This document provides an algebra cheat sheet that summarizes many common algebraic properties, formulas, and concepts. It covers topics such as arithmetic operations, properties of inequalities and absolute value, exponent properties, factoring formulas, solving equations, graphing functions, and common algebraic errors. The cheat sheet is a concise 3-page reference for the basics of algebra.

1. Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations
Properties of Inequalities
If a < b then a + c < b + c and a − c < b − c
a b
If a < b and c > 0 then ac < bc and <
c c
a b
If a < b and c < 0 then ac > bc and >
c c
 b  ab
a  =
c c
ab + ac = a ( b + c )
a
  a
b =
c
bc
a
ac
=
b b
 
c
a c ad + bc
+ =
b d
bd
a c ad − bc
− =
b d
bd
a −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
c c
a
  ad
b =
 c  bc
 
d
ab + ac
= b + c, a ≠ 0
a
a+b ≤ a + b
(a )
n m
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 = an
m n
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
b
ab = 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

2. 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 = 3 because 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
x 2 − a 2 = ( x + a )( x − a )
Quadratic Formula
Solve ax 2 + bx + c = 0 , a ≠ 0
x 2 + 2ax + a 2 = ( x + a )
2
x 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
⇒
−b < 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 + =
4 4
 2
 2
3
29

x−  =
2
4

(5) Use Square 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

3. 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 with point ( 0, b ) and
slope m.
Slope
Slope of the line containing the two
points ( x1 , y1 ) and ( x2 , y2 ) is
y2 − y1 rise
=
x2 − x1 run
Slope – intercept form
The equation of the line with slope m
and y-intercept ( 0,b ) is
y = mx + b
Point – Slope form
The equation of the line with slope m
and passing through the point ( x1 , y1 ) is
m=
y = y1 + m ( x − x1 )
Parabola/Quadratic Function
2
2
y = a ( x − h) + k
f ( x) = a ( x − h) + k
The graph is a parabola that opens up if
a > 0 or down if a < 0 and has a vertex
at ( h, k ) .
Parabola/Quadratic Function
y = ax 2 + bx + c f ( x ) = ax 2 + bx + c
The graph is a parabola that opens up if
a > 0 or down if a < 0 and has a vertex
 b
 b 
at  − , f  −   .
 2a  2 a  
Parabola/Quadratic Function
x = ay 2 + by + c g ( y ) = ay 2 + by + c
The graph is a parabola that opens right
if a > 0 or left if a < 0 and has a vertex
  b  b 
at  g  −  , −  .
  2a  2 a 
Circle
2
2
( x − h) + ( y − k ) = r 2
Graph is a circle with radius r and center
( h, k ) .
Ellipse
( x − h)
2
( y −k)
+
2
=1
a2
b2
Graph is an ellipse with center ( h, k )
with vertices a units right/left from the
center and vertices b units up/down from
the center.
Hyperbola
( x − h)
2
( y −k)
−
2
( x − h)
2
=1
a2
b2
Graph is a hyperbola that opens left and
right, has a center at ( h, k ) , vertices a
units left/right of center and asymptotes
b
that pass through center with slope ± .
a
Hyperbola
(y −k)
2
=1
b2
a2
Graph is a hyperbola that opens up and
down, has a center at ( h, k ) , vertices b
units up/down from the center and
asymptotes that pass through center with
b
slope ± .
a
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
−
© 2005 Paul Dawkins

4. Common Algebraic Errors
Error
Reason/Correct/Justification/Example
2
2
≠ 0 and ≠ 2
0
0
Division by zero is undefined!
−32 ≠ 9
−32 = −9 ,
(x )
(x )
2 3
2 3
≠ x5
a
a a
≠ +
b+c b c
1
≠ x −2 + x −3
2
3
x +x
− a ( x − 1) ≠ − ax − a
x2 + a2 ≠ x + a
x+a ≠ x + a
( x + a)
≠ x n + a n and
n
= 9 Watch parenthesis!
= x2 x2 x2 = x6
( x + a)
≠ x2 + a2
2
2
1
1
1 1
=
≠ + =2
2 1+1 1 1
A more complex version of the previous
error.
a + bx a bx
bx
= +
= 1+
a
a a
a
Beware of incorrect canceling!
− a ( x − 1) = − ax + a
Make sure you distribute the “-“!
a + bx
≠ 1 + bx
a
( x + a)
( −3 )
n
x+a ≠ n x + n a
= ( x + a )( x + a ) = x 2 + 2ax + a 2
2
5 = 25 = 32 + 42 ≠ 32 + 42 = 3 + 4 = 7
See previous error.
More general versions of previous three
errors.
2 ( x + 1) = 2 ( x 2 + 2 x + 1) = 2 x 2 + 4 x + 2
2
2 ( x + 1) ≠ ( 2 x + 2 )
2
( 2 x + 2)
2
2
2
≠ 2 ( x + 1)
( 2 x + 2)
= 4 x2 + 8x + 4
Square first then distribute!
See the previous example. You can not
factor out a constant if there is a power on
the parethesis!
2
1
− x2 + a2 ≠ − x2 + a2
a
ab
≠
b c
 
c
a
  ac
b ≠
c
b
− x2 + a2 = ( − x2 + a 2 ) 2
Now see the previous error.
a
 
a
1
 a  c  ac
=   =    =
 b   b   1  b  b
   
c c
a a
   
 b  =  b  =  a  1  = a
  
c
 c   b  c  bc
 
1
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins