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Algebra Cheat Sheet (Reduced) - Pauls Online Math Notes

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<strong>Algebra</strong> <strong>Cheat</strong> <strong>Sheet</strong><br />

Basic Properties & Facts<br />

Arithmetic Operations<br />

Properties of Inequalities<br />

æbö<br />

ab If a < b thena+ c< b+ c and a- c< b-c<br />

ab+ ac= ab ( + c)<br />

aç ÷ =<br />

è c ø c<br />

a b<br />

If a < b and c > 0 then ac< bc and <<br />

æaö<br />

c c<br />

ç ÷<br />

èb<br />

ø a a ac<br />

a b<br />

= =<br />

If a < b and c< 0 then ac> bc and ><br />

c bc æb<br />

ö b<br />

c c<br />

ç<br />

c<br />

÷<br />

è ø<br />

Properties of Absolute Value<br />

a c ad + bc a c ad -bc<br />

ìa<br />

if a³<br />

0<br />

+ = - =<br />

a =í<br />

b d bd b d bd<br />

î - a if a < 0<br />

a-b b- a a+<br />

b a b a ³ 0<br />

- a = a<br />

= = +<br />

c-d d -c c c c<br />

a a<br />

ab = ab<br />

=<br />

æaö<br />

b b<br />

ab+ ac<br />

ç<br />

b<br />

÷<br />

ad<br />

= b+ c, a ¹ 0<br />

è ø<br />

=<br />

a+ b £ a + b Triangle Inequality<br />

a<br />

æ c ö bc<br />

ç<br />

d<br />

÷<br />

è ø<br />

Distance Formula<br />

Exponent Properties<br />

If P<br />

n<br />

1<br />

= ( x1,<br />

y1)<br />

and P2 = ( x2,<br />

y2)<br />

are two<br />

n m n+ m a n-m<br />

1<br />

aa = a = a = points the distance between them is<br />

m<br />

m-n<br />

a a<br />

n<br />

( )<br />

m<br />

nm<br />

0<br />

a a a a<br />

( ab)<br />

a<br />

-n<br />

= = 1, ¹ 0<br />

n n n æaö<br />

a<br />

= ab<br />

ç ÷ =<br />

èbø<br />

b<br />

1 1 n<br />

= = a<br />

n<br />

-n<br />

a<br />

a<br />

-n n n<br />

n<br />

n<br />

m<br />

n<br />

n<br />

n<br />

n<br />

1<br />

m<br />

( ) ( ) 1<br />

m n<br />

æaö æbö<br />

b<br />

ç ÷ = ç ÷ = a = a = a<br />

èbø è aø<br />

a<br />

Properties of Radicals<br />

1<br />

n n<br />

n n n<br />

m n nm n<br />

n<br />

n<br />

a = a ab = a b<br />

a<br />

a = a =<br />

b<br />

n<br />

a = a,if<br />

n is odd<br />

n<br />

a = a ,if n is even<br />

n<br />

n<br />

a<br />

b<br />

( , ) = ( - ) + ( - )<br />

2 2<br />

1 2 2 1 2 1<br />

d P P x x y y<br />

Complex Numbers<br />

i = - i =- - a = i a a³<br />

( a+ bi) + ( c+ di) = a+ c+ ( b+<br />

d)<br />

i<br />

( a+ bi) -( c+ di) = a- c+ ( b-d)<br />

i<br />

( + )( + ) = - + ( + )<br />

2 2<br />

( a+ bi)( a- bi)<br />

= a + b<br />

( )<br />

2<br />

1 1 , 0<br />

a bi c di ac bd ad bc i<br />

a+ bi = a + b<br />

2 2<br />

a+ bi = a-bi<br />

( )( )<br />

a+ bi a+ bi = a+<br />

bi<br />

Complex Modulus<br />

Complex Conjugate<br />

2<br />

Logarithms and Log Properties<br />

Definition<br />

y<br />

y= log x is equivalent to x=<br />

b<br />

b<br />

Example<br />

3<br />

log 125= 3 because 5 = 125<br />

5<br />

Special Logarithms<br />

lnx=<br />

log x natural log<br />

e<br />

logx<br />

= log10<br />

x common log<br />

where e= 2.718281828K<br />

Factoring Formulas<br />

2 2<br />

x - a = x+ a x-a<br />

( )( )<br />

( )<br />

2 2<br />

2<br />

x + 2ax+ a = x+<br />

a<br />

2<br />

2 ( )<br />

( ) ( )( )<br />

3<br />

3 3 ( )<br />

- + = -<br />

2 2<br />

x ax a x a<br />

+ + + = + +<br />

2<br />

x a b x ab x a x b<br />

3 2 2 3<br />

x + ax + ax+ a = x+<br />

a<br />

3 2 2 3<br />

3<br />

x - 3ax + 3ax- a = x-a<br />

( )<br />

( )( )<br />

( )( )<br />

( )( )<br />

x + a = x+ a x - ax+<br />

a<br />

3 3 2 2<br />

x - a = x- a x + ax+<br />

a<br />

3 3 2 2<br />

x - a = x - a x + a<br />

2n 2n n n n n<br />

If n is odd then,<br />

x - a = x- a x + ax + L+<br />

a<br />

x<br />

-1 -2 -1<br />

( )( )<br />

n n n n n<br />

n<br />

+ a<br />

n<br />

n-1 n-2 2 n-3 n-1<br />

( x a)( x ax ax L a )<br />

= + - + - +<br />

2<br />

Solve 2x<br />

-6x- 10=<br />

0<br />

2<br />

(1) Divide by the coefficient of the x<br />

2<br />

x -3x- 5=<br />

0<br />

(2) Move the constant to the other side.<br />

2<br />

x - 3x=<br />

5<br />

(3) Take half the coefficient of x, square<br />

it and add it to both sides<br />

2 2<br />

2 æ 3ö æ 3ö<br />

9 29<br />

x - 3x+ ç - ÷ = 5+ ç - ÷ = 5+ =<br />

è 2ø è 2ø<br />

4 4<br />

Logarithm Properties<br />

log b= 1 log 1=<br />

0<br />

log<br />

log<br />

b<br />

x<br />

logb<br />

x<br />

bb = x b = x<br />

b<br />

r<br />

( )<br />

x = rlog<br />

x<br />

( )<br />

log xy = log x+<br />

log y<br />

b b b<br />

æ xö logbç ÷ = logb x - logb<br />

y<br />

è yø<br />

The domain of log b<br />

x is x> 0<br />

Factoring and Solving<br />

Quadratic Formula<br />

2<br />

Solve ax + bx+ c= 0 , a ¹ 0<br />

If b<br />

If b<br />

If b<br />

2<br />

2<br />

2<br />

b<br />

b<br />

2<br />

- b± b -4ac<br />

x =<br />

2a<br />

- 4ac<br />

> 0 - Two real unequal solns.<br />

- 4ac<br />

= 0 - Repeated real solution.<br />

- 4ac<br />

< 0 - Two complex solutions.<br />

Square Root Property<br />

2<br />

If x = p then x=±<br />

p<br />

Absolute Value Equations/Inequalities<br />

If b is a positive number<br />

p = b Þ p=- b or p=<br />

b<br />

p < b Þ - b< p<<br />

b<br />

p > b Þ p<br />

b<br />

Completing the Square<br />

(4) Factor the left side<br />

æ 3ö<br />

29<br />

ç x- ÷ =<br />

è 2ø<br />

4<br />

(5) Use Square Root Property<br />

3 29 29<br />

x- =± =±<br />

2 4 2<br />

(6) Solve for x<br />

2<br />

3 29<br />

x= ±<br />

2 2<br />

For a complete set of online <strong>Algebra</strong> notes visit http://tutorial.math.lamar.edu.<br />

© 2005 Paul Dawkins<br />

For a complete set of online <strong>Algebra</strong> notes visit http://tutorial.math.lamar.edu.<br />

© 2005 Paul Dawkins


Functions and Graphs<br />

Constant Function<br />

Parabola/Quadratic Function<br />

y= a or f ( x)<br />

= a<br />

2<br />

x= ay + by+ c<br />

2<br />

g( y)<br />

= ay + by+<br />

c<br />

Graph is a horizontal line passing<br />

through the point ( 0,a ) .<br />

The graph is a parabola that opens right<br />

if a > 0 or left if a < 0 and has a vertex<br />

Line/Linear Function<br />

æ æ b ö b ö<br />

at ç gç - ,<br />

y= mx+ b or f ( x)<br />

= mx+<br />

b<br />

2a<br />

÷ -<br />

2a<br />

÷<br />

è è ø ø .<br />

Graph is a line with point ( 0,b ) and<br />

Circle<br />

slope m.<br />

( x- h) 2 + ( y- k)<br />

2 = r<br />

2<br />

Slope<br />

Slope of the line containing the two<br />

Graph is a circle with radius r and center<br />

( hk , ) .<br />

points ( x1,<br />

y<br />

1)<br />

and ( x2,<br />

y<br />

2)<br />

is<br />

y2 - y1<br />

rise<br />

Ellipse<br />

m= = x<br />

2<br />

- x<br />

1<br />

run<br />

( x-h) 2 ( y-k)<br />

2<br />

+ = 1<br />

2 2<br />

Slope – intercept form<br />

a b<br />

The equation of the line with slope m Graph is an ellipse with center ( hk , )<br />

and y-intercept ( 0,b ) is<br />

with vertices a units right/left from the<br />

y = mx+<br />

b<br />

Point – Slope form<br />

center and vertices b units up/down from<br />

the center.<br />

The equation of the line with slope m<br />

and passing through the point ( x1,<br />

y<br />

1)<br />

is Hyperbola<br />

y= y1+ m( x-<br />

x1)<br />

( x-h) 2 ( y-k)<br />

2<br />

- = 1<br />

2 2<br />

a b<br />

Graph is a hyperbola that opens left and<br />

Parabola/Quadratic Function<br />

2 2 right, has a center at<br />

y= a( x- h) + k f ( x) = a( x- h)<br />

( hk , ) , vertices a<br />

+ k<br />

units left/right of center and asymptotes<br />

b<br />

The graph is a parabola that opens up if that pass through center with slope ± .<br />

a > 0 or down if a < 0 and has a vertex<br />

a<br />

Hyperbola<br />

at ( hk , ) .<br />

( y-k) 2 ( x-h)<br />

2<br />

- = 1<br />

2 2<br />

Parabola/Quadratic Function<br />

b a<br />

2 2<br />

y = ax + bx+ c f ( x)<br />

= ax + bx+<br />

c<br />

Graph is a hyperbola that opens up and<br />

hk , , vertices b<br />

The graph is a parabola that opens up if<br />

a > 0 or down if a < 0 and has a vertex<br />

æ b æ b öö<br />

at ç - , f ç - ÷<br />

2a<br />

2a<br />

÷ .<br />

è è øø<br />

down, has a center at ( )<br />

units up/down from the center and<br />

asymptotes that pass through center with<br />

b<br />

slope ± .<br />

a<br />

Common <strong>Algebra</strong>ic Errors<br />

Error<br />

Reason/Correct/Justification/Example<br />

2<br />

0<br />

0 ¹ and 2 ¹ 2<br />

Division by zero is undefined!<br />

0<br />

2<br />

- 3 ¹ 9<br />

( x<br />

2 ) 3<br />

¹ x<br />

5<br />

( ) 3<br />

a a a<br />

¹ +<br />

b+<br />

c b c<br />

1<br />

¹ x + x<br />

2 3<br />

x + x<br />

-2 -3<br />

a + bx<br />

¹ 1+<br />

bx<br />

a<br />

2<br />

- 3 =- 9, (- 3) 2<br />

= 9 Watch parenthesis!<br />

x = xxx = x<br />

a( x 1)<br />

( )<br />

- - ¹-ax- a<br />

2 2 2 2 6<br />

1 1 1 1<br />

= ¹ + = 2<br />

2 1+<br />

1 1 1<br />

A more complex version of the previous<br />

error.<br />

a+<br />

bx a bx bx<br />

= + = 1+<br />

a a a a<br />

Beware of incorrect canceling!<br />

-a x- 1 =- ax+<br />

a<br />

Make sure you distribute the “-“!<br />

( ) 2 2 2<br />

x+ a ¹ x + a<br />

( ) ( )( )<br />

2 2<br />

x + a ¹ x+<br />

a<br />

2 2 2<br />

x+ a = x+ a x+ a = x + 2ax+<br />

a<br />

2 2 2 2<br />

5= 25 = 3 + 4 ¹ 3 + 4 = 3+ 4 = 7<br />

x+ a ¹ x+ a<br />

See previous error.<br />

+ ¹ + and n x a n x n More general versions of previous three<br />

+ ¹ + a<br />

errors.<br />

( ) n n n<br />

x a x a<br />

( x+ ) ¹ ( x+<br />

)<br />

2 2<br />

2 1 2 2<br />

( 2x+ 2) ¹ 2( x+<br />

1)<br />

2 2<br />

( ) ( )<br />

2 2 2<br />

2 x+ 1 = 2 x + 2x+ 1 = 2x + 4x+<br />

2<br />

( ) 2 2<br />

2x+ 2 = 4x + 8x+<br />

4<br />

Square first then distribute!<br />

See the previous example. You can not<br />

factor out a constant if there is a power on<br />

the parethesis!<br />

2 2 2 2<br />

- x + a ¹- x + a<br />

( ) 1<br />

- x 2 + a 2 = - x 2 + a<br />

2 2<br />

Now see the previous error.<br />

æaö<br />

a ab<br />

¹<br />

a<br />

ç<br />

1<br />

÷<br />

æaöæcö<br />

ac<br />

æb<br />

ö c<br />

=<br />

è ø<br />

=<br />

ç<br />

c<br />

÷<br />

b b<br />

ç ÷ç ÷ =<br />

æ ö æ ö è1øèbø<br />

b<br />

è ø<br />

ç ÷ ç ÷<br />

ècø ècø<br />

æaö æa<br />

ö<br />

æaö<br />

ç<br />

ç ÷<br />

b<br />

÷ ç<br />

b<br />

÷<br />

è ø æaöæ1ö<br />

a<br />

èb<br />

ø ac<br />

=<br />

è ø<br />

= ¹ c c<br />

ç<br />

b<br />

֍<br />

c<br />

÷ =<br />

æ ö è øè ø bc<br />

c b<br />

ç ÷<br />

è1ø<br />

For a complete set of online <strong>Algebra</strong> notes visit http://tutorial.math.lamar.edu.<br />

© 2005 Paul Dawkins<br />

For a complete set of online <strong>Algebra</strong> notes visit http://tutorial.math.lamar.edu.<br />

© 2005 Paul Dawkins

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