2 7 

= 0 

________ + 81 x 2 ______ +81 x 2 

The solutions are ___ 13 

7 and - ___ 13 

7 . 

___ 4 

81 = ____ 81 x 2 

28. 81 x 2 + 17 = 81 

81 

________ - 17 ____ -17 

___ 4 

81 = x 2 

____ 81 x 2 

± √ ___ 

81 = ___ 64 

81 

4 

81 = x 

x 2 = ___ 64 

81 

± 

2__ 

9 = x 

x = ± 

√ ___ 64 


81 

9 and - 2__ 

9 . 

x = ± 

8__ 

9 

24. -4 x 2 - 49 = 0 


9 and - 9 . 

_________ 

+4 x 2 _____ +4 x 2 

____ -49 

4 = ___ 4 x 2 

29. 4 x 2 = 88 

4 

___ 4 x 2 

49 

-___ 

4 = x 2 

4 = ___ 88 

4 

± √ ___ 

x 2 = 22 

49 

4 ≠ x 


25. 64 x 2 - 5 = 20 

30. x 2 - 29 = 0 

_______ + 5 ___ +5 

____ 64 x 2 

64 = ___ 

______ + 29 ____ +29 

25 

x 2 = 29 

64 

x 2 = ___ 25 

64 

x = ± √ ___ 25 

64 

31. x 2 + 40 = 144 

______ - 40 ____ -40 

8 . x 2 = 104 

x = ± 

5__ 

8 


8 and - 5__ 

x = ± √ 22 

x ≈ ±4.69 


x = ± √ 29 

x ≈ ±5.39 


x = ± √ 104 

x ≈ ±10.20 




32. 3 x 2 - 84 = 0 

_______ + 84 +84 ____ 

___ 3 x 2 

3 = ___ 84 

3 

x 2 = 28 

x = ± √ 28 

x ≈ ±5.29 


33. 50 - x 2 = 0 

_______ + x 2 + ____ x 2 

50 = x 2 

± √ 50 = x 

x ≈ ±7.07 


34. 2 x 2 - 10 = 64 

_______ + 10 +10 ____ 

___ 2 x 2 

2 = ___ 74 

2 

x 2 = 37 

x = ± √ 37 

x ≈ ±6.08 


35. h = -16 t 2 + 600 

0 = -16 t 2 + 600 

_____ +16 t 2 +16 _____ t 2 

____ 16 t 2 

16 = ____ 600 

16 

t 2 = ___ 75 

2 

t = ± √ ___ 75 

2 

t ≈ ±6.1 

Negative numbers are not reasonable for time, 

so t ≈ 6.1 is the only solution that makes sense. 

Therefore, it takes the sack 6.1 s to reach the 

ground. 

36. A = s 2 

196 = x 2 

± √ 196 = x 

±14 = x 


so x = 14 is the only solution that makes sense. 

Therefore, the dimensions of the square are 14 m 

by 14 m. 

37. 2ab = 36 

2(2b)b = 36 

4 b 2 = 36 

___ 4 b 2 

4 = ___ 36 

4 

b 2 = 9 

b = ± √ 9 

b = ±3 

Case 1 b = 3 

a = 2b 

a = 2(3) 

38. 

Case 2 b = -3 

a = 2b 

a = 2(-3) 

a = -6 

a = 6 

The solutions for a and b are 6 and 3, and 

-6 and -3. 

a__ 

x = x__ 

b 

2__ ___ 

x = x 

18 

36 = x 2 

√ 36 = x 

6 = x 

The geometric mean of 2 and 18 is 6. 

39. Locate 35 on the y-axis. Move right to the graph of 

the function. Then move down to the x-axis to find 

the corresponding value of x. 

x ≈ 4.2 

The other side length is 2x, or 8.4. 

The dimensions of the rectangle are about 4.2 ft by 

8.4 ft. 

40. L = 9.78 t 2 

60 = 9.78 t 2 

____ 60 

9.78 = _____ 9.78 t 2 

9.78 

_____ 1000 

163 = t 2 

± 

√ _____ 1000 

163 = t 

t ≈ ±2.5 



Therefore, the period of the pendulum is about 

2.5 s. 

41. A; 100 was added to both sides of the equation 

instead of subtracted. 




44. h = -16 t 2 + 60 

0 = -16 t 2 + 60 

_____ +16 t 2 __________ 

+16 t 2 

____ 16 t 2 

16 = ___ 60 

16 

t 2 = ___ 15 

4 

t = ± √ ___ 15 

4 

t ≈ ±1.9 



Therefore, the soccer ball returns to the ground after 

3.75 s. 

45a. a must be greater than 0. 

b. a must be equal to 0. c. a must be less than 0. 

46a. d = 16 t 2 

b. 

4 = 16 t 2 

___ 4 

16 = ____ 16 t 2 

16 

1__ 

4 = t 2 

± √ 1__ 

4 = t 

± 

1__ 

2 = t 


so t ≈ 1__ is the only solution that makes sense. 

2 

Therefore, the golf ball will take 1__ s to drop 4 ft. 

2 

d t 2 t = 16 

0 0 

 

1 16 

2 64 

 

3 144 

 

4 256 

 

c. From the table, the golf ball will drop 16 ft in 1 s. 

d. From the table, it will take the golf ball 2 s to drop 

64 ft. 

47. x 2 + a = 0 

x 2 - 1__ 

2 = 0 

+ 1__ + 

1__ 

48. x 2 + a = 0 

x 2 + 1__ 

2 = 0 

- 1__ - 

1__ 

______ 2 ___ 2 

1__ 

2 

x = ± √ 1__ 

2 

√ 2 

x = ± 

___ 

2 

___ 2 

2 is irrational. 

______ 2 ___ 2 

x 2 = 

No; √ 

 

 

x 2 = - 

1__ 

2 

x ≠ ± √ 1__ 

2 

No; there are no real 

solutions. 

49. x 2 + a = 0 

x 2 - 1__ 

50. x 2 + a = 0 

4 = 0 

x 2 + 1__ 

+ 1__ 

4 = 0 

+ 

1__ 

- 1__ - 

1__ 

______ 4 ___ 4 

______ 4 ___ 4 

x 2 = 

1__ 

x 2 = - 

1__ 

4 

x = ± √ 1__ 

4 

x = ± √ 1__ 

4 

4 

x = ± 

1__ 

No; there are no real 

2 

solutions. 

Yes; ± 

1__ 

2 is rational. 

51. x 2 + 4 = 0 is equivalent to x 2 = -4, which has no 

real solutions. x 2 - 4 = 0 is equivalent to x 2 = 4, 

which has two solutions (±2). 


52. D 

V = 3.14 r 2 h 

1256 = 3.14 r 2 (100) 

1256 = 314 r 2 

_____ 

_____ 

1256 

314 = 314 r 2 

314 

4 = r 2 

± √ 4 = r 

±2 = r 

Negative numbers are not reasonable for length, so 

r = 2 is the only solution that makes sense. 

53. H; since 1__ 

2 x 2 = 20 → x 2 = 40, 6 2 = 36, 7 2 = 49, 

and 36 < 40 < 49, the value of x is between 6 and 

7. 

____ 

___ 

54. A; since 81 x 2 - 169 = 0 → x 2 = 169 

81 , or x = ± 13 

9 , 

there are two rational solutions. 


55. 288 x 2 - 19 = -1 

_________ + 19 ____ +19 

_____ 288 x 2 

288 = ____ 18 

288 

x 2 = ___ 1 

16 

x = ± √ ___ 1 

16 

x = ± 

1__ 

4 


4 and - 1__ 

4 . 

56. -75 x 2 = -48 

______ -75 x 2 

= ____ -48 

-75 -75 

x 2 = ___ 16 

25 

x = ± √ ___ 16 

25 

x = ± 

4__ 

5 


5 and - 4__ 

5 . 



57. x 2 = ____ 128 

242 

x 2 = ____ 64 

121 

x = ± √ ____ 64 

121 

8 

x = ± 

___ 

11 

The solutions are ___ 8 

11 and - ___ 8 

11 . 

58a. a 2 + b 2 = c 2 

9 2 + 12 2 = c 2 

81 + 144 = c 2 

225 = c 2 

± √ 225 = c 

±15 = c 


so c = 15 is the only solution that makes sense. 

Therefore, the length of the hypotenuse is 15 cm. 

b. Note that for an isosceles triangle, a = b. 

a 2 + b 2 = c 2 

a 2 + a 2 = 10 2 

2 a 2 = 100 

___ 2 a 2 

2 = ____ 100 

2 

a 2 = 50 

a = ± √ 50 

a ≈ ±7.1 


so a ≈ 7.1 is the only solution that makes sense. 

Therefore, the length of each leg of the isosceles 

triangle is approximately 7.1 cm. 


59. P of = P of △ 

2l + 2w = a + b + c 

2(x + 2) + 2(x - 7) = (x + 1) + (x - 1) + (x + 3) 

2x + 4 + 2x - 14 = 3x + 3 

4x - 10 = 3x + 3 

________ 

-3x _______ -3x 

x - 10 = 3 

______ + 10 ____ +10 

x = 13 

60. 2x - y = 8 

_______ -2x -2x ____ 

-y = -2x + 8 

-1(-y) = -1(-2x + 8) 

y = 2x - 8 

6x - 2y = 10 

y + 4 = 2(3 - x) 

________ 

-6x ____ -6x 

y + 4 = 6 - 2x 

-2y = -6x + 10 

____ -2y -6x + 10 

= ________ 

y + 4 = -2x + 6 

____ - 4 _______ - 4 

-2 -2 y = -2x + 2 

y = 3x - 5 

The lines described by y = -2x + 3 and y + 4 = 

2(3 - x) represent parallel lines. They each have 

slope -2. 

61. x 2 - 6x + 8 = 0 

(x - 2)(x - 4) = 0 

x - 2 = 0 or x - 4 = 0 

x = 2 or x = 4 


62. x 2 + 5x - 6 = 0 

(x + 6)(x - 1) = 0 

x + 6 = 0 or x - 1 = 0 

x = -6 or x = 1 


63. x 2 - 5x = 14 

_______ - 14 ____ -14 

x 2 - 5x - 14 = 0 

(x + 2)(x - 7) = 0 

x + 2 = 0 or x - 7 = 0 

x = -2 or x = 7 


CONNECTING ALGEBRA TO GEOMETRY: 

THE DISTANCE FORMULA, 

PAGES 642–643 

TRY THIS, PAGES 642–643 

1. ⎪ x 2 

- x 1 ⎥ 

= ⎪4 - (-1)⎥ 

= ⎪5⎥ = 5 

3. ⎪ y 2 

- y 1 ⎥ 

= ⎪5 - (-1)⎥ 

= ⎪6⎥ = 6 

5. ⎪ y 2 

- y 1 ⎥ 

= ⎪-4.3 - 0.5⎥ 

= ⎪-4.8⎥ = 4.8 

7. ⎪ x 2 

- x 1 ⎥ 

= ⎪3 - (-3)⎥ 

= ⎪6⎥ = 6 

9. ⎪ x 2 

- x 1 ⎥ 

= ⎪3 - (-2)⎥ 

= ⎪5⎥ = 5 

2. ⎪ y 2 

- y 1 ⎥ 

= ⎪-8 - (-2)⎥ 

= ⎪-6⎥ = 6 

4. ⎪ x 2 

- x 1 ⎥ 

= ⎪25 - 14⎥ 

= ⎪11⎥ = 11 

6. ⎪ x 2 

- x 1 ⎥ 

= ⎪3.8 - 1.4⎥ 

= ⎪2.4⎥ = 2.4 

8. ⎪ y 2 

- y 1 ⎥ 

= ⎪4 - (-3)⎥ 

= ⎪7⎥ = 7 

10. Let (-8, 3) be ( x 1 

, y 1 

) and (-11, -4) be ( x 2 

, y 2 

). 

D = √ 

( x 2 

- x 1 

) 2 + ( y 2 

- y 1 

) 2 

= √ 

(-11 - (-8)) 2 + (-4 - 3) 2 

= √ 

(-3) 2 + (-7) 2 

= √ 9 + 49 

= √ 58 ≈ 7.62 

11. Let (30, -15) be ( x 1 

, y 1 

) and (-55, 40) be ( x 2 

, y 2 

). 

D = √ 

( x 2 

- x 1 

) 2 + ( y 2 

- y 1 

) 2 

= √ 

(-55 - 30) 2 + (40 - (-15)) 2 

= √ 

(-85) 2 + (55) 2 

= √ 

7225 + 3025 

= √ 10,250 ≈ 101.24 



12. Let (1.2, 0.5) be ( x 1 

, y 1 

) and (2.5, 0.6) be ( x 2 

, y 2 

). 

D = √ 

( x 2 

- x 1 

) 2 + ( y 2 

- y 1 

) 2 

= √ 

(2.5 - 1.2) 2 + (0.6 - 0.5) 2 

= √ 

(1.3) 2 + (0.1) 2 

= √ 

1.69 + 0.01 

= √ 1.7 ≈ 1.30 

13. Let (1, 3) be ( x 1 

, y 1 

) and (3, -2) be ( x 2 

, y 2 

). 

D = √ 

( x 2 

- x 1 

) 2 + ( y 2 

- y 1 

) 2 

= √ 

(3 - 1) 2 + (-2 - 3) 2 

= √ 

(2) 2 + (-5) 2 

= √ 4 + 25 

= √ 29 ≈ 5.39 

14. Let (3, 2) be ( x 1 

, y 1 

) and (-3, -1) be ( x 2 

, y 2 

). 

D = √ 

( x 2 

- x 1 

) 2 + ( y 2 

- y 1 

) 2 

= √ 

(-3 - 3) 2 + (-1 - 2) 2 

= √ 

(-6) 2 + (-3) 2 

= √ 36 + 9 

= √ 45 ≈ 6.71 

15. Let (1, -3) be ( x 1 

, y 1 

) and (-1, 1) be ( x 2 

, y 2 

). 

D = √ 

( x 2 

- x 1 

) 2 + ( y 2 

- y 1 

) 2 

= √ 

(-1 - 1) 2 + (1 - (-3)) 2 

= √ 

(-2) 2 + (4) 2 

= √ 4 + 16 

= √ 20 ≈ 4.47 

ALGEBRA LAB: MODEL COMPLETING 

THE SQUARE, PAGE 644 


1. x 2 + 4x + 4 = (x + 2) 2 2. x 2 + 2x + 1 = (x + 1) 2 

3. x 2 + 10x + 25 = (x + 5) 2 

4. x 2 + 8x + 16 = (x + 4) 2 5. 36 

9-8 COMPLETING THE SQUARE, 

PAGES 645–651 


1a. b = 12 

( ___ 12 

2 ) 2 = 6 2 = 36 

x 2 + 12x + 36 

c. b = 8 

( 8__ 

2) 2 = 4 2 = 16 

8x + x 2 + 16 

b. b = -5 

( ___ -5 

2 ) 2 = ___ 25 

4 

x 2 - 5x + ___ 25 

4 

2a. b = 10 

( ___ 10 

2 ) 2 = 5 2 = 25 

x 2 + 10x = -9 

x 2 + 10x + 25 = -9 + 25 

(x + 5) 2 = 16 

x + 5 = ±4 

x + 5 = 4 or x + 5 = -4 

x = -1 or x = -9 


b. b = -8 

( ___ -8 

2 ) 2 = (-4) 2 = 16 

t 2 - 8t - 5 = 0 

t 2 - 8t = 5 

t 2 - 8t + 16 = 5 + 16 

(t - 4) 2 = 21 

t - 4 = ± √ 21 

t - 4 = √ 21 or t - 4 = - 

√ 

21 

t = 4 + √ 21 or t = 4 - √ 21 

The solutions are 4 + √ 21 and 4 - √ 21 . 

3a. 3 x 2 - 5x - 2 = 0 

___ 3 x 2 

3 - ___ 5x 

3 - 2__ 

3 = 0 

x 2 - 5__ 

3 x - 2__ 

3 = 0 

x 2 - 5__ 

b = - 

5__ 

3 

( 

5__ - 2 

___ 3 

) 

2 

3 x = 2__ 

3 

2 

= ( - 

5__ 

6) 

x 2 - 5__ 

x 2 - 5__ ___ 

= ___ 25 

36 

3 x = 2__ 

3 

3 x + 25 

36 = 2__ 

3 + ___ 25 

36 

( x - 5__ 

6) 2 = ___ 49 

36 

x - 5__ 

6 = 7__ 

6 

x - 5__ 

6 = ± 7__ 

6 

or x - 

5__ 

6 = - 7__ 

6 

x = 2 or x = - 

1__ 

3 

The solutions are 2 and - 

1__ 

3 . 



. 4 t 2 - 4t + 9 = 0 

___ 4 t 2 

4 - __ 4t 

4 + 9__ 

4 = 0 

t 2 - t + 9__ 

4 = 0 

t 2 - t = - 

9__ 

4 

b = -1 

( ___ -1 

2 ) 2 = 1__ 

4 

t 2 - t = - 

9__ 

4 

t 2 - t + 1__ 9__ 

4 = - 4 + 1__ 

4 

( t - 1__ 

2) 2 = -2 

There is no real number whose square is negative, 

so there are no real solutions. 

4. Let x represent the width of the room. 

lw = A 

(x + 8)(x) = 400 

x 2 + 8x = 400 

b = 8 

( 8__ 

2) 2 = 4 2 = 16 

x 2 + 8x = 400 

x 2 + 8x + 16 = 400 + 16 

(x + 4) 2 = 416 

x + 4 ≈ ±20.4 

x + 4 ≈ 20.4 or x + 4 ≈ -20.4 

x ≈ 16.4 or x ≈ -24.4 


x ≈ 16.4 is the only solution that makes sense. 

The width is 16.4 ft, and the length is 

16.4 + 8 ≈ 24.4 ft. The dimensions of the room are 

about 24.4 ft by 16.4 ft. 


1. Possible answer: Subtract c from both sides. Then 

add ( b__ 

2) 2 to both sides. 

2. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



1. Divide 4 by 2 and square the result: ( 4__ 2) 2 = 4. Add 4 

to both sides of the equation: 5 = x 2 + 4x + 4. Write 

the equation as 5 = (x + 2) 2 . Take the square root 

of both sides: ± √ 5 = x + 2. Solve for x: 

x = -2 ± √ 5 . 

2. b = 14 

( ___ 14 

2 ) 2 = 7 2 = 49 

x 2 + 14x + 49 

4. b = -3 

( ___ -3 

2 ) 2 = 9__ 

4 

x 2 - 3x + 9__ 

4 

5. b = 6 

( 6__ 

2) 2 = 3 2 = 9 

x 2 + 6x = -5 

x 2 + 6x + 9 = -5 + 9 

(x + 3) 2 = 4 

x + 3 = ±2 

x + 3 = 2 or x + 3 = -2 

x = -1 or x = -5 


6. b = -8 

( ___ -8 

2 ) 2 = (-4) 2 = 16 

x 2 - 8x = 9 

x 2 - 8x + 16 = 9 + 16 

(x - 4) 2 = 25 

x - 4 = ±5 

x - 4 = 5 or x - 4 = -5 

x = 9 or x = -1 


7. b = 1 

( 2) 1__ 2 = 1__ 

4 

x 2 + x = 30 

x 2 + x + 1__ = 30 + 

1__ 

4 4 

( x + 2) 1__ 2 = ____ 121 

4 

x + 1__ 

2 = ± ___ 11 

2 

x + 1__ 

2 = ___ 11 or x + 

1__ 

2 2 = - ___ 11 

2 

x = 5 or x = -6 


3. b = -4 

( ___ -4 

2 ) 2 = (-2) 2 = 4 

x 2 - 4x + 4 



8. b = 2 

( 2__ 2) 2 = 1 2 = 1 

x 2 + 2x = 21 

x 2 + 2x + 1 = 21 + 1 

(x + 1) 2 = 22 

x + 1 = ± √ 22 

x + 1 = √ 22 or x + 1 = - √ 22 

x = -1 + √ 22 or x = -1 - √ 22 

The solutions are -1 + √ 22 and -1 - √ 22 . 

9. b = -10 

( ____ -10 

2 ) 2 = (-5) 2 = 25 

x 2 - 10x = -9 

x 2 - 10x + 25 = -9 + 25 

(x - 5) 2 = 16 

x - 5 = ±4 

x - 5 = 4 or x - 5 = -4 

x = 9 or x = 1 


10. b = 16 

( ___ 16 

2 ) 2 = 8 2 = 64 

x 2 + 16x = 92 

x 2 + 16x + 64 = 92 + 64 

(x + 8) 2 = 156 

x + 8 = ±2 √ 39 

x + 8 = 2 √ 39 or x + 8 = -2 √ 39 

x = -8 + 2 √ 39 or x = -8 - 2 √ 39 

The solutions is -8 ± √ 156 . 

11. - x 2 - 5x = -5 

____ - x 2 

-1 - ___ 5x 

-1 = ___ -5 

-1 

x 2 + 5x = 5 

b = 5 

( 5__ 

___ 

2) 2 = 25 

4 

x 2 + 5x = 5 

x 2 + 5x + ___ 25 

4 = 5 + ___ 25 

4 

( x + 5__ 

2) 2 = ___ 45 

4 

x + 5__ 

2 = ± ____ 3 √ 5 

2 

x + 5__ 

2 = ____ 3 √ 5 or x + 5__ 

2 

2 = - ____ 3 √ 5 

2 

x = 

_________ 

-5 + 3 √ 5 or x = 

_________ 

-5 - 3 √ 5 

2 

2 

The solutions are 

_________ 

-5 + 3 √ 5 and 

_________ 

-5 - 3 √ 5 . 

2 

2 

12. - x 2 - 3x + 2 = 0 

____ - x 2 

-1 - ___ 3x 

-1 + ___ 2 

-1 = 0 

x 2 + 3x - 2 = 0 

x 2 + 3x = 2 

b = 3 

( 2) 3__ 2 = 9__ 

4 

x 2 + 3x = 2 

x 2 + 3x + 9__ 

4 = 2 + 9__ 

4 

( x + 3__ ___ 

2) 2 = 17 

4 

x + 3__ 

2 = ± ____ √ 17 

2 

x + 3__ 

2 = ____ √ 17 or x + 3__ 

2 

2 = - ____ √ 17 

2 

-3 + √ 

x = 

_________ 17 or x = 

_________ 

-3 - √ 17 

2 

2 

-3 + √ 


_________ 17 and 

_________ 

-3 - √ 17 . 

2 

2 

13. -6x = 3 x 2 + 9 

-3 x 2 - 6x = 9 

_____ -3 x 2 

-3 - ___ 6x 

-3 = ___ 9 

-3 

x 2 + 2x = -3 

b = 2 

( 2__ 2) 2 = 1 2 = 1 

x 2 + 2x = -3 

x 2 + 2x + 1 = -3 + 1 

(x + 1) 2 = -2 



14. 2 x 2 - 6x = -10 

___ 2 x 2 

2 - ___ 6x 

2 = - ___ 10 

2 

x 2 - 3x = -5 

b = -3 

( ___ -3 

4 

2 ) 2 = 9__ 

x 2 - 3x = -5 

x 2 - 3x + 9__ = -5 + 

9__ 

4 4 

( x - 3__ 

2) 2 = -___ 

11 

4 





15. - x 2 + 8x - 6 = 0 

____ - x 2 

-1 + ___ 8x 

-1 - ___ 6 

-1 = 0 

x 2 - 8x + 6 = 0 

x 2 - 8x = -6 

b = -8 

( ___ -8 

2 ) 2 = (-4) 2 = 16 

x 2 - 8x = -6 

x 2 - 8x + 16 = -6 + 16 

(x - 4) 2 = 10 

x - 4 = ± √ 10 

x - 4 = √ 10 or x - 4 = - √ 10 

x = 4 + √ 10 or x = 4 - √ 10 


16. 4 x 2 + 16 = -24x 

4 x 2 + 24x + 16 = 0 

4 x 2 + 24x = -16 

___ 4 x 2 

4 + ____ 24x 

4 = ____ -16 

4 

x 2 + 6x = -4 

b = 6 

( 6__ 

2) 2 = (3) 2 = 9 

x 2 + 6x = -4 

x 2 + 6x + 9 = -4 + 9 

(x + 3) 2 = 5 

x + 3 = ± √ 5 

x + 3 = √ 5 or x + 3 = - √ 5 

x = -3 + √ 5 or x = -3 - √ 5 


17. lw = A 

(x + 4)(x) = 80 

x 2 + 4x = 80 

b = 4 

( 4__ 2) 2 = 2 2 = 4 

x 2 + 4x = 80 

x 2 + 4x + 4 = 80 + 4 

(x + 2) 2 = 84 

x + 2 ≈ ±9.2 

x + 2 ≈ 9.2 or x + 2 ≈ -9.2 

x ≈ 7.2 or x ≈ -11.2 



The width is 7.2 m, and the length is 

7.2 + 4 ≈ 11.2 m. 

20. b = 11 

( ___ 11 

2 ) 2 = ____ 121 

4 

x 2 + 11x + ____ 121 

4 

21. b = -10 

( ____ -10 

2 ) 2 = (-5) 2 = 25 

x 2 - 10x = 24 

x 2 - 10x + 25 = 24 + 25 

(x - 5) 2 = 49 

x - 5 = ±7 

x - 5 = 7 or x - 5 = -7 

x = 12 or x = -2 


22. b = -6 

( ___ -6 

2 ) 2 = (-3) 2 = 9 

x 2 - 6x = -9 

x 2 - 6x + 9 = -9 + 9 

(x - 3) 2 = 0 

x - 3 = 0 

x = 3 


23. b = 15 

( ___ 15 

2 ) 2 = ____ 225 

4 

x 2 + 15x = -26 

x 2 + 15x + ____ 225 = -26 + ____ 225 

4 4 

( x + ___ 15 

2 ) 2 = ____ 121 

4 

x + ___ 15 

2 = ± ___ 11 

2 

x + ___ 15 

2 = ___ 11 or x + ___ 15 

2 2 = - ___ 11 

2 

x = -2 or x = -13 


24. b = 6 

( 6__ 

2) 2 = 3 2 = 9 

x 2 + 6x = 16 

x 2 + 6x + 9 = 16 + 9 

(x + 3) 2 = 25 

x + 3 = ±5 

x + 3 = 5 or x + 3 = -5 

x = 2 or x = -8 



18. b = -16 

( ____ -16 

2 ) 2 = (-8) 2 = 64 

x 2 - 16x + 64 

19. b = -2 

( ___ -2 

2 ) 2 = (-1) 2 = 1 

x 2 - 2x + 1 



25. b = -2 

( ___ -2 

2 ) 2 = (-1) 2 = 1 

x 2 - 2x = 48 

x 2 - 2x + 1 = 48 + 1 

(x - 1) 2 = 49 

x - 1 = ±7 

x - 1 = 7 or x - 1 = -7 

x = 8 or x = -6 


26. b = 12 

( ___ 12 

2 ) 2 = 6 2 = 36 

x 2 + 12x = -36 

x 2 + 12x + 36 = -36 + 36 

(x + 6) 2 = 0 

x + 6 = 0 

x = -6 


27. - x 2 + x + 6 = 0 

____ - x 2 

-1 + ___ x 

-1 + ___ 6 

-1 = 0 

x 2 - x - 6 = 0 

x 2 - x = 6 

b = -1 

( ___ -1 

4 

2 ) 2 = 1__ 

x 2 - x = 6 

x 2 - x + 1__ 

( x - 1__ ___ 

4 = 6 + 1__ 

4 

2) 2 = 25 

4 

x - 1__ 5__ 

2 = ± 2 

x - 1__ 

2 = 5__ or x - 

1__ 5__ 

2 2 = - 2 

x = 3 or x = -2 


29. - x 2 - x + 1 = 0 

____ - x 2 

-1 - ___ x 

-1 + ___ 1 

-1 = 0 

b = 1 

( 2) 1__ 2 = 1__ 

4 

x 2 + x - 1 = 0 

x 2 + x = 1 

x 2 + x = 1 

x 2 + x + 1__ 

4 = 1 + 1__ 

4 

( x + 2) 1__ 2 = 5__ 

4 

x + 1__ ___ 5 

2 = ± √ 

2 

x + 1__ 

2 = ___ √ 5 or x + 1__ 

2 

2 = - ___ √ 5 

2 

x = 

________ -1 + √ 5 or x = 

________ -1 - √ 5 

2 

2 


________ -1 + √ 5 and 

________ -1 - √ 5 . 

2 

2 

30. 3 x 2 - 6x - 9 = 0 

___ 3 x 2 

3 - ___ 6x 

3 - 9__ 

3 = 0 

x 2 - 2x - 3 = 0 

x 2 - 2x = 3 

b = -2 

( ___ -2 

2 ) 2 = (-1) 2 = 1 

x 2 - 2x = 3 

x 2 - 2x + 1 = 3 + 1 

(x - 1) 2 = 4 

x - 1 = ±2 

x - 1 = 2 or x - 1 = -2 

x = 3 or x = -1 


28. 2 x 2 = -7x - 29 

2 x 2 + 7x = -29 

___ 2 x 2 

2 + ___ 7x 

2 = ____ -29 

2 

x 2 + 7__ 

2 x = - ___ 29 

2 

b = 7__ 

2 

7_ 2 

2__ 

= ( 7__ 

( 2) 

___ 

4) 2 = 49 

16 

x 2 + 7__ 

x 2 + 7__ 

2 x + ___ 49 

( x + 7__ 

___ 

2 x = - 29 

2 

16 = - ___ 29 

2 + ___ 49 

16 

4) 2 183 

= -____ 

16 





31. - x 2 = 15x + 30 

- x 2 - 15x = 30 

____ - x 2 

-1 - ____ 15x 

-1 = ___ 30 

-1 

x 2 + 15x = -30 

b = 15 

( ___ 15 

2 ) 2 = ____ 225 

4 

x 2 + 15x = -30 

x 2 + 15x + ____ 225 = -30 + ____ 225 

4 4 

( x + ___ 15 

2 ) 2 = ____ 105 

4 

x + ___ 15 

2 = ± _____ √ 105 

2 

x + ___ 15 

2 = _____ √ 105 or x + ___ 15 

2 

2 = - _____ √ 105 

2 

-15 + √ 

x = 

___________ 105 or x = 

___________ 

-15 - √ 105 

2 

2 

-15 + √ 


___________ 105 and 

___________ 

-15 - √ 105 . 

2 

2 

32. 2 x 2 + 20x - 10 = 0 

___ 2 x 2 

2 + ____ 20x 

2 - ___ 10 

2 = 0 

x 2 + 10x - 5 = 0 

x 2 + 10x = 5 

b = 10 

( ___ 10 

2 ) 2 = 5 2 = 25 

x 2 + 10x = 5 

x 2 + 10x + 25 = 5 + 25 

(x + 5) 2 = 30 

x + 5 = ± √ 30 

x + 5 = √ 30 or x + 5 = - √ 30 

x = -5 + √ 30 or x = -5 - √ 30 


33. bh = A 

(2x + 8)(x) = 64 

2 x 2 + 8x = 64 

___ 2 x 2 

2 + ___ 8x 

2 = ___ 64 

2 

x 2 + 4x = 32 

b = 4 

( 4__ 2) 2 = 2 2 = 4 

x 2 + 4x = 32 

x 2 + 4x + 4 = 32 + 4 

(x + 2) 2 = 36 

x + 2 = ±6 

x + 2 = 6 or x + 2 = -6 

x = 4 or x = -8 


x = 4 is the only solution that makes sense. 

The height of the parallelogram is 4 in. 

34. 3 x 2 + x = 10 

___ 3 x 2 

3 + x__ 

3 = ___ 10 

3 

x 2 + 1__ 

3 x = ___ 10 

3 

b = 1__ 

3 

1_ 2 

3__ 

= ( 1__ 

( 2) 

___ 

6) 2 = 1 

x 2 + 1__ 

x 2 + 1__ 

3 x + ___ 1 

( x + 1__ 

x + 1__ 

36 

___ 

3 x = 10 

3 

36 = ___ 10 

3 + ___ 1 

36 

6) 2 = ____ 121 

36 

x + 1__ 

6 = ± ___ 11 

6 

6 = ___ 11 or x + 

1__ 

6 6 = - ___ 11 

6 

x = 5__ or x = -2 

3 

The solutions are 5__ and -2. 

3 

35. x 2 = 2x + 6 

x 2 - 2x = 6 

b = -2 

( ___ -2 

2 ) 2 = (-1) 2 = 1 

x 2 - 2x = 6 

x 2 - 2x + 1 = 6 + 1 

(x - 1) 2 = 7 

x - 1 = ± √ 7 

x - 1 = √ 7 or x - 1 = - √ 7 

x = 1 + √ 7 or x = 1 - √ 7 




36. 2 a 2 = 5a + 12 

2 a 2 - 5a = 12 

___ 2 a 2 

2 - ___ 5a 

2 = ___ 12 

2 

( x + 4) 5__ = 49 

16 

x + 5__ 

4 = ± 7__ 

4 

x + 5__ 

4 = 7__ or x + 

5__ 

4 4 = - 7__ 

4 

x = 1__ or x = -3 

2 


2 and -3. 38. 4x = 7 - x 2 

x 2 + 4x = 7 

b = 4 

a 2 - 5__ 

( 2) 4__ 2 a = 6 

= 2 2 = 4 

b = - 

5__ 

2 

(___ 

2 

2 

) = 

2 ( - 

5__ 

4) 2 = 25 

16 

a 2 - 5__ 

2 a = 6 

a 2 - 5__ 

2 a + 25 

16 = 6 + 25 

16 

( a - 4) 5__ = 121 

16 

a - 5__ 

b = 8 

4 = ± ___ 11 

4 

( 

a - 5__ 

4 = 11 or a - 

5__ 

4 4 = - ___ 

2) 8__ = 4 2 = 16 

11 

4 

a = 4 or a = - 

3__ 

2 


3__ 

2 . 

___ 2 x 2 

2 + 5x 

2 = 3__ 

2 

x 2 + 5__ 

2 x = 3__ 

2 

b = 5__ 

2 

5_ 2 

( 2__ 

= 

2) ( 4) 5__ = 25 

____ 16 t 2 

16 

16 - 320t 

16 = 32 

16 

x 2 + 5__ 

2 x = 3__ 

t 2 - 20t = 2 

2 

b = 20 

x 2 + 5__ 

2 x + 25 

16 = 3__ 

2 + 25 

( 20 

16 

37. 2 x 2 + 5x = 3 

x 2 + 4x = 7 

x 2 + 4x + 4 = 7 + 4 

(x + 2) 2 = 11 

x + 2 = ± √ 11 

x + 2 = √ 11 or x + 2 = - √ 11 

x = -2 + √ 11 or x = -2 - √ 11 


39. 8x = - x 2 + 20 

x 2 + 8x = 20 

x 2 + 8x = 20 

x 2 + 8x + 16 = 20 + 16 

(x + 4) 2 = 36 

x + 4 = ±6 

x + 4 = 6 or x + 4 = -6 

x = 2 or x = -10 


40. h = -16 t 2 + 320t + 32 

0 = -16 t 2 + 320t + 32 

16 t 2 = 320t + 32 

16 t 2 - 320t = 32 

2 ) 2 = 10 2 = 100 

t 2 - 20t = 2 

t 2 - 20t + 100 = 2 + 100 

(t - 10) 2 = 102 

t - 10 ≈ ±10.1 

t - 10 ≈ 10.1 or t - 10 ≈ -10.1 

t ≈ 20.1 or t ≈ -0.1 

Negative numbers are not reasonable for time, so 

t ≈ 20.1 is the only solution that makes sense. 

Therefore, it takes the rocket approximately 20.1 s 

to return to the ground. 

41. b = 18 

( ___ 18 

2 ) 2 = 9 2 = 81 

x 2 + 18x + 81 

42. b = -100 

( _____ -100 

2 ) 2 = (-50) 2 = 2500 

x 2 - 100x + 2500 



43. b = -7 

( ___ -7 

2 ) 2 = ___ 49 

4 

x 2 - 7x + ___ 49 

4 

45. ___ 81 

4 = ( 2) 9__ 2 

b = 9 

x 2 - 9x + ___ 81 

4 

44. 4 = 2 2 = ( 4__ 2) 2 

b = 4 

x 2 + 4x + 4 

46. ___ 1 

36 = ( 1__ 

b = 1__ 

3 

x 2 + 1__ 

1_ 

6) 2 = ( 3__ 

___ 

3 x + 1 

36 

47a. lw = A 

(10 + 2x)(34 + 2x) = 640 

b. (10 + 2x)(34 + 2x) = 640 

10(34) + 10(2x) + 2x(34) + 2x(2x) = 640 

340 + 20x + 68x + 4 x 2 = 640 

4 x 2 + 88x + 340 = 640 

4 x 2 + 88x = 300 

___ 4 x 2 

4 + ____ 88x 

4 = ____ 300 

4 

x 2 + 22x = 75 

b = 22 

( ___ 22 

2 ) 2 = 11 2 = 121 

x 2 + 22x = 75 

x 2 + 22x + 121 = 75 + 121 

(x + 11) 2 = 196 

x + 11 = ±14 

x + 11 = 14 or x + 11 = -14 

x = 3 or x = -25 


so x = 3 is the only solution that makes sense. 

Therefore, the width of the roped-off area is 3 ft. 

2) 

2 

48a. 2 x 2 - 3x - 2 = 0 

___ 2 x 2 

2 - ___ 3x 

2 - 2__ 

2 = 0 

b. 

x 2 - 3__ 

2 x - 1 = 0 

x 2 - 3__ 

2 x = 1 

b = - 

3__ 

( 

2 

2 

2 

) 

2 

___ 

3__ - 

= ( - 

3__ 

x 2 - 3__ 

x 2 - 3__ ___ 

___ 

4) 2 = 9 

16 

2 x = 1 

2 x + 9 

16 = 1 + ___ 9 

16 

( x - 3__ 

4) 2 = ___ 25 

16 

x - 3__ 

4 = 5__ 

4 

x - 3__ 5__ 

4 = ± 4 

or x - 

3__ 5__ 

4 = - 4 

x = 2 or x = - 

1__ 

2 


1__ 

2 . 

c. Use the Trace and Zoom functions to find where 

the graph crosses the x-axis, or use the Intersect 

function to find where the graph intersects the line 

y = 0. 

d. Possible answer: Completing the square takes 

several steps, but it produces the exact solutions. 

Creating the graph with a graphing calculator is 

fast, but you may only get approximate solutions. 

49. If (x + 2) 2 = 81, then x + 2 = 9 or x + 2 = -9. The 

correct answer is x = 7 or x = -11. 

50. 5 x 2 - 50x = 55 

___ 5 x 2 

5 - ____ 50x 

5 = ___ 55 

5 

x 2 - 10x = 11 

b = -10 

( ____ -10 

2 ) 2 = (-5) 2 = 25 

x 2 - 10x = 11 

x 2 - 10x + 25 = 11 + 25 

(x - 5) 2 = 36 

x - 5 = ±6 

x - 5 = 6 or x - 5 = -6 

x = 11 or x = -1 




51. 3 x 2 + 36x = -27 

___ 3 x 2 

3 + ____ 36x 

3 = ____ -27 

3 

x 2 + 12x = -9 

b = 12 

( ___ 12 

2 ) 2 = 6 2 = 36 

x 2 + 12x = -9 

x 2 + 12x + 36 = -9 + 36 

(x + 6) 2 = 27 

x + 6 = ±3 √ 3 

x + 6 = 3 √ 3 or x + 6 = -3 √ 3 

x = -6 + 3 √ 3 or x = -6 - 3 √ 3 

The solutions are -6 + 3 √ 3 and -6 - 3 √ 3 . 

52. 28x - 2 x 2 = 26 

_____ -2 x 2 

-2 + ____ 28x 

-2 = ___ 26 

-2 

x 2 - 14x = -13 

b = -14 

( ____ -14 

2 ) 2 = (-7) 2 = 49 

x 2 - 14x = -13 

x 2 - 14x + 49 = -13 + 49 

(x - 7) 2 = 36 

x - 7 = ±6 

x - 7 = 6 or x - 7 = -6 

x = 13 or x = 1 


53. -36x = 3 x 2 + 108 

-3 x 2 - 36x = 108 

_____ -3 x 2 

-3 - ____ 36x 

-3 = ____ 108 

-3 

x 2 + 12x = -36 

b = 12 

( ___ 12 

2 ) 2 = 6 2 = -36 

x 2 + 12x = -36 

x 2 + 12x + 36 = -36 + 36 

(x + 6) 2 = 0 

x + 6 = 0 

x = -6 


54. 0 = 4 x 2 + 32x + 44 

-4 x 2 = 32x + 44 

-4 x 2 - 32x = 44 

_____ -4 x 2 

-4 - ____ 32x 

-4 = ___ 44 

-4 

x 2 + 8x = -11 

b = 8 

( 8__ 

2) 2 = 4 2 = 16 

x 2 + 8x = -11 

x 2 + 8x + 16 = -11 + 16 

(x + 4) 2 = 5 

x + 4 = ± √ 5 

x + 4 = √ 5 or x + 4 = - √ 5 

x = -4 + √ 5 or x = -4 - √ 5 


55. 16x + 40 = -2 x 2 

2 x 2 + 16x + 40 = 0 

2 x 2 + 16x = -40 

___ 2 x 2 

2 + ____ 16x 

2 = ____ -40 

2 

x 2 + 8x = -20 

b = 8 

( 8__ 

2) 

2 

= 4 2 = 16 

x 2 + 8x = -20 

x 2 + 8x + 16 = -20 + 16 

(x + 4) 2 = -4 



56. x 2 + 5x + 6 = 10x 

x 2 - 5x + 6 = 0 

x 2 - 5x = -6 

b = -5 

( ___ -5 

2 ) 2 = ___ 25 

4 

x 2 - 5x = -6 

x 2 - 5x + ___ 25 = -6 + ___ 25 

4 4 

( x - 5__ 

2) 2 = 1__ 

4 

x - 5__ 

x - 5__ 

2 = 1__ 

2 

2 = ± 1__ 

2 

or x - 

5__ 

2 = - 1__ 

2 

x = 3 or x = 2 




57. x 2 + 3x + 18 = -3x 

x 2 + 6x + 18 = 0 

x 2 + 6x = -18 

b = 6 

( 6__ 

2) 2 = 3 2 = 9 

x 2 + 6x = -18 

x 2 + 6x + 9 = -18 + 9 

(x + 3) 2 = -9 



58. 4 x 2 + x + 1 = 3 x 2 

x 2 + x + 1 = 0 

x 2 + x = -1 

b = 1 

( 1__ 2) 2 = 1__ 

4 

x 2 + x = -1 

= -1 + 

1__ 

4 

x 2 + x + 1__ 

4 

( x + 1__ 

2) 2 = - 

3__ 

4 



59. Possible answer: Completing the square to solve 

x 2 + 11x + 18 = 0 involves fractions, whereas 

completing the square to solve x 2 + 20x - 21 = 0 

does not. 

60. Add 2xy. The perfect-square trinomial of 

(x + y)(x + y) simplifies to x 2 + xy + xy + y 2 , or 

x 2 + 2xy + y 2 . This is the equivalent of 

x 2 + y 2 + 2xy. 

61a. h(t) = -16 t 2 + vt + c 

0 = -16 t 2 + 64t + 32 

b. -16 t 2 + 64t + 32 = 0 

_____ -16 t 2 

-16 + ____ 64t 

-16 + ____ 32 

-16 = 0 

t 2 - 4t - 2 = 0 

t 2 - 4t = 2 

b = -4 

( ___ -4 

2 ) 2 = (-2) 2 = 4 

4 should be added to both sides. 

62. When solving an equation of the form 

a x 2 + bx + c = 0, you must first divide the equation 

by the coefficient a. 


___ 

63. B; since ( 16 

2 ) 2 = 8 2 = 64, B is correct. 

64. J; since b must have the form ax, the answer is G or 

J; since ( ___ 10 

2 ) 2 = 5 2 = 25, J is correct. 

65. B 

3 x 2 + 2x - 4 = 0 

___ 3 x 2 

3 + ___ 2x 

3 - 4__ 

3 = 0 

x 2 + 2__ 

3 x - 4__ 

3 = 0 

b = 2__ 

3 

2_ 2 

3__ 

( 2) 

x 2 + 2__ 

3 x = 4__ 

3 

= ( 1__ 3) 2 = 1__ 

9 

x 2 + 2__ 

3 x = 4__ 

3 

x 2 + 2__ 

3 x + 1__ 

9 = 4__ 

( x + 1__ 

3 + 1__ 

___ 

9 

3) 2 = 13 

9 

x + 1__ 

3 ≈ ±1.20 

x + 1__ ≈ 1.20 or x + 

1__ 

3 3 ≈ -1.20 

x ≈ 0.86 or x ≈ -1.54 

Since 0.86 is closer to 1 than -1.54 is to 0, B is 

correct. 

66. x 2 - 8x - 20 = 0 Given 

x 2 - 8x = 20 

Add 20 to both sides. 

x 2 - 8x + 16 = 20 + 16 Complete the square. 

(x - 4) 2 Fa ctor the perfectsquare 

trinomial. 

= 36 

Ta ke the square root of 

x - 4 = 6 or x - 4 = -6 

both sides. 

x = 10 or x = -2 Solve for x. 

c. t 2 - 4t = 2 

t 2 - 4t + 4 = 2 + 4 

(t - 2) 2 = 6 

t - 2 ≈ ±2.4 

t - 2 ≈ 2.4 or t - 2 ≈ -2.4 

t ≈ 4.4 or t ≈ -0.4 



Therefore, it takes the ball approximately 4.4 s to 

reach the fairway. 




67. 6 x 2 + 5x = 6 

___ 6 x 2 

6 + 5x 

6 = 6__ 

6 

x 2 + 5__ 

6 x = 1 

b = 5__ 

6 

5_ 2 

( 6__ 

= 

2) ( 12) 5 = 25 

144 

x 2 + 5__ 

6 x = 1 

x 2 + 5__ 

6 x + 25 

144 = 1 + 25 

144 

( x + 12) 5 = 169 

144 

x + 5 

12 = ± ___ 13 

12 

x + 5 

12 = 13 

12 or x + 5 

12 = - ___ 13 

12 

x = 2__ 

3__ 

3 or x = - 2 


3 and - 2 . 

68. 7x + 3 = 6 x 2 

-6 x 2 + 7x + 3 = 0 

-6 x 2 + 7x = -3 

_____ -6 x 2 

-6 + 7x 

-6 = -3 

-6 

x 2 - 7__ 

6 x = 1__ 

2 

b = - 

7__ 

6 

( 

2 

___ 6 

) = 

2 ( -___ 

7 

12) 2 = 49 

144 

x 2 - 7__ 

6 x = 1__ 

2 

x 2 - 7__ 

6 x + 49 

144 = 1__ 

2 + 49 

144 

( x - 12) 7 = 121 

144 

x - 7 

12 = ± ___ 11 

12 

x - 7 

12 = 11 

12 or x - 7 

12 = - ___ 11 

12 

x = 3__ 

2 or x = - 1__ 

3 

3 . 69. 

70. 

71. 

4x = 1 - 3 x 2 

3 x 2 + 4x = 1 

___ 3 x 2 

3 + 4x 

3 = 1__ 

3 

x 2 + 4__ 

3 x = 1__ 

3 

b = 4__ 

3 

4_ 2 

( 3__ 

= 

2) ( 2__ 3) 2 = 4__ 

9 

x 2 + 4__ 

3 x = 1__ 

3 

x 2 + 4__ 

3 x + 4__ 

9 = 1__ 

3 + 

( x + 2__ 3) 2 = 7__ 

9 

x + 2__ 

3 = ± ___ √ 7 

3 

x + 2__ 

3 = ___ √ or 

3 

x = 

________ -2 + √ or 

3 


_________ 

-2 

Divide both terms by 

Then add half of b__ 

a squared: 

a x 2 + bx = 0 

___ a x 2 

a 

+ bx 

a = 0__ 

a 

x 2 + b__ 

a x = 0 

b 1 

= b__ 

a 

b_ 2 

( 

a__ 

= 

2) ( 2a) b = 

___ b 2 

4 a 2 

x 2 + b__ 

a x = 0 

x 2 + b__ 

a x + ___ b 2 

___ b 2 

4 a 2 4 a 

( x + 2a) b = 

___ b 2 

4 a 

x + b 

2a = ± ___ 

2a 

x + b 

2a = b 

2a or x + 4__ 

9 

x + 2__ 

3 = - ___ √ 7 

3 

x = 

________ -2 - √ 3 

7 

and 

________ -2 - √ 3 

3 

a: x 2 + b__ 

a x. 

+ b__ 

a x + ( 2a) b . 

b 

2a = - ___ b 

2a 


2 and - 1__ 

x = 0 or x = - 

b__ 

a 


b__ 

a . 



72. a 2 + b 2 = c 2 

x 2 + (x + 4) 2 = (20) 2 

x 2 + (x) 2 + 2(x)(4) + (4) 2 = 400 

2 x 2 + 8x + 16 = 400 

2 x 2 + 8x = 384 

___ 2 x 2 

2 + ___ 8x 

2 = ____ 384 

2 

A = 1__ 

2 bh 

x 2 + 4x = 192 

= 1__ (x)(x + 4) 

2 

= 1__ ( x 2 + 4x) 

2 

= 1__ 

2 (192) 

= 96 

The area of the triangle is 96 cm 2 . 


73. Plot (0, -3). Count 4 units up and 1 unit right and 

plot another point. Draw the line connecting the two 

points. 

 

 

 

 

74. Plot (0, 4). Count 2 units down and 3 units right and 


points. 

 

 

 

 

 

75. Plot (0, -2). Count 2 units down and 1 unit right and 


points. 

 

 

 

 

76. Plot (0, 0). Count 4 units down and 3 units right and 


points. 

 

 

 

 

77. (a - b) 2 = a 2 - 2ab + b 2 

(x - 4) 2 = (x) 2 - 2(x)(4) + (4) 2 

= x 2 - 8x + 16 

 

 

 

 

 

 

 

 

 

78. (a - b)(a + b) = a 2 - b 2 

(x - 4)(x + 4) = (x) 2 - (4) 2 

= x 2 - 16 

79. (a - b) 2 = a 2 - 2ab + b 2 

(4 - t) 2 = (4) 2 - 2(4)(t) + (t) 2 

= 16 - 8t + t 2 

80. (a + b) 2 = a 2 + 2ab + b 2 

(2z + 3) 2 = (2z) 2 + 2(2z)(3) + (3) 2 

= 4 z 2 + 12z + 9 

81. (a - b)(a + b) = a 2 - b 2 

( 8 b 

2 - 2 ) (8 b 2 + 2) = ( 8 b 

2) 2 - (2) 2 

= 64 b 4 - 4 

82. (a - b)(a + b) = a 2 - b 2 

(2x - 6)(2x + 6) = (2x) 2 - (6) 2 

83. 5 x 2 = 5 

5 x 2 

___ 

5 = 5__ 

5 

= 4 x 2 - 36 

x 2 = 1 

x = ± √ 1 

x = ±1 


84. x 2 + 3 = 12 

_____ - 3 ___ -3 

x 2 = 9 

x = ± √ 9 

x = ±3 


85. 5 x 2 = 80 

___ 5 x 2 

5 = ___ 80 

5 

x 2 = 16 

x = ± √ 16 

x = ±4 


86. 9 x 2 = 64 

___ 9 x 2 

9 = ___ 64 

9 

x 2 = ___ 64 

9 

x = ± √ ___ 64 

9 

x = ± 

8__ 

3 


3 and - 8__ 

3 . 

87. 25 + x 2 = 250 

________ -25 ____ -25 

x 2 = 225 

x = ± √ 225 

x = ±15 




88. 64 x 2 + 3 = 147 

_______ - 3 ___ -3 

____ 64 x 2 

64 = ____ 144 

64 

x 2 = 9__ 

4 

x = ± √ 9__ 

4 

x = ± 

3__ 

2 


2 and - 3__ 

2 . 

89. 12 = 5 x 2 

___ 

___ 

12 

5 = 5 x 2 

5 

___ 12 

5 = x 2 

± √ ___ 12 

5 = x 2 

x ≈ ±1.55 


90. 3 x 2 - 4 = 15 

______ + 4 ___ +4 

___ 3 x 2 

3 = ___ 19 

3 

x 2 = ___ 19 

3 

x = ± √ ___ 19 

3 

x ≈ ±2.52 


91. x 2 - 7 = 19 

_____ + 7 ___ +7 

x 2 = 26 

x = ± √ 26 

x ≈ ±5.10 


92. 6 + x 2 = 72 

_______ -6 ___ -6 

x 2 = 66 

x = ± √ 66 

x ≈ ±8.12 


93. 10 x 2 - 10 = 12 

________ + 10 ____ +10 

____ 10 x 2 

10 = ___ 22 

10 

x 2 = ___ 11 

5 

x = ± √ ___ 11 

5 

x ≈ ±1.48 


94. 2 x 2 + 2 = 33 

______ - 2 ___ -2 

___ 2 x 2 

2 = ___ 31 

2 

x 2 = ___ 31 

2 

x = ± √ ___ 31 

2 

x ≈ ±3.94 


9-9 THE QUADRATIC FORMULA AND THE 

DISCRIMINANT, PAGES 652–659 


1a. a = -3, b = 5, c = 2 

x = 

-5 ± √ 

 

__________________ 

5 2 - 4(-3)(2) 

2(-3) 

________________ 

x = -5 ± √ 

25 - (-24) 

-6 

_________ 

-5 ± √ 

x = 49 

-6 

x = _______ -5 ± 7 

-6 

x = _______ -5 + 7 or x = _______ -5 - 7 

-6 

-6 

x = - 

1__ 

or x = 2 

3 

b. 2 - 5 x 2 = -9x 

-5 x 2 = -9x - 2 

0 = 5 x 2 - 9x - 2 

a = 5, b = -9, c = -2 

x = 

-(-9) ± √ 

 

_______________________ 

(-9) 2 - 4(5)(-2) 

2(5) 

_______________ 

x = 9 ± √ 

81 - (-40) 

10 

_________ 

x = 9 ± √ 121 

10 

x = ______ 9 ± 11 

10 

x = ______ 9 + 11 or x = ______ 9 - 11 

10 

10 

x = 2 or x = - 

1__ 

5 

2. a = 2, b = -8, c = 1 

x = 

-(-8) ± √ 

 

______________________ 

(-8) 2 - 4(2)(1) 

2(2) 

___________ 64 - 8 

x = 8 ± √ 

4 

x = 

________ 8 ± √ 56 

4 

x = 

________ 8 + √ 56 or x = 

________ 8 - √ 56 

4 

4 

x ≈ 3.87 or x ≈ 0.13 



3a. a = 2, b = -2, c = 3 

b 2 - 4ac 

= (-2) 2 - 4(2)(3) 

= 4 - 24 

= -20 

b 2 - 4ac is negative. 


solutions. 

c. a = 1, b = -9, c = 4 

b 2 - 4ac 

= (-9) 2 - 4(1)(4) 

= 81 - 16 

= 65 

b 2 - 4ac is positive. 

There are two real solutions. 

4. h = -16 t 2 + vt + c 

20 = -16 t 2 + 20t + 1 

0 = -16 t 2 + 20t - 19 

b. a = 1, b = 4, c = 4 

b 2 - 4ac 

= 4 2 - 4(1)(4) 

= 16 - 16 

= 0 

b 2 - 4ac is zero. 

There is one real 

solution. 

a = -16, b = 20, c = -19 

b 2 - 4ac 

= (20) 2 - 4(-16)(-19) 

= 400 - 1216 

= -816 

The disciminant is negative, so the equation has no 

real solutions. The weight will not ring the bell. 

5a. x + 7x + 10 = 0 

(x + 5)(x + 2) = 0 

x + 5 = 0 or x + 2 = 0 

x = -5 or x = -2 

b. -14 + x 2 = 5x 

x 2 - 5x - 14 = 0 

(x + 2)(x - 7) = 0 

x + 2 = 0 or x - 7 = 0 

x = -2 or x = 7 

c. a = 2, b = 4, c = -21 

x = 

-4 ± √ 

 

___________________ 

4 2 - 4(2)(-21) 

2(2) 

_________________ 

x = -4 ± √ 

16 - (-168) 

4 

__________ 

-4 ± √ 

x = 184 

4 

-4 + √ 

x = 

__________ 184 or x = 

__________ 

-4 - √ 184 

4 

4 

x ≈ 2.39 or x ≈ -4.39 

3. If the discriminant is negative, there are no real 

solutions and the object will not reach the given 

height. If the discriminant is 0 or positive, there are 

1 or 2 solutions and the object will reach its given 

height. 

4. 

 

 

 

 

 

 

 

 



1. no 

2. a = 1, b = -5, c = 4 

x = 

-(-5) ± √ 

 

______________________ 

(-5) 2 - 4(1)(4) 

2(1) 

____________ 25 - 16 

x = 5 ± √ 

2 

x = 

_______ 5 ± √ 9 

2 

x = _____ 5 ± 3 

2 

x = _____ 5 + 3 

2 

x = 4 or x = 1 

or x = _____ 5 - 3 

2 

3. 2 x 2 = 7x - 3 

2 x 2 - 7x = -3 

2 x 2 - 7x + 3 = 0 

a = 2, b = -7, c = 3 

x = 

-(-7) ± √ 

 

______________________ 

(-7) 2 - 4(2)(3) 

2(2) 

____________ 49 - 24 

x = 7 ± √ 

4 

x = 

________ 7 ± √ 25 

4 

x = _____ 7 ± 5 

4 

x = _____ 7 + 5 

4 

x = 3 or x = 1__ 

2 

or x = _____ 7 - 5 

4 

 

 

 


1. Possible answer: If b 2 - 4ac > 0, there are 2 real 

solutions. If b 2 - 4ac = 0, there is 1 real solution. 

If b 2 - 4ac < 0, there are no real solutions. 

2. x = -1 or x = -4; possible answer: factoring 

method; least number of steps 



4. a = 1, b = -6, c = -7 

x = -(-6) ± √ 

________________________ 

(-6) 2 - 4(1)(-7) 

2(1) 

_______________ 

x = 6 ± √ 

36 - (-28) 

2 

________ 

x = 6 ± √ 64 

2 

x = _____ 6 ± 8 

2 

x = _____ 6 + 8 or x = _____ 6 - 8 

2 

2 

x = 7 or x = -1 

5. x 2 = -14x - 40 

x 2 + 14x = -40 

x 2 + 14x + 40 = 0 

a = 1, b = 14, c = 40 

x = 

-14 ± √ 

 

____________________ 

14 2 - 4(1)(40) 

2(1) 

_________________ 

-14 ± √ 196 - 160 

x = 

2 

-14 ± √ 

x = 

__________ 36 

2 

x = ________ -14 ± 6 

2 

x = ________ -14 + 6 

2 

x = -4 or x = -10 

or x = ________ -14 - 6 

2 

6. 3 x 2 - 2x = 8 

3 x 2 - 2x - 8 = 0 

a = 3, b = -2, c = -8 

x = 

-(-2) ± √ 

 

_______________________ 

(-2) 2 - 4(3)(-8) 

2(3) 

______________ 

x = 2 ± √ 4 - (-96) 

6 

_________ 

x = 2 ± √ 100 

6 

x = ______ 2 ± 10 

6 

x = ______ 2 + 10 or x = ______ 2 - 10 

6 

6 

x = 2 or x = - 

4__ 

3 

7. a = 4, b = -4, c = -3 

x = 

-(-4) ± √ 

 

_______________________ 

(-4) 2 - 4(4)(-3) 

2(4) 

_______________ 

x = 4 ± √ 

16 - (-48) 

8 

________ 

x = 4 ± √ 64 

8 

x = _____ 4 ± 8 

8 

x = _____ 4 + 8 

8 

x = 3__ 

2 

or x = _____ 4 - 8 

8 

or x = - 

1__ 

2 

8. a = 2, b = 0, c = -6 

x = 

-0 ± √ 

 

__________________ 

0 2 - 4(2)(-6) 

2(2) 

______________ 

x = 0 ± √ 0 - (-48) 

4 

____ 

√ 48 

x = ± 

4 

x = 

____ √ 48 

4 or x = - ____ √ 48 

4 

x ≈ 1.73 or x ≈ -1.73 

9. a = 1, b = 6, c = 3 

x = 

-6 ± √ 

 

_________________ 

6 2 - 4(1)(3) 

2(1) 

______________ 

-6 ± √ 36 - 12 

x = 

2 

-6 ± √ 

x = 

_________ 24 

2 

-6 + √ 

x = 

_________ 24 or x = 

_________ 

-6 - √ 24 

2 

2 

x ≈ -0.55 or x ≈ -5.45 

10. a = 1, b = -7, c = 2 

x = 

-(-7) ± √ 

 

______________________ 

(-7) 2 - 4(1)(2) 

2(1) 

___________ 49 - 8 

x = 7 ± √ 

2 

x = 

________ 7 ± √ 41 

2 

x = 

________ 7 + √ 41 or x = 

________ 7 - √ 41 

2 

2 

x ≈ 6.70 or x ≈ 0.30 

11. 3 x 2 = -x + 5 

3 x 2 + x = 5 

3 x 2 + x - 5 = 0 

a = 3, b = 1, c = -5 

x 

= -1 ± √ 

 

____________________ 

1 2 - 4(3)(-5) 

2(3) 

_______________ 

x = -1 ± √ 1 - (-60) 

6 

_________ 

-1 ± √ 

x = 61 

6 

-1 + √ 

x = 

_________ 61 or x = 

_________ 

-1 - √ 61 

6 

6 

x ≈ 1.14 or x ≈ -1.47 



12. a = 1, b = -4, c = -7 

x = 

-(-4) ± √ 

 

_______________________ 

(-4) 2 - 4(1)(-7) 

2(1) 

_______________ 

x = 4 ± √ 

16 - (-28) 

2 

________ 

x = 4 ± √ 44 

2 

x = 

________ 4 + √ 44 or x = 

________ 4 - √ 44 

2 

2 

x ≈ 5.32 or x ≈ -1.32 

13. a = 2, b = 1, c = -5 

x = 

-1 ± √ 

 

__________________ 

1 2 - 4(2)(-5) 

2(2) 

_______________ 

x = -1 ± √ 1 - (-40) 

4 

_________ 

-1 ± √ 

x = 41 

4 

-1 + √ 

x = 

_________ 41 or x = 

_________ 

-1 - 

√ 41 

4 

4 

x ≈ 1.35 or x ≈ -1.85 

14. a = 2, b = 4, c = 3 

b 2 - 4ac 

= 4 2 - 4(2)(3) 

= 16 - 24 

= -8 



solutions. 

16. a = 2, b = -11, c = 6 

b 2 - 4ac 

= (-11) 2 - 4(2)(6) 

= 121 - 48 

= 73 


There are two real 

solutions. 

18. 3 x 2 = 5x - 1 

3 x 2 - 5x = -1 

3 x 2 - 5x + 1 = 0 

a = 3, b = -5, c = 1 

b 2 - 4ac 

= (-5) 2 - 4(3)(1) 

= 25 - 12 

= 13 



solutions. 

15. a = 1, b = 4, c = 4 

b 2 - 4ac 

= 4 2 - 4(1)(4) 

= 16 - 16 

= 0 



solution. 

17. a = 1, b = 1, c = 1 

b 2 - 4ac 

= 1 2 - 4(1)(1) 

= 1 - 4 

= -3 



solutions. 

19. -2x + 3 = 2 x 2 

-2 x 2 - 2x + 3 = 0 

a = -2, b = -2, c = 3 

b 2 - 4ac 

= (-2) 2 - 4(-2)(3) 

= 4 + 24 

= 28 



solutions. 

20. 2 x 2 + 12x = -18 

2 x 2 + 12x + 18 = 0 

a = 2, b = 12, c = 18 

b 2 - 4ac 

= 12 2 - 4(2)(18) 

= 144 - 144 

= 0 



solution. 

22. 8x = 1 - x 2 

0 = - x 2 - 8x + 1 

a = -1, b = -8, c = 1 

b 2 - 4ac 

= (-8) 2 - 4(-1)(1) 

= 64 + 4 

= 68 



23. h = -4.9 t 2 + 102t + 100 

600 = -4.9 t 2 + 102t + 100 

0 = -4.9 t 2 + 102t - 500 

21. 5 x 2 + 3x = -4 

5 x 2 + 3x + 4 = 0 

a = 5, b = 3, c = 4 

b 2 - 4ac 

= 3 2 - 4(5)(4) 

= 9 - 80 

= -71 



solutions. 

a = -4.9, b = 102, c = -500 

b 2 - 4ac 

= 102 2 - 4(-4.9)(-500) 

= 10,404 - 9800 

= 604 

The discriminant is positive, so the equation has two 

solutions. The rocket will reach a height of 600 m. 

24. x 2 + x - 12 = 0 

(x + 4)(x - 3) = 0 

x + 4 = 0 or x - 3 = 0 

x = -4 or x = 3 

25. x 2 + 6x + 9 = 0 

(x + 3) 2 = 0 

x + 3 = 0 

x = -3 

26. 2 x 2 - x - 1 = 0 

(2x + 1)(x - 1) = 0 

2x + 1 = 0 or x - 1 = 0 

2x = -1 or x = 1 

x = - 

1__ 

2 

27. 4 x 2 + 4x + 1 = 0 

(2x + 1) 2 = 0 

2x + 1 = 0 

2x = -1 

x = - 

1__ 

2 



28. a = 2, b = 1, c = -7 

x = 

-1 ± √ 

 

__________________ 

1 2 - 4(2)(-7) 

2(2) 

_______________ 

x = -1 ± √ 1 - (-56) 

4 

_________ 

-1 ± √ 

x = 57 

4 

-1 + √ 

x = 

_________ 57 or x = 

_________ 

-1 - √ 57 

4 

4 

x ≈ 1.64 or x ≈ -2.14 

29. 9 = 2 x 2 + 3x 

9__ 

2 = ___ 2 x 2 

2 + ___ 3x 

2 

9__ 

2 = x 2 + 3__ 

b = 3__ 

2 

3_ 2 

2__ 

= ( 3__ 

( 2) 

9__ 

2 + 9 

2 x 

___ 

4) 2 = 9 

16 

9__ 

2 = x 2 + 3__ 

___ 

16 = x 2 + 3__ 

___ 81 

16 = ( x + 3__ 

± 

9__ 

x + 3__ 

4 = 9__ 

4 

4 = x + 3__ 

4 

2 x 

___ 

2 x + 9 

16 

4) 2 

or x + 

3__ 9__ 

4 = - 4 

x = 3__ or x = -3 

2 


30. 3 x 2 = 13x - 4 

3 x 2 - 13x = -4 

3 x 2 - 13x + 4 = 0 

a = 3, b = -13, c = 4 

x = 

-(-13) ± √ 

 

________________________ 

(-13) 2 - 4(3)(4) 

2(3) 

______________ 

13 ± √ 169 - 48 

x = 

6 

13 ± √ 

x = 

__________ 121 

6 

13 ± 11 

x = _______ 

6 

13 + 11 

x = _______ or x = 

6 

x = 4 

or x = 1__ 

3 

_______ 13 - 11 

6 

31. a = 1, b = -10, c = 9 

x = 

-(-10) ± √ 

 

________________________ 

(-10) 2 - 4(1)(9) 

2(1) 

______________ 

10 ± √ 100 - 36 

x = 

2 

10 ± √ 

x = 

_________ 64 

2 

x = ______ 10 ± 8 

2 

x = ______ 10 + 8 

2 

x = 9 or x = 1 

32. 1 = 3 x 2 + 2x 

0 = 3 x 2 + 2x - 1 

or x = ______ 10 - 8 

2 

a = 3, b = 2, c = -1 

x = 

-2 ± √ 

 

__________________ 

2 2 - 4(3)(-1) 

2(3) 

_______________ 

x = -2 ± √ 4 - (-12) 

6 

_________ 

-2 ± √ 

x = 16 

6 

x = _______ -2 ± 4 

6 

x = _______ -2 + 4 or x = _______ -2 - 4 

6 

6 

x = 1__ or x = -1 

3 

33. a = 1, b = -4, c = 1 

x = 

-(-4) ± √ 

 

______________________ 

(-4) 2 - 4(1)(1) 

2(1) 

___________ 16 - 4 

x = 4 ± √ 

2 

x = 

________ 4 ± √ 12 

2 

x = 

________ 4 + √ 12 or x = 

________ 4 - √ 12 

2 

2 

x ≈ 3.73 or x ≈ 0.27 

34. a = 3, b = 0, c = -5 

x = 

-0 ± √ 

 

__________________ 

0 2 - 4(3)(-5) 

2(3) 

______________ 

x = 0 ± √ 0 - (-60) 

6 

____ 

√ 60 

x = ± 

6 

x = 

____ √ 60 

6 or x = - ____ √ 60 

6 

x ≈ 1.29 or x ≈ -1.29 



35. 2 x 2 + 7x = -4 

2 x 2 + 7x + 4 = 0 

a = 2, b = 7, c = 4 

x = 

 

_________________ 

-7 ± √ 7 2 - 4(2)(4) 

2(2) 

______________ 

-7 ± √ 49 - 32 

x = 

4 

-7 ± √ 

x = 

_________ 17 

4 

-7 + √ 

x = 

_________ 17 or x = 

_________ 

-7 - √ 17 

4 

4 

x ≈ -0.72 or x ≈ -2.78 

36. a = 3, b = -6, c = 3 

b 2 - 4ac 

= (-6) 2 - 4(3)(3) 

= 36 - 36 

= 0 



solution. 

38. a = 7, b = 6, c = 2 

b 2 - 4ac 

= 6 2 - 4(7)(2) 

= 36 - 56 

= -20 



39. h = -16 x 2 + 12x 

4 = -16 x 2 + 12x 

0 = -16 x 2 + 12x - 4 

37. a = 1, b = -3, c = -8 

b 2 - 4ac 

= (-3) 2 - 4(1)(-8) 

= 9 + 32 

= 41 



solutions. 

a = -16, b = 12, c = -4 

b 2 - 4ac 

= 12 2 - 4(-16)(-4) 

= 144 - 256 

= -112 

The disciminant is negative, so the equation has no 

real solutions. The gymnast’s hands will not reach a 

height of 10 ft. 

42. x 2 - 12 = -x 

x 2 + x - 12 = 0 

(x + 4)(x - 3) = 0 

x + 4 = 0 or x - 3 = 0 

x = -4 or x = 3 

43. 2x = 3 + 2 x 2 

0 = 2 x 2 - 2x + 3 

a = 2, b = -2, c = 3 

b 2 - 4ac 

= (-2) 2 - 4(2)(3) 

= 4 - 24 

= -20 



44. x 2 = 2x + 9 

x 2 - 2x = 9 

x 2 - 2x - 9 = 0 

a = 1, b = -2, c = -9 

b 2 - 4ac 

= (-2) 2 - 4(1)(-9) 

= 4 + 36 

= 40 



x = 

-(-2) ± √ 

 

______________________ 

(-2) 2 - 4(1)(9) 

2(1) 

______________ 

x = 2 ± √ 4 - (-36) 

2 

________ 

x = 2 ± √ 40 

2 

x = 

________ 2 + √ 40 or x = 

________ 2 - √ 40 

2 

2 

x ≈ 4.16 or x ≈ -2.16 

40. x 2 + 4x + 3 = 0 

(x + 3)(x + 1) = 0 

x + 3 = 0 or x + 1 = 0 

x = -3 or x = -1 

41. b = 2 

( 2__ 2) 2 = 1 2 = 1 

x 2 + 2x = 15 

x 2 + 2x + 1 = 15 + 1 

(x + 1) 2 = 16 

x + 1 = ±4 

x + 1 = 4 or x + 1 = -4 

x = 3 or x = -5 



45. 2 = 7x + 4 x 2 

0 = 4 x 2 + 7x - 2 

a = 4, b = 7, c = -2 

b 2 - 4ac 

= 7 2 - 4(4)(-2) 

= 49 + 32 

= 81 



x = 

-7 ± √ 

 

__________________ 

7 2 - 4(4)(-2) 

2(4) 

________________ 

x = -7 ± √ 

49 - (-32) 

8 

_________ 

-7 ± √ 

x = 81 

8 

x = _______ -7 ± 9 

8 

x = _______ -7 + 9 or x = _______ -7 - 9 

8 

8 

x = 1__ or x = -2 

4 

46. -7 = x 2 

0 = x 2 + 0x + 7 

a = 1, b = 0, c = 7 

b 2 - 4ac 

= 0 2 - 4(1)(7) 

= 0 - 28 

= -28 



47. -12x = -9 x 2 - 4 

0 = -9 x 2 + 12x - 4 

a = -9, b = 12, c = -4 

b 2 - 4ac 

= 12 2 - 4(-9)(-4) 

= 144 - 144 

= 0 


There is one real solution. 

x = 

-12 ± √ 

 

______________________ 

12 2 - 4(-9)(-4) 

2(-9) 

_________________ 

-12 ± √ 144 - 144 

x = 

-18 

x = 

_________ 

-12 ± √ 0 

-18 

x = ________ -12 ± 0 

-18 

x = ____ -12 

-18 

x = 2__ 

3 

48. x 2 - 14 = 0 

x 2 + 0x - 14 = 0 

a = 1, b = 0, c = -14 

b 2 - 4ac 

= 0 2 - 4(1)(-14) 

= 0 + 56 

= 56 



x = 

-0 ± √ 

 

___________________ 

0 2 - 4(1)(-14) 

2(1) 

___________ 0 + 56 

x = 0 ± √ 

2 

√ 56 

x = ± 

____ 

2 

x = 

____ √ 56 

2 or x = - ____ √ 56 

2 

x ≈ 3.74 or x ≈ -3.74 

49. a = 2, b = -1, c = -21 

b 2 - 4ac 

= (-1) 2 - 4(2)(-21) 

= 1 + 168 

= 169 


There are two x-intercepts. 

x = 

-(-1) ± √ 

 

________________________ 

(-1) 2 - 4(2)(-21) 

2(2) 

_______________ 

x = 1 ± √ 

1 - (-168) 

4 

_________ 

x = 1 ± √ 169 

4 

x = ______ 1 ± 13 

4 

x = ______ 1 + 13 or x = ______ 1 - 13 

4 

4 

x = 7__ or x = -3 

2 

50. a = 5, b = 12, c = 8 

b 2 - 4ac 

= 12 2 - 4(5)(8) 

= 144 - 160 

= -16 





51. a = 1, b = -10, c = 25 

b 2 - 4ac 

= (-10) 2 - 4(1)(25) 

= 100 - 100 

= 0 


There is one x-intercept. 

52. 

x = 

-(-10) ± √ 

 

_________________________ 

(-10) 2 - 4(1)(25) 

2(1) 

_______________ 

10 ± √ 100 - 100 

x = 

2 

x = 

________ 10 ± √ 0 

2 

x = ______ 10 ± 0 

2 

x = ___ 10 

2 

x = 5 

Quadratic Equation 

Discriminant 

Number of 

Solutions 

x 2 + 12x - 20 = 0 224 2 

8x + x 2 = -16 0 1 

0.5 x 2 + x - 3 = 0 7 2 

-3 x 2 - 2x = 1 -8 0 

53a. h(t) = -4.9 t 2 + 3.5t + 10 

1 = -4.9 t 2 + 3.5t + 10 

0 = -4.9 t 2 + 3.5t + 9 

a = -4.9, b = 3.5, c = 9 

t = 

-3.5 ± √ 

 

_______________________ 

3.5 2 - 4(-4.9)(9) 

2(-4.9) 

_______________________ 

t = -3.5 ± √ 

12.25 - (-176.4) 

-9.8 

______________ 

-3.5 ± √ 

t = 188.65 

-9.8 

-3.5 + √ 

t = 

______________ 188.65 or t = 

______________ 

-3.5 - √ 188.65 

-9.8 

-9.8 

t ≈ -1.04 or t ≈ 1.76 



Therefore, it takes the diver about 1.76 s to reach 

a point 1 m above the water. 

b. 2 solutions 

c. No; negative solutions do not make sense because 

the variable represents time. 

54. 2; 0; 1; x 2 = k in standard form is 1 x 2 + 0x - k, so 

a = 1, b = 0, c = -k and b 2 - 4ac = 4k. k > 0: 

positive discriminant, k < 0: negative discriminant, 

and k = 0: 0 discriminant. 

55. The discriminant tells how many solutions there are. 

If an equation looks difficult to solve, first use the 

discriminant to find out if it has any solutions. If it 

doesn’t, stop right there. 

56a. No; the discriminant of 130 = -16 t 2 + 80t + 20 is 

negative, so the equation has no real solutions. 

b. 116 = -16 t 2 + 80t + 20 

0 = -16 t 2 + 80t - 96 

a = -16, b = 80, c = -96 

b 2 - 4ac 

= 80 2 - 4(-16)(-96) 

= 6400 - 6144 

= 256 

The discriminant is positive, so the equation has 

two solutions. 

t = 

-80 ± √ 

 

________________________ 

80 2 - 4(-16)(-98) 

2(-16) 

___________________ 

-80 ± √ 

6400 - 6144 

t = 

-32 

-80 ± √ 

t = 

___________ 256 

-32 

-80 ± 16 

t = _________ 

-32 

-80 + 16 

t = _________ or t = _________ 

-80 - 16 

-32 

-32 

t = 2 or t = 3 

The golf ball will be at a height of 116 ft at 2 s and 

3 s. 

c. a = -16, b = 80, c = 20 

t = 

-80 ± √ 

 

______________________ 

80 2 - 4(-16)(20) 

2(-16) 

_____________________ 

t = -80 ± √ 

6400 - (-1280) 

-32 

____________ 

-80 ± √ 

t = 7680 

-32 

-80 + √ 

t = 

____________ 7680 or t = 

____________ 

-80 - √ 7680 

-32 

-32 

t ≈ -0.24 or t ≈ 5.24 




57. A; since (-3) 2 - 4(4)(1) = 9 - 16 = -7 < 0, there 

are no real solutions. 

58. G; if b 2 > 4ac, then b 2 - 4ac > 0 so the equation 

has two real solutions. So II creates two solutions, 

therefore G or J is correct. If a = b, and c = b, then 

b 2 - 4ac = b 2 - 4(b)(b) = -3 b 2 . Since b 2 > 0, 

b 2 - 4ac = -3 b 2 < 0, so III does not create two 

solutions. 

59a. a = 1, b = 2, c = 1 

b 2 - 4ac 

= 2 2 - 4(1)(1) 

= 4 - 4 

= 0 


There is one real solution. 



. Graph y = x 2 + 2x + 1. The axis of symmetry is 

x = -1. The vertex is (-1, 0). Another point is 

(1, 4). Graph the points and reflect them across the 

axis of symmetry. 

y 

6 

4 

2 

x 

-4 -2 

0 The only zero appears to be 

2 

-1. 

c. 0 = x 2 + 2x + 1 

0 = (x + 1)(x + 1) 

0 = x + 1 

-1 = x 


d. a = 1, b = 2, c = 1 

x = 

-2 ± √ 

 

_________________ 

2 2 - 4(1)(1) 

2(1) 

____________ 

-2 ± √ 4 - 4 

x = 

2 

x = 

________ -2 ± √ 0 

2 

x = _______ -2 ± 0 

2 

x = ___ -2 

2 

x = -1 

e. Possible answer: The easiest was solving by 

factoring: 0 = (x + 1)(x + 1) because the equation 

includes a perfect-square trinomial. 


60. P = 2l + 2w 

80 = 2l + 2w 

____ -2w _______ - 2w 

_______ 80 - 2w 

= ___ 2l 

2 2 

40 - w = l 

A = lw 

A = (40 - w)w 

A = 40w - w 2 

400 = 40w - w 2 

0 = - w 2 + 40w - 400 

a = -1, b = 40, c = -400 

w = 

-40 ± √ 

 

________________________ 

40 2 - 4(-1)(-400) 

2(-1) 

___________________ 

-40 ± √ 

1600 - 1600 

w = 

-2 

w = 

_________ 

-40 ± √ 0 

-2 

w = ________ -40 + 0 

-2 

w = ____ -40 

-2 

w = 20 

l = 40 - w 

l = 40 - 20 

l = 20 

The area of the pen is represented by 40w - w 2 . 

The length and width of the pen are each 20 yd. 

61a. Yes; the three boundaries can be represented by x, 

2x, and 1000 - 3x. The area is represented by 

1__ 

2 (2x)(x + 1000 - 3x) or -2 x 2 + 1000x. So 

-2 x 2 + 1000x = 125,000. Rewrite this in 

standard form: 2 x 2 - 1000x + 125,000 = 0. The 

discriminant is 0, so there is exactly one solution. 

b. From above, a = 2, b = -1000, c = 125,000. 

x = 

-(-1000) ± √ 

 

__________________________________ 

(-1000) 2 - 4(2)(125,000) 

2(2) 

___________________________ 

1000 ± √ 

1,000,000 - 1,000,000 

x = 

4 

x = 

__________ 

1000 ± √ 0 

4 

x = ________ 1000 ± 0 

4 

x = _____ 1000 

4 

x = 250 

So the western boundary is x = 250 ft, the northern 

boundary is 2x = 500 ft, and the eastern boundary 

is 1000 - 3x = 250 ft. Since the parallel sides of 

the trapazoid are equal in length, the field must be 

a parallelogram. 




MULTI-STEP TEST PREP, PAGE 660 

62. b = -2 

( ___ -2 

2 ) 2 = (-1) 2 1. h = -16 t 2 + 80t 

= 1 

0 = -16 t 2 + 80t 

x 2 0 = -16t(t - 5) 

- 2x - 24 = 0 

x 2 -16t = 0 or t - 5 = 0 

- 2x = 24 

t = 0 or t = 5 

x 2 - 2x + 1 = 24 + 1 

The golf ball is in the air for 5 s. 

(x - 1) 2 = 25 

2. a = -16, b = 80 

x - 1 = ±5 

x - 1 = 5 or x - 1 = -5 

t = -___ 

b 

2a = - ______ 80 

2(-16) = - ____ 80 

-32 = 2.5 

x = 6 or x = -4 


h = -16 t 2 + 80t 

= -16(2.5) 2 + 80(2.5) 

63. b = 6 

( 6__ 

2) 2 = 3 2 = 9 

x 2 + 6x = 40 

x 2 + 6x + 9 = 40 + 9 

(x + 3) 2 = 49 

4. h = -16 t 2 + 80t 

x + 3 = ±7 

x + 3 = 7 or x + 3 = -7 

= -16(3.5) 2 + 80(3.5) 

x = 4 or x = -10 

= -196 + 280 = 84 


The golf ball is 84 ft high after 3.5 s. 

64. -3 x 2 + 12x = 15 

_____ -3 x 2 

-3 + ____ 12x 

-3 = ___ 15 

-3 

x 2 - 4x = -5 

READY TO GO ON? PAGE 661 

b = -4 

( ___ -4 

2 ) 2 = (-2) 2 = 4 

x 2 - 4x = -5 

x 2 - 4x + 4 = -5 + 4 

y 

(x - 2) 2 x 

= -1 

-4 

0 

2 4 


-4 


65. s 2 r 3 + 5 r 3 + 5t + s 2 t 

= r 3 ( s 2 + 5) + t (5 + s 2 -12 

) 

= r 3 ( s 2 + 5) + t ( s 2 + 5) 

= ( r 3 + t) ( s 2 + 5) 

66. b 3 - 4 b 2 + 2b - 8 67. n 5 - 6 n 4 - 2n + 12 

= b 2 (b - 4) + 2(b - 4) = n 4 (n - 6) - 2(n - 6) 

= ( b 2 + 2) (b - 4) 

= ( n 4 - 2) (n - 6) 

y 

8 

68. ⎪0.2⎥ = 0.2 ⎪1.5⎥ = 1.5 ⎪1⎥ = 1 

g(x) = 1.5 x 2 + 4, h(x) = x 2 - 8, f(x) = 0.2 x 2 

4 

69. ⎪ - 

1__ 

5⎥ = 1__ 

5 ⎪ 1__ 6⎥ = 1__ 

x 

-6 -4 -2 

0 

2 

6 

f(x) = - 

1__ 

5 x 2 + 5, g(x) = 1__ 

-4 

6 x 2 -8 

= -100 + 200 = 100 

The maximum height of the golf ball is 100 ft. 

3. From above, the maximum height occurs when 

t = 2.5. So the golf ball reaches its maximum height 

after 2.5 s. 

5. At 1 s and 4 s; Using the table function in a graphing 

calculator, X = 1 and 4 when Y 1 

= 64. 

1. Graph y = x 2 - 9. The axis of symmetry is x = 0. 



symmetry. 


3 and -3. 

2. Graph y = x 2 + 3x - 4. The axis of symmetry is 

x = -1.5. The vertex is (-1.5, -6.25). The 

y-intercept is -4. Another point is (-1, -6). Graph 


symmetry. 

The zeros appear to be 1 and 

-4. 



3. 4 x 2 + 8x = 32 

_______ - 32 ____ -32 

4 x 2 + 8x - 32 = 0 

Graph y = 4 x 2 + 8x - 32. The axis of symmetry is 

x = -1. The vertex is (-1, -36). The y-intercept is 



-4 

6 

4 

2 

y 

0 -2 2 

x 


2 and -4. 

4. Graph y = -5 x 2 + 30x + 35. 

The zeros appear to be -1 and 7. 

The firework leaves the ground at 0 s and returns 

to the ground at 7 s. The firework takes 7 s to reach 

the ground. 

-4 

90 

60 

30 

y 

0 4 12 

5. (x + 1)(x + 3) = 0 

x + 1 = 0 or x + 3 = 0 

x = -1 or x = -3 


6. (x - 6)(x - 3) = 0 

x - 6 = 0 or x - 3 = 0 

x = 6 or x = 3 


7. x(x + 3) = 18 

x(x) + x(3) = 18 

x 2 + 3x = 18 

______ - 18 ____ -18 

x 2 + 3x - 18 = 0 

(x + 6)(x - 3) = 0 

x + 6 = 0 or x - 3 = 0 

x = -6 or x = 3 


8. (x + 2)(x - 5) = 60 

x(x) + x(-5) + 2(x) + 2(-5) = 60 

x 2 - 5x + 2x - 10 = 60 

x 2 - 3x - 10 = 60 

___________ - 60 ____ -60 

x 2 - 3x - 70 = 0 

(x + 7)(x - 10) = 0 

x + 7 = 0 or x - 10 = 0 

x = -7 or x = 10 


x 

9. x 2 - 4x - 32 = 0 

(x + 4)(x - 8) = 0 

x + 4 = 0 or x - 8 = 0 

x = -4 or x = 8 

The solutions are -4 and 8 

10. x 2 - 8x + 15 = 0 

(x - 3)(x - 5) = 0 

x - 3 = 0 or x - 5 = 0 

x = 3 or x = 5 


11. x 2 + x = 6 

______ - 6 ___ -6 

x 2 + x - 6 = 0 

(x + 3)(x - 2) = 0 

x + 3 = 0 or x - 2 = 0 

x = -3 or x = 2 


12. -8x - 33 = - x 2 

________ + x 2 ____ + x 2 

x 2 - 8x - 33 = 0 

(x + 3)(x - 11) = 0 

x + 3 = 0 or x - 11 = 0 

x = -3 or x = 11 


13. h = -16 t 2 + 64t 

0 = -16 t 2 + 64t 

0 = -16t(t - 4) 

-16t = 0 or t - 4 =0 

t = 0 or t = 4 

It takes the soccer ball 4 s to return to the ground. 

14. 3 x 2 = 48 

___ 3 x 2 

3 = ___ 48 

3 

x 2 = 16 

x = ± √ 16 

x = ±4 


15. 36 x 2 - 49 = 0 

________ + 49 ____ +49 

____ 36 x 2 

36 = ___ 49 

36 

x 2 = ___ 49 

36 

x = ± √ ___ 49 

36 

x = ± 

7__ 

6 


6 and - 7__ 

6 . 

16. -12 = x 2 - 21 

____ +21 _______ + 21 

9 = x 2 

± √ 9 = x 

±3 = x 




17. 3 x 2 + 5 = 21 

______ - 5 ___ -5 

___ 3 x 2 

3 = ___ 16 

3 

x 2 = ___ 16 

3 

x 2 = ± √ ___ 16 

3 

x ≈ ±2.31 


18. b = -12 

( ____ -12 

2 ) 2 = (-6) 2 = 36 

x 2 - 12x + 36 

20. b = 9 

( 9__ 

___ 

2) 2 = 81 

4 

x 2 + 9x + ___ 81 

4 

21. b = 2 

( 2__ 2) 2 = 1 2 = 1 

x 2 + 2x = 3 

x 2 + 2x + 1 = 3 + 1 

(x + 1) 2 = 4 

x + 1 = ±2 

x + 1 = 2 or x + 1 = -2 

x = 1 or x = -3 


22. x 2 - 5 = 2x 

x 2 - 2x - 5 = 0 

x 2 - 2x = 5 

b = -2 

( ___ -2 

2 ) 2 = (-1) 2 = 1 

x 2 - 2x = 5 

x 2 - 2x + 1 = 5 + 1 

(x - 1) 2 = 6 

x - 1 ≈ ±2.45 

x - 1 ≈ 2.45 or x - 1 ≈ -2.45 

x ≈ 3.45 or x ≈ -1.45 

23. b = 7 

( 7__ 

___ 

2) 2 = 49 

4 

x 2 + 7x = 8 

x 2 + 7x + ___ 49 

4 = 8 + ___ 49 

4 

( x + 2) 7__ 2 = ___ 81 

4 

x + 7__ 

2 = 9__ 

2 

x + 7__ 9__ 

2 = ± 2 

or x + 

7__ 9__ 

2 = - 2 

x = 1 or x = -8 

19. b = 4 

( 4__ 2) 2 = 2 2 = 4 

x 2 + 4x + 4 

24. lw = A 

(x - 4)(x) = 42 

x 2 - 4x = 42 

b = -4 

( ___ -4 

2 ) 2 = (-2) 2 = 4 

x 2 - 4x = 42 

x 2 - 4x + 4 = 42 + 4 

(x - 2) 2 = 46 

x - 2 ≈ ±6.8 

x - 2 ≈ 6.8 or x - 2 ≈ -6.8 

x ≈ 8.8 or x ≈ -4.8 



The width is 8.8 ft, and the length is 8.8 - 4 ≈ 4.8 ft. 

25. a = 1, b = 5, c = 1 

x = 

-5 ± √ 

 

_________________ 

5 2 - 4(1)(1) 

2(1) 

_____________ 

-5 ± √ 25 - 4 

x = 

2 

-5 ± √ 

x = 

_________ 21 

2 

-5 + √ 

x = 

_________ 21 or x = 

_________ 

-5 - √ 21 

2 

2 

x ≈ -0.21 or x ≈ -4.79 

26. 3 x 2 + 1 = 2x 

3 x 2 - 2x + 1 = 0 

a = 3, b = -2, c = 1 

b 2 - 4ac 

= (-2) 2 - 4(3)(1) 

= 4 - 12 

= -8 



27. 5x + 8 = 3 x 2 

-3 x 2 + 5x + 8 = 0 

a = -3, b = 5, c = 8 

x = 

-5 ± √ 

 

__________________ 

5 2 - 4(-3)(8) 

2(-3) 

________________ 

x = -5 ± √ 

25 - (-96) 

-6 

__________ 

-5 ± √ 

x = 121 

-6 

-5 ± 11 

x = _______ 

-6 

-5 + 11 

x = _______ or x = 

-6 

x = -1 

or x = 8__ 

3 

_______ -5 - 11 

-6 



28. a = 2, b = -3, c = 4 

b 2 - 4ac 

= (-3) 2 - 4(2)(4) 

= 9 - 32 

= -23 



solutions. 

30. a = 1, b = 4, c = -5 

b 2 - 4ac 

= 4 2 - 4(1)(-5) 

= 16 + 20 

= 36 



29. a = 1, b = 2, c = 1 

b 2 - 4ac 

= 2 2 - 4(1)(1) 

= 4 - 4 

= 0 



solution. 

STUDY GUIDE REVIEW, PAGES 662–665 

1. vertex 2. minimum; maximum 

3. zero of a function 4. discriminant 

5. completing the square 

LESSON 9-1, PAGE 662 



and c = -5. 


a is 0. 


in the form y = a x 2 + bx + c, where a = - 

1__ 

2 , b = 0, 

and c = 0. 

9. This is not a quadratic function because it contains 

a power greater than 2. 

10. 

11. 

x y = 6 x 2 

 

-2 24 

-1 6 

0 0 

1 6 

2 24 

 

x y = -4 x 2 

 

 

-2 -16 

 

-1 -4 

0 0 

1 -4 

2 -16 

 

 

 

12. 

13. 

x 

y = 1__ 

-2 1 

-1 1__ 

4 

0 0 

1 1__ 

4 

2 1 

4 x 2 

 

 

 

x y = -3 x 2 

 

 

-2 -12 

 

-1 -3 

0 0 

1 -3 

2 -12 

14. y = 5 x 2 - 12 

a = 5 


15. y = - x 2 + 3x - 7 

a = -1 


16. The vertex is (-2, -4) and the minimum is -4. 




19. The zeros are 4 and 8. 

x = _____ 4 + 8 = ___ 12 

2 2 = 6 

y = - x 2 + 12x - 32 

= - (6) 2 + 12(6) - 32 

= -36 + 72 - 32 = 4 

The axis of symmetry is x = 6. The vertex is (6, 4). 


x = _______ -4 + 2 = ___ -2 

2 2 = -1 

y = 2 x 2 + 4x - 16 

= 2 (-1) 2 + 4(-1) - 16 

= 2 - 4 - 16 = -18 

The axis of symmetry is x = -1. The vertex is 

(-1, -18). 

 

 

 

 




21. a = 1, b = 6 

b 

x = -___ 

2a = - ____ 6 6__ 

2(1) = - 2 = -3 

y = x 2 + 6x + 6 

= (-3) 2 + 6(-3) + 6 

= 9 - 18 + 6 = -3 


y = x 2 + 6x + 6 → y = x 2 + 6x + (6) 


(0, 6). 

Let x = -2. 

y = x 2 + 6x + 6 

= (-2) 2 + 6(-2) + 6 

= 4 - 12 + 6 = -2 

Let x = -1. 

y = x 2 + 6x + 6 

= (-1) 2 + 6(-1) + 6 

= 1 - 6 + 6 = 1 

Two other points are (-2, -2) and (-1, 1). 




 

 

22. a = 1, b = -4 

b 

x = -___ 

2a = - ____ -4 

2(1) = 4__ 

2 = 2 

y = x 2 - 4x - 12 

= (2) 2 - 4(2) - 12 

= 4 - 8 - 12 = -16 


 

 

 

 

y = x 2 - 4x - 12 → y = x 2 - 4x + (-12) 


(0, -12). 

Let x = -1. 

y = x 2 - 4x - 12 

= (-1) 2 - 4(-1) - 12 

= 1 + 4 - 12 = -7 

Let x = 1. 

y = x 2 - 4x - 12 

= (1) 2 - 4(1) - 12 

= 1 - 4 - 12 = -15 

Two other points are (-1, -7) and (1, -15). 




 

 

 

 

 

 

 

 

23. a = 1, b = -8 

b 

x = -___ 

2a = - ____ -8 

2(1) = 8__ 

2 = 4 

y = x 2 - 8x + 7 

= (4) 2 - 8(4) + 7 

= 16 - 32 + 7 = -9 


y = x 2 - 8x + 7 → y = x 2 - 8x + (7) 


(0, 7). 

Let x = 1. 

y = x 2 - 8x + 7 

= (1) 2 - 8(1) + 7 

= 1 - 8 + 7 = 0 

Let x = 2. 

y = x 2 - 8x + 7 

= (2) 2 - 8(2) + 7 

= 4 - 16 + 7 = -5 





 

 

 

 

 

24. a = 2, b = -6 

b 

x = -___ 

2a = - ____ -6 

2(2) = 6__ 

y = 2 x 2 - 6x - 8 

= 2 ( 2) 3__ 2 -6 ( 3__ 

2) - 8 

= 2 ( 4) 9__ - 9 - 8 

= 9__ 

___ 

2 - 9 - 8 = - 25 

2 

2 , - ___ 25 

2 ) . 


 

4 = 3__ 

2 

y = 2 x 2 - 6x - 8 → y = 2 x 2 - 6x + (-8) 


(0, -8). 

Let x = -1. 

y = 2 x 2 - 6x - 8 

= 2 (-1) 2 - 6(-1) - 8 

= 2(1) + 6 - 8 

= 2 + 6 - 8 = 0 

 

 

Let x = 1. 

y = 2 x 2 - 6x - 8 

= 2 (1) 2 - 6(1) - 8 

= 2(1) - 6 - 8 

= 2 - 6 - 8 = -12 

Two other points are (-1, 0) and (1, -12). 




 

 

 

 

 

 

 



25. 3 x 2 + 6x = y - 3 

________ + 3 _____ + 3 

3 x 2 + 6x + 3 = y 

a = 3, b = 6 

b 

x = -___ 

2a = - ____ 6 6__ 

2(3) = - 6 = -1 

y = 3 x 2 + 6x + 3 

= 3 (-1) 2 + 6(-1) + 3 

= 3(1) - 6 + 3 

= 3 - 6 + 3 = 0 

The vertex is at (-1, 0). 

y = 3 x 2 + 6x + 3 → y = 3 x 2 + 6x + (3) 


(0, 3). 

Let x = 1. 

y = 3 x 2 + 6x + 3 

= 3 (1) 2 + 6(1) + 3 

= 3(1) + 6 + 3 

= 3 + 6 + 3 = 12 

Let x = 2. 

y = 3 x 2 + 6x + 3 

= 3 (2) 2 + 6(2) + 3 

= 3(4) + 12 + 3 

= 12 + 12 + 3 = 27 





 

 

 

 

 

 

 

 

26. 2 - 4 x 2 + y = 8x - 10 

___________ 

-2 _______ -2 

-4 x 2 + y = 8x - 12 

________ +4 x 2 _______ +4 x 2 

y = 4 x 2 + 8x - 12 

a = 4, b = 8 

b 

x = -___ 

2a = - ____ 8 8__ 

2(4) = - 8 = -1 

y = 4 x 2 + 8x - 12 

= 4 (-1) 2 + 8(-1) - 12 

= 4(1) - 8 - 12 

= 4 - 8 - 12 = -16 


y = 4 x 2 + 8x - 12 → y = 4 x 2 + 8x + (-12) 


(0, -12). 

Let x = 1. 

y = 4 x 2 + 8x - 12 

= 4 (1) 2 + 8(1) - 12 

= 4(1) + 8 - 12 

= 4 + 8 - 12 = 0 


Let x = 2. 

y = 4 x 2 + 8x - 12 

= 4 (2) 2 + 8(2) - 12 

= 4(4) + 16 - 12 

= 16 + 16 - 12 = 20 




 

 

 

 

 

 



27. a = -5, b = 20 

b 

x = -___ 

2a = - _____ 20 

2(-5) = - ____ 20 

-10 = 2 

y = -5 x 2 + 20x 

= -5 (2) 2 + 20(2) 

= -5(4) + 40 

= -20 + 40 = 20 


y = -5 x 2 + 20x → y = -5 x 2 + 20x + (0) 


(0, 0). 

Let x = 1. 

y = -5 x 2 + 20x 

= -5 (1) 2 + 20(1) 

= -5(1) + 20 

= -5 + 20 = 15 

Another point is (1, 15). 




 

 

 

 

 

 

 

 

The vertex is (2, 20). So at 2 s, the drop of water 

reaches its maximum height of 20 m. The graph 

shows the zeros of the function are 0 and 4. At 0 s, 

the drop of water has not left the sprinkler and at 4 s 

it reaches the ground. The drop of water takes 4 s to 

reach the ground. 


28. ⎪2⎥ = 2 ⎪4⎥ = 4 

g(x) = 4 x 2 , f(x) = 2 x 2 

29. 6 = 6 -6 = 6 

Since the absolute values are equal, the graphs 

have the same width. 

30. ⎪1⎥ = 1 ⎪ 1__ 

3⎥ = ⎪ 1__ 

3⎥ 3 = 3 

h(x) = 3 x 2 , f(x) = x 2 , g(x) = 1__ 

3 x 2 


axis of symmetry, and also opens upward, but its 

vertex is translated 5 units up to (0, 5). 

32. The graph of g(x) has the same axis of symmetry, 

and also opens upward, but it is narrower, and its 

vertex is translated 1 unit down to (0, -1). 

33. The graph of g(x) is narrower, has the same axis of 





x = -2. The vertex is (-2, -1). The y-intercept is 3. 



 

 

 

The zeros appear to be -3 and -1. 





 

The only zero appears to be -3. 

36. -4 x 2 = 3 

_____ +4 x 2 _____ +4 x 2 

0 = 4 x 2 + 3 

Graph y = 4 x 2 + 3. The axis of symmetry is x = 0. 

The vertex is (0, 3). Another point is (1, 7). Graph 


symmetry. 

 

The function appears to have no zeros. 

37. x 2 + 5 = 6x 

_______ - x 2 ____ - x 2 

5 = - x 2 + 6x 

___ -5 ________ 

-5 

0 = - x 2 + 6x - 5 

 

 

 

Graph y = - x 2 + 6x - 5. The axis of symmetry is 

x = 3. The vertex is (3, 4). The y-intercept is -5. 



 

 

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



38. -4 x 2 = 64 - 32x 

_____ +4 x 2 _________ 

+4 x 2 

0 = 4 x 2 - 32x + 64 

Graph y = 4 x 2 - 32x + 64. The axis of symmetry is 

x = 4. The vertex is (4, 0). The y-intercept is 64. 




39. 9 = 9 x 2 

___ -9 ____ -9 

0 = 9 x 2 - 9 

 

 

 

 

 

 

 

Graph y = 9 x 2 - 9. The axis of symmetry is 

x = 0. The vertex is (0, -9). Another point is (2, 27). 


symmetry. 

 

 

 

The zeros appear to be -1 and 1. 

40. -3 x 2 + 2x = 5 

_____ +3 x 2 _____ +3 x 2 

2x = 3 x 2 + 5 

____ -2x ________ 

-2x 

0 = 3 x 2 - 2x + 5 

 

Graph y = 3 x 2 - 2x + 5. The axis of symmetry is 

x = 1__ 

3 . The vertex is ( 1__ 

3 , 4 2__ . The y-intercept is 5. 

3) 

Another point is (-1, 10). Graph the points and 


 

The function appears to have no zeros. 


41. x 2 + 6x + 5 = 0 

(x + 5)(x + 1) = 0 

x + 5 = 0 or x + 1 = 0 

x = -5 or x = -1 


 

 

 

 

 

 

 

 

 

 

 

42. x 2 + 9x + 14 = 0 

(x + 7)(x + 2) = 0 

x + 7 = 0 or x + 2 = 0 

x = -7 or x = -2 


43. x 2 - 2x - 15 = 0 

(x + 3)(x - 5) = 0 

x + 3 = 0 or x - 5 = 0 

x = -3 or x = 5 


44. 2 x 2 - 2x - 4 = 0 

2 ( x 2 - x - 2) = 0 

2(x + 1)(x - 2) = 0 

2 ≠ 0, x + 1 = 0 or x - 2 = 0 

x = -1 or x = 2 


45. x 2 + 10x + 25 = 0 

(x + 5)(x + 5) = 0 

x + 5 = 0 

x = -5 


46. 4 x 2 - 36x = -81 

_________ + 81 ____ +81 

4 x 2 - 36x + 81 = 0 

(2x - 9)(2x - 9) = 0 

2x - 9 = 0 

2x = 9 

x = 4.5 

The only solution is 4.5. 

47. Let x represent the width of the rectangle. 

A = lw 

48 = (x + 2)(x) 

48 = x 2 + 2x 

____ -48 _______ -48 

0 = x 2 + 2x - 48 

0 = (x + 8)(x - 6) 

x + 8 = 0 or x - 6 = 0 

x = -8 ✗ or x = 6 

The only solution is 6. The width of the rectangle is 

6 ft. 


48. 5 x 2 = 320 

___ 5 x 2 

5 = ____ 320 

5 

x 2 = 64 

x = ± √ 64 

x = ±8 


49. - x 2 + 144 = 0 

_________ 

+ x 2 ____ + x 2 

144 = x 2 

± √ 144 = x 

±12 = x 




50. x 2 = -16 

x ≠ ± √ 16 


51. x 2 + 7 = 7 

_____ - 7 -7 ___ 

x 2 = 0 

x = ± √ 0 

x = 0 


52. 2 x 2 = 50 

___ 2 x 2 

2 = ___ 50 

2 

x 2 = 25 

x = ± √ 25 

x = ±5 


53. 4 x 2 = 25 

___ 4 x 2 

4 = ___ 25 

4 

x 2 = ___ 25 

4 

x = ± √ ___ 25 

4 

x = ± 

5__ 

2 


2 and - 5__ 

2 . 

54. A = lw 

32 = (2w)w 

32 = 2 w 2 

___ 

____ 

32 

2 = 2 w 2 

2 

16 = w 2 

± √ 16 = w 

±4 = w 


so w = 4 is the only solution that makes sense. 

Therefore, the width of the rectangle is 4 ft. 


55. b = 2 

( 2__ 2) 2 = 1 2 = 1 

x 2 + 2x = 48 

x 2 + 2x + 1 = 48 + 1 

(x + 1) 2 = 49 

x + 1 = ±7 

x + 1 = 7 or x + 1 = -7 

x = 6 or x = -8 


56. b = 4 

( 4__ 2) 2 = 2 2 = 4 

x 2 + 4x = 21 

x 2 + 4x + 4 = 21 + 4 

(x + 2) 2 = 25 

x + 2 = ±5 

x + 2 = 5 or x + 2 = -5 

x = 3 or x = -7 


57. 2 x 2 - 12x + 10 = 0 

___ 2 x 2 

2 - ____ 12x 

2 + ___ 10 

2 = 0 

x 2 - 6x + 5 = 0 

x 2 - 6x = -5 

b = 6 

( ___ -6 

2 ) 2 = (-3) 2 = 9 

x 2 - 6x = -5 

x 2 - 6x + 9 = -5 + 9 

(x - 3) 2 = 4 

x - 3 = ±2 

x - 3 = 2 or x - 3 = -2 

x = 5 or x = 1 


58. b = -10 

( ____ -10 

2 ) 2 = (-5) 2 = 25 

x 2 - 10x = -20 

x 2 - 10x + 25 = -20 + 25 

(x - 5) 2 = 5 

x - 5 = ± √ 5 

x - 5 = √ 5 or x - 5 = - √ 5 

x = 5 + √ 5 or x = 5 - √ 5 


59. Let x represent the width of the room. 

lw = A 

(x + 4)(x) = 192 

x 2 + 4x = 192 

b = 4 

( 4__ 2) 2 = 2 2 = 4 

x 2 + 4x = 192 

x 2 + 4x + 4 = 192 + 4 

(x + 2) 2 = 196 

x + 2 = ±14 

x + 2 = 14 or x + 2 = -14 

x = 12 or x = -16 


x = 12 is the only solution that makes sense. 

The width is 12 ft, and the length is 12 + 4 = 16 ft. 

The dimensions of the room are 16 ft by 12 ft. 




60. a = 1, b = -5, c = -6 

x = 

-(-5) ± √ 

 

_______________________ 

(-5) 2 - 4(1)(-6) 

2(1) 

_______________ 

x = 5 ± √ 

25 - (-24) 

2 

________ 

x = 5 ± √ 49 

2 

x = _____ 5 ± 7 

2 

x = _____ 5 + 7 or x = _____ 5 - 7 

2 

2 

x = 6 or x = -1 

61. a = 2, b = -9, c = -5 

x = 

-(-9) ± √ 

 

_______________________ 

(-9) 2 - 4(2)(-5) 

2(2) 

_______________ 

x = 9 ± √ 

81 - (-40) 

4 

_________ 

x = 9 ± √ 121 

4 

x = ______ 9 ± 11 

4 

x = ______ 9 + 11 or x = ______ 9 - 11 

4 

4 

x = 5 or x = - 

1__ 

2 

62. a = 4, b = -8, c = 4 

x = 

-(-8) ± √ 

 

______________________ 

(-8) 2 - 4(4)(4) 

2(4) 

____________ 64 - 64 

x = 8 ± √ 

8 

x = 

_______ 8 ± √ 0 

8 

x = _____ 8 ± 0 

8 

x = 8__ 

8 

x = 1 

63. x 2 - 6x = -7 

x 2 - 6x + 7 = 0 

a = 1, b = -6, c = 7 

x = 

-(-6) ± √ 

 

______________________ 

(-6) 2 - 4(1)(7) 

2(1) 

____________ 36 - 28 

x = 6 ± √ 

2 

x = 

_______ 6 ± √ 8 

2 

64. a = 1, b = -12, c = 36 

b 2 - 4ac 

= (-12) 2 - 4(1)(36) 

= 144 - 144 

= 0 



solution. 

65. a = 3, b = 0, c = 5 

b 2 - 4ac 

= 0 2 - 4(3)(5) 

= 0 - 60 

= -60 



solutions. 

66. 2 x 2 - 13x = -20 

2 x 2 - 13x + 20 = 0 

a = 2, b = -13, c = 20 

b 2 - 4ac 

= (-13) 2 - 4(2)(20) 

= 169 - 160 

= 9 



67. 6 x 2 - 20 = 15x + 1 

6 x 2 - 15x - 20 = 1 

6 x 2 - 15x - 21 = 0 

a = 6, b = -15, c = -21 

b 2 - 4ac 

= (-15) 2 - 4(6)(-21) 

= 225 + 504 

= 729 



CHAPTER TEST, PAGE 666 


are constant. 

2. 3 x 2 + y = 4 + 3 x 2 

________ -3 x 2 _______ - 3 x 2 

y = 4 


a is 0. 

3. y = -2 x 2 + 7x - 5 

a = -2 


Therefore, the parabola has a maximum. 


5. ________ -10 + 0 = ____ -10 

2 2 = -5 


6. a = 1, b = 6 

b 

x = -___ 

2a = - ____ 6 6__ 

2(1) = - 2 = -3 

y = x 2 + 6x + 8 

= (-3) 2 + 6(-3) + 8 

= 9 - 18 + 8 = -1 




7. a = 1, b = -4 

x = -____ 

-4 

2(1) = 4__ 

2 = 2 

y = x 2 - 4x + 2 

= (2) 2 - 4(2) + 2 

= 4 - 8 + 2 = -2 


y = x 2 - 4x + 2 → y = x 2 - 4x + (2) 


(0, 2) 

Let x = -1. 

y = x 2 - 4x + 2 

= (-1) 2 - 4(-1) + 2 

= 1 + 4 + 2 = 7 

Let x = 1. 

y = x 2 - 4x + 2 

= (1) 2 - 4(1) + 2 

= 1 - 4 + 2 = -1 

Two other points are (-1, 7) and (1, -1). 




 

 

 

 

 




9. The graph of h(x) is wider, has the same axis 

of symmetry, opens upward, and its vertex is 

translated 1 unit up to (0, 1). 


of symmetry, opens upward, and its vertex is 


11a. h(t) = -16 t 2 + 40; h(t) = -16 t 2 + 60 

The graph of the hammer that is dropped from a 

height of 40 ft is a vertical translation of the graph 

of the hammer that is dropped from a height of 

60 ft. The y-intercept of the graph of the hammer 

that is dropped from 60 ft is 20 units higher. 

b. about 1.6 s and about 1.9 s 

12. Graph y = -5 x 2 + 110x. 

 

 

 

 

 

 

 

 

 

 

 

 

 


The rocket leaves the ground at 0 s and returns to 

the ground at 22 s. 

The rocket takes about 22 s to return to the ground. 

 

 

13. x 2 + 6x + 5 = 0 

(x + 5)(x + 1) = 0 

x + 5 = 0 or x + 1 = 0 

x = -5 or x = -1 


14. x 2 - 12x = -36 

________ + 36 ____ +36 

x 2 - 12x + 36 = 0 

(x - 6)(x - 6) = 0 

x - 6 = 0 

x = 6 


15. x 2 - 81 = 0 

(x + 9)(x - 9) = 0 

x + 9 = 0 or x - 9 = 0 

x = -9 or x = 9 


16. -2 x 2 = -72 

_____ -2 x 2 

-2 = ____ -72 

-2 

x 2 = 36 

x = ± √ 36 

x = ±6 


17. 9 x 2 - 49 = 0 

_______ + 49 ____ +49 

___ 9 x 2 

9 = ___ 49 

9 

x 2 = ___ 49 

9 

x = ± 

√ ___ 49 

9 

x = ± 

7__ 

3 


3 and - 7__ 

3 . 

18. 3 x 2 + 12 = 0 

_______ - 12 ____ -12 

___ 3 x 2 

3 = ____ -12 

3 

x 2 = -4 

x ≠ ± √ 4 


19. b = 10 

( ___ 10 

2 ) 2 = 5 2 = 25 

x 2 + 10x = -21 

x 2 + 10x + 25 = -21 + 25 

(x + 5) 2 = 4 

x + 5 = ±2 

x + 5 = 2 or x + 5 = -2 

x = -3 or x = -7 




20. b = -6 

( ___ -6 

2 ) 2 = (-3) 2 = 9 

x 2 - 6x + 4 = 0 

x 2 - 6x = -4 

x 2 - 6x + 9 = -4 + 9 

(x - 3) 2 = 5 

x - 3 = ± √ 5 

x - 3 = √ 5 or x - 3 = - √ 5 

x = 3 + √ 5 or x = 3 - √ 5 


21. 2 x 2 + 16x = 0 

___ 2 x 2 

2 + ____ 16x 

2 = 0 

x 2 + 8x = 0 

b = 8 

( 8__ 

2) 2 = 4 2 = 16 

x 2 + 8x = 0 

x 2 + 8x + 16 = 0 + 16 

(x + 4) 2 = 16 

x + 4 = ±4 

x + 4 = 4 or x + 4 = -4 

x = 0 or x = -8 


22. Let x represent the width of the patio. 

lw = A 

(x + 6)(x) = 150 

x 2 + 6x = 150 

b = 6 

( 6__ 

2) 2 = 3 2 = 9 

x 2 + 6x = 150 

x 2 + 6x + 9 = 150 + 9 

(x + 3) 2 = 159 

x + 3 ≈ ±12.6 

x + 3 ≈ 12.6 or x + 3 ≈ -12.6 

x ≈ 9.6 or x ≈ -15.6 



The width is 9.6 ft, and the length is 

9.6 + 6 = 15.6 ft. The dimensions of the patio are 

about 15.6 ft by 9.6 ft. 

23. a = 1, b = 3, c = -40 

x = 

-3 ± √ 

 

___________________ 

3 2 - 4(1)(-40) 

2(1) 

________________ 

x = -3 ± √ 

9 - (-160) 

2 

__________ 

-3 ± √ 

x = 169 

2 

-3 ± 13 

x = ________ 

2 

-3 + 13 

x = ________ or x = 

2 

x = 5 or x = -8 

________ -3 - 13 

2 

24. 2 x 2 + 7x = -5 

2 x 2 + 7x + 5 = 0 

a = 2, b = 7, c = 5 

x = 

-7 ± √ 

 

_________________ 

7 2 - 4(2)(5) 

2(2) 

______________ 

-7 ± √ 49 - 40 

x = 

4 

x = 

________ -7 ± √ 9 

4 

x = _______ -7 ± 3 

4 

x = _______ -7 + 3 

4 

x = -1 or x = - 

5__ 

2 

or x = _______ -7 - 3 

4 

25. a = 8, b = 3, c = -1 

x = 

-3 ± √ 

 

__________________ 

3 2 - 4(8)(-1) 

2(8) 

_______________ 

x = -3 ± √ 9 - (-32) 

16 

_________ 

-3 ± √ 

x = 41 

16 

-3 + √ 

x = 

_________ 41 or x = 

_________ 

-3 - √ 41 

16 

16 

x ≈ 0.21 or x ≈ -0.59 

26. a = 4, b = -4, c = 1 

b 2 - 4ac 

= (-4) 2 - 4(4)(1) 

= 16 - 16 

= 0 



solution. 

28. a = 1__ 

2 , b = 0, c = 8 

b 2 - 4ac 

= 0 2 - 4 ( 1__ 2) (8) 

= 0 - 16 

= -16 



27. a = 2, b = 5, c = -25 

b 2 - 4ac 

= 5 2 - 4(2)(-25) 

= 25 + 200 

= 225 



solutions. 

COLLEGE ENTRANCE EXAM PRACTICE, 

PAGE 667 

1. D; since the parabola opens downward, it is 

necessary that a < 0, so A is incorrect. The vertex is 

b 

(-2, 9). Since for B and D -___ 

2a = - _____ -4 

2(-1) = -2, B 

and D have their vertex with an x-value -2 but, C 

and E do not. So it must be B or D. Since for D, 

f(-2) = -(-2) 2 - 4(-2) + 5 = 9, D is correct. 



2. B; rearranging and factoring gives 

(3x - 4)(3x + 2) = 0. So the two solutions are 4__ 

3 

and - 

2__ 

. The sum is therefore 

2__ 

3 3 . 

3. A; since a < 0, the parabola opens downward and 

since b 2 - 4ac < 0, it does not intersect the x-axis. 

Therefore, the graph is below the x-axis so it does 

not have points in the first or second quadrants, so 

it also does not have points in all quadrants. So I is 

true and III is false. Since the graph intersects the 

y-axis at c, it will have point in both quadrant 3 and 

4 so II is incorrect. Therefore A is correct. 

4. C; the axis of symmetry is the average of the zeros 

or x = _______ -2 + 4 = 2__ 

2 2 = 1. 

5. C; since a = 1, b = -7, c = 1, then 

b 2 - 4ac = (-7) 2 - 4(1)(1) = 45 > 0, so there are 

two real solutions.

Answers to Holt Chapter 9

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