Primaria

Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations

Properties of Inequalities
If a < b then a + c < b + c and a − c < b − c
ab
If a < b and c > 0 then ac < bc and <
cc
ab
If a < b and c < 0 then ac > bc and >
cc

 b  ab
a  =
c c

ab + ac = a ( b + c )
a
 a
b =
c
bc

a
ac
=
b b

c

a c ad + bc
+=
bd
bd

a c ad − bc
−=
bd
bda −b b−a
=
c−d d −c

Properties of Absolute Value
if a ≥ 0
a
a =
if a < 0
 −a
a ≥0
−a = a

a+b a b
=+
c
cc
a
  ad
b =
 c  bc

d

ab + ac
= b + c, a ≠ 0
a

a+b ≤ a + b

(a )

nm

an
1
= a n−m = m−n
m
a
a

( ab )

a 0 = 1, a ≠ 0
n

n

a −n =
a

b

= a nm

−n

1
an
n

bn
b
=  = n
a
a

n
m

1

a = anmn

a = nm a

( x2 − x1 ) + ( y2 − y1 )
2

2

n

Complex Numbers
i = −1

( ) = (a )

a=a

Properties of Radicals
n

d ( P , P2 ) =
1

a
a
 =n
b
b
1
= an
−n
a

= a nb n

Triangle Inequality

Distance Formula
If P = ( x1 , y1 ) and P2 = ( x2 , y2 ) are two
1
points the distance between them is

Exponent Properties
a n a m = a n+m

a
a
=
b
bab = a b

1
m

n

n

1
m

i 2 = −1

−a = i a , a ≥ 0

( 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 2 + b 2

n

ab = n a n b

a + bi = a 2 + b 2

n

a na
=
b nb

( a + bi ) = a − bi Complex Conjugate
2
( a + bi ) ( a + bi ) = a + bi

n

a n= a, if n is odd

n

Complex Modulus

a n = a , if n is even

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

© 2005 Paul Dawkins

Logarithms and Log Properties
Definition
y = log b x is equivalent to x = b y

Logarithm Properties
log b b = 1
log b 1 = 0
log b b x = x

b logb x = x

log b ( x r ) = r log b x

Example
log 5 125 = 3because 53 = 125

log b ( xy ) = log b x + log b y

Special Logarithms
ln x = log e x
natural log

x
log b   = log b x − log b y
 y

log x = log10 x common log
where e = 2.718281828K

The domain of log b x is x > 0

Factoring and Solving
Factoring Formulas
x2 − a2 = ( x + a ) ( x − a )

Quadratic Formula
Solve ax 2 + bx + c = 0 , a ≠ 0

x 2 + 2ax + a 2 = ( x + a )

2x 2 − 2ax + a 2 = ( x − a )

2

−b ± b 2 − 4ac
2a
2
If b − 4ac > 0 - Two real unequal solns.
If b 2 − 4ac = 0 - Repeated real solution.
If b 2 − 4ac < 0 - Two complex solutions.
x=

x 2 + ( a + b ) x + ab = ( x + a )( x + b )
x3 + 3ax 2 + 3a 2 x + a 3 = ( x + a )
x3 − 3ax 2 + 3a 2 x − a 3 = ( x − a )

3

3

Square Root Property
If x 2 = p then x = ± p

x3 + a3 = ( x + a ) (x 2 − ax + a 2 )
x3 − a 3 = ( x − a ) ( x 2 + ax + a 2 )
x −a
2n

2n

= (x −a
n

n

)( x

n

+a

n

)

If n is odd then,
x n − a n = ( x − a ) ( x n −1 + ax n − 2 + L + a n −1 )
xn + a n

Absolute Value Equations/Inequalities
If b is a positive number
p =b
⇒
p = −b or p = b
p b

⇒

p < −b or

p>b

= ( x + a ) ( x n −1 − ax n − 2 + a 2 x n −3 − L + a n −1 )Completing the Square
(4) Factor the left side

Solve 2 x − 6 x − 10 = 0
2

2

(1) Divide by the coefficient of the x 2
x 2 − 3x − 5 = 0
(2) Move the constant to the other side.
x 2 − 3x = 5
(3) Take half the coefficient of x, square
it and add it to both sides
2

2

9 29
 3
 3
x 2 − 3x +  −  = 5 +  −  = 5 + =
44
 2
 2

3
29

x−  =
2
4

(5) UseSquare Root Property
3
29
29
x− = ±
=±
2
4
2
(6) Solve for x
3
29
x= ±
2
2

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

© 2005 Paul Dawkins

Functions and Graphs
Constant Function
y = a or f ( x ) = a
Graph is a horizontal line passing
through the point ( 0, a ) .
Line/Linear Function
y = mx + b or f ( x ) = mx + b

Graph is a line...