Math

Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations ab + ac = a ( b + c ) a   a b = c bc a c ad + bc + = b d bd a −b b−a = c−d d −c ab + ac = b + c, a ≠ 0 a Exponent Properties a n a m = a n+m an 1 = a n−m = m−n m a a a 0 = 1, a ≠ 0 a a   = n b b 1 = an −n a
n n n

 b  ab a  = c c a ac = b b   c a c ad − bc − = b d bd a+b a b = + c c c a   ad b = c  bc   d

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 Properties of Absolute Value if a ≥ 0 a a = if a < 0  −a a ≥0 −a = a ab = a b a+b ≤ a + b a a = b b Triangle Inequality

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

(a )

n m

= a nm = a nb n 1 an bn b =  = n a a

( x2 − x1 ) + ( y2 − y1 )
2

2

( ab )

n

Complex Numbers i = −1
n
1 m

a −n = a   b
−n

i 2 = −1

−a = i a , a ≥ 0

a = a

n m

( ) = (a )
1 m

n

Properties of Radicals
n

( 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
a + bi = a 2 + b 2 Complex Modulus

a = an a = nm a

1

n

ab = n a n b a na = b nb

m n

n

n n

a n = a, if n is odd a n = a , if n is even

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

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 Example log 5 125 = 3 because 53 = 125 Special Logarithms ln x = log e x natural log log x = log10 x common log where e = 2.718281828K Factoring Formulas x 2 − a 2 = ( x + a )( x − a ) x 2 + 2ax + a 2 = ( x + a ) x 2 − 2ax + a 2 = ( x − a )
2 2

Logarithm Properties log b b = 1 log b 1 = 0 log b bx = x log b ( x r ) = r log b x b logb x = x

log b ( xy ) = log b x + log b y x log b   = log b x − log b y  y The domain of log b x is x > 0

Factoring and Solving
Quadratic Formula Solve ax 2 + bx + c = 0 , a ≠ 0 −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= Square Root Property Ifx 2 = p then x = ± p Absolute Value Equations/Inequalities If b is a positive number p =b ⇒ p = −b or p = b p b ⇒ ⇒ −b < p < b p < −b or p>b

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

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 = ( x + a ) ( x n −1 − ax n − 2 + a 2 x n −3 − L + a n −1 )
2

Solve 2 x − 6 x − 10 = 0

Completing the Square (4) Factor the left side 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
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 9 29  3  3 x 2 − 3x +  −  = 5 +  −  = 5 + = 4 4  2  2
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 horizontalline passing through the point ( 0, a ) . Line/Linear Function y = mx + b or f ( x ) = mx + b 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...