Algebra Cheat Sheet (Reduced) - Pauls Online Math Notes
Algebra Cheat Sheet (Reduced) - Pauls Online Math Notes
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