# Algebra Cheat Sheet

```Algebra Cheat Sheet
Basic Properties &amp; Facts
Arithmetic Operations
Properties of Inequalities
If a &lt; b then a + c &lt; b + c and a - c &lt; b - c
a b
If a &lt; b and c &gt; 0 then ac &lt; bc and &lt;
c c
a b
If a &lt; b and c &lt; 0 then ac &gt; bc and &gt;
c c
&aelig; b &ouml; ab
a&ccedil; &divide; =
&egrave;c&oslash; c
ab + ac = a ( b + c )
&aelig;a&ouml;
&ccedil; &divide; a
&egrave;b&oslash; =
c
bc
a
ac
=
&aelig;b&ouml; b
&ccedil; &divide;
&egrave;c&oslash;
+ =
b d
bd
- =
b d
bd
a -b b-a
=
c-d d -c
a+b a b
= +
c
c c
&aelig;a&ouml;
&egrave;b&oslash; =
&aelig; c &ouml; bc
&ccedil; &divide;
&egrave;d&oslash;
ab + ac
= b + c, a &sup1; 0
a
Properties of Absolute Value
if a &sup3; 0
&igrave;a
a =&iacute;
if a &lt; 0
&icirc; -a
a &sup3;0
-a = a
a+b &pound; a + b
a n a m = a n+m
an
1
= a n-m = m-n
m
a
a
(a )
a 0 = 1, a &sup1; 0
( ab )
n
a -n =
&aelig;a&ouml;
&ccedil; &divide;
&egrave;b&oslash;
-n
= a nm
n
1
an
n
bn
&aelig;b&ouml;
=&ccedil; &divide; = n
a
&egrave;a&oslash;
n
m
1
a = an
m n
a = nm a
( x2 - x1 ) + ( y2 - y1 )
2
2
n
Complex Numbers
i = -1
( ) = (a )
a = a
n
d ( P1 , P2 ) =
a
&aelig;a&ouml;
&ccedil; &divide; = n
b
&egrave;b&oslash;
1
= an
-n
a
= a nb n
Triangle Inequality
Distance Formula
If P1 = ( x1 , y1 ) and P2 = ( x2 , y2 ) are two
points the distance between them is
Exponent Properties
n m
a
a
=
b
b
ab = a b
1
m
n
n
1
m
i 2 = -1
-a = i a , a &sup3; 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
a n = a , if n is even
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
Complex Modulus
&copy; 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
Example
log 5 125 = 3 because 53 = 125
b logb x = x
log b ( x r ) = r log b x
log b ( xy ) = log b x + log b y
Special Logarithms
ln x = log e x
natural log
&aelig;x&ouml;
log b &ccedil; &divide; = log b x - log b y
&egrave; y&oslash;
log x = log10 x common log
where e = 2.718281828K
The domain of log b x is x &gt; 0
Factoring and Solving
Factoring Formulas
x 2 - a 2 = ( x + a )( x - a )
Solve ax 2 + bx + c = 0 , a &sup1; 0
x 2 + 2ax + a 2 = ( x + a )
2
x 2 - 2ax + a 2 = ( x - a )
2
-b &plusmn; b 2 - 4ac
2a
2
If b - 4ac &gt; 0 - Two real unequal solns.
If b 2 - 4ac = 0 - Repeated real solution.
If b 2 - 4ac &lt; 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 = &plusmn; 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
&THORN;
p = -b or p = b
p &lt;b
&THORN;
-b &lt; p &lt; b
p &gt;b
&THORN;
p &lt; -b or
p&gt;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
&aelig; 3&ouml;
&aelig; 3&ouml;
x 2 - 3x + &ccedil; - &divide; = 5 + &ccedil; - &divide; = 5 + =
4 4
&egrave; 2&oslash;
&egrave; 2&oslash;
3&ouml;
29
&aelig;
&ccedil;x- &divide; =
2&oslash;
4
&egrave;
(5) Use Square Root Property
3
29
29
x- = &plusmn;
=&plusmn;
2
4
2
(6) Solve for x
3
29
x= &plusmn;
2
2
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
&copy; 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 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 )
2
2
y = a ( x - h) + k
f ( x) = a ( x - h) + k
The graph is a parabola that opens up if
a &gt; 0 or down if a &lt; 0 and has a vertex
at ( h, k ) .
y = ax 2 + bx + c f ( x ) = ax 2 + bx + c
The graph is a parabola that opens up if
a &gt; 0 or down if a &lt; 0 and has a vertex
&aelig; b
&aelig; b &ouml;&ouml;
at &ccedil; - , f &ccedil; - &divide; &divide; .
&egrave; 2a &egrave; 2 a &oslash; &oslash;
x = ay 2 + by + c g ( y ) = ay 2 + by + c
The graph is a parabola that opens right
if a &gt; 0 or left if a &lt; 0 and has a vertex
&aelig; &aelig; b &ouml; b &ouml;
at &ccedil; g &ccedil; - &divide; , - &divide; .
&egrave; &egrave; 2a &oslash; 2 a &oslash;
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 &plusmn; .
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 &plusmn; .
a
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
-
&copy; 2005 Paul Dawkins
Common Algebraic Errors
Error
Reason/Correct/Justification/Example
2
2
&sup1; 0 and &sup1; 2
0
0
Division by zero is undefined!
-32 &sup1; 9
-32 = -9 ,
(x )
(x )
2 3
2 3
&sup1; x5
a
a a
&sup1; +
b+c b c
1
&sup1; x -2 + x -3
2
3
x +x
- a ( x - 1) &sup1; - ax - a
2
x+a &sup1; x + a
n
= 9 Watch parenthesis!
= x2 x2 x2 = x6
( x + a)
&sup1; x2 + a2
x2 + a2 &sup1; x + a
( x + a)
2
1
1
1 1
=
&sup1; + =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
&sup1; 1 + bx
a
( x + a)
( -3 )
&sup1; x n + a n and
n
x+a &sup1; n x + n a
= ( x + a )( x + a ) = x 2 + 2ax + a 2
2
5 = 25 = 32 + 42 &sup1; 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) &sup1; ( 2 x + 2 )
2
( 2 x + 2)
2
2
2
&sup1; 2 ( x + 1)
( 2 x + 2)
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 parenthesis!
1
- x2 + a2 &sup1; - x2 + a2
a
ab
&sup1;
&aelig;b&ouml; c
&ccedil; &divide;
&egrave;c&oslash;
&aelig;a&ouml;
&ccedil; &divide; ac
&egrave;b&oslash; &sup1;
c
b
- x2 + a2 = ( - x2 + a 2 ) 2
Now see the previous error.
&aelig;a&ouml;
&ccedil; &divide;
a
1
&aelig; a &ouml;&aelig; c &ouml; ac
= &egrave; &oslash; = &ccedil; &divide;&ccedil; &divide; =
&aelig; b &ouml; &aelig; b &ouml; &egrave; 1 &oslash;&egrave; b &oslash; b
&ccedil; &divide; &ccedil; &divide;
&egrave;c&oslash; &egrave;c&oslash;
&aelig;a&ouml; &aelig;a&ouml;
&ccedil; &divide; &ccedil; &divide;
&egrave; b &oslash; = &egrave; b &oslash; = &aelig; a &ouml;&aelig; 1 &ouml; = a
&ccedil; &divide;&ccedil; &divide;
c
&aelig; c &ouml; &egrave; b &oslash;&egrave; c &oslash; bc
&ccedil; &divide;
&egrave;1&oslash;
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
&copy; 2005 Paul Dawkins
```