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Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations
b ab ab ac a b c a c c
a
b a^ a^ ac
c bc b b
c
a c ad bc a c ad bc
b d bd b d bd
a b b a a b a b
c d d c c c c
a
ab ac (^) b ad b c a a c bc
d
Exponent Properties
( ) (^ )^
1 1
0
n (^) m m m
n n m n m n m m m n
n m nm
n (^) n n (^) n n
n
n n n n
n n (^) n n n n
a a a a a a a
a a a a
a a ab a b b b
a a a a
a b b a a a b a a
− −
−
=^ =^ =^ =
Properties of Radicals
1
, if is odd
, if is even
n (^) n n n n
n m n nm (^) n
n
n n
n n
a a ab a b
a a a a b (^) b
a a n
a a n
Properties of Inequalities
If then and
If and 0 then and
If and 0 then and
a b a c b c a c b c
a b a b c ac bc c c
a b a b c ac bc c c
Properties of Absolute Value
if 0
if 0
a a a a a
^ ≥
−^ <
Triangle Inequality
a a a
a a ab a b b b
a b a b
Distance Formula
If P 1 = ( x 1 , y 1 )and P 2 = ( x 2 , y 2 )are two
points the distance between them is
2 2 d P P 1 , 2 (^) = x 2 (^) − x 1 (^) + y 2 (^) − y 1
Complex Numbers
2
2 2
2 2
2
Complex Modulus
Complex Conjugate
i i a i a a
a bi c di a c b d i
a bi c di a c b d i
a bi c di ac bd ad bc i
a bi a bi a b
a bi a b
a bi a bi
a bi a bi a bi
Logarithms and Log Properties
Definition
log is equivalent to
y y = (^) bx x = b
Example
3 log 125 5 = 3 because 5 = 125
Special Logarithms
10
ln log natural log
log log common log
x (^) ex
x x
where e = 2.718281828K
Logarithm Properties
( )
( )
log
log 1 log 1 0
log
log log
log log log
log log log
b
b b
x x b
r b b
b b b
b b b
b
b x b x
x r x
xy x y
x x y y
=^ −
The domain of log b x is x > 0
Factoring and Solving
Factoring Formulas
( ) ( )
( )
( )
( ) ( ) ( )
2 2
2 2 2
2 2 2
2
x a x a x a
x ax a x a
x ax a x a
x a b x ab x a x b
( )
( )
( )( )
( ) (^) ( )
3 2 2 3 3
3 2 2 3 3
3 3 2 2
3 3 2 2
x ax a x a x a
x ax a x a x a
x a x a x ax a
x a x a x ax a
( )( )
2 n 2 n n n n n x − a = x − a x + a
If n is odd then,
( ) ( )
( )( )
1 2 1
1 2 2 3 1
n n n n n
n n
n n n n
x a x a x ax a
x a
x a x ax a x a
− − −
− − − −
L
L
Quadratic Formula
Solve
2 ax + bx + c = 0 , a ≠ 0
2 4
b b ac x a
If
2 b − 4 ac > 0 - Two real unequal solns.
If
2 b − 4 ac = 0 - Repeated real solution.
If
2 b − 4 ac < 0 - Two complex solutions.
Square Root Property
If
2 x = p then x = ± p
Absolute Value Equations/Inequalities
If b is a positive number
or
or
p b p b p b
p b b p b
p b p b p b
Completing the Square
Solve
2 2 x − 6 x − 10 = 0
(1) Divide by the coefficient of the
2 x 2 x − 3 x − 5 = 0
(2) Move the constant to the other side.
2 x − 3 x = 5
(3) Take half the coefficient of x , square
it and add it to both sides
2 2 2 3 3 9 29 3 5 5 2 2 4 4
x x
(4) Factor the left side
2 3 29
x
−^ =
(5) Use Square Root Property
x − = ± = ±
(6) Solve for x
x = ±
Common Algebraic Errors
Error Reason/Correct/Justification/Example
≠ and
≠ (^) Division by zero is undefined!
2 − 3 ≠ 9
2 − 3 = − 9 , (^) ( )
2 − 3 = 9 Watch parenthesis!
( )
2 3 5 x ≠ x ( )
2 3 2 2 2 6 x = x x x = x
a a a
b c b c
2 3 2 3
x x x x
− − ≠ +
A more complex version of the previous
error.
a bx
a
≠ 1 + bx
a bx a bx bx
a a a a
Beware of incorrect canceling!
− a x ( − (^1) )≠ − ax − a
− a x (^) ( − (^1) )= − ax + a
Make sure you distribute the “-“!
( )
(^2 2 ) x + a ≠ x + a ( ) ( ) ( )
(^2 2 ) x + a = x + a x + a = x + 2 ax + a
2 2 x + a ≠ x + a
2 2 2 2 5 = 25 = 3 + 4 ≠ 3 + 4 = 3 + 4 = 7
x + a ≠ x + a See previous error.
( )
n (^) n n x + a ≠ x + a and
n n n x + a ≠ x + a
More general versions of previous three
errors.
( ) ( )
2 2 2 x + 1 ≠ 2 x + 2
( ) (^) ( )
(^2 2 ) 2 x + 1 = 2 x + 2 x + 1 = 2 x + 4 x + 2
( )
(^2 ) 2 x + 2 = 4 x + 8 x + 4
Square first then distribute!
( ) ( )
2 2 2 x + 2 ≠ 2 x + 1
See the previous example. You can not
factor out a constant if there is a power on
the parethesis!
2 2 2 2 − x + a ≠ − x + a
( )
1 (^2 2 2 2 ) − x + a = − x + a
Now see the previous error.
a ab
b c
c
a
a a c ac
b b (^) b b
c c
^ ^
a
b ac
c b
a a
b b a a
c c b c bc
^ ^
