Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations ab + ac = a ( b + c ) æaö ç ÷ a èbø = c bc a c ad + bc + = b d bd a-b b-a = c-d d -c ab + ac = b + c, a ¹ 0 a Exponent Properties a na m = a n+ m an 1 = a n -m = m -n am a a 0 = 1, a ¹ 0 a æaö ç ÷ = n b èbø 1 n =a a-n
n n n
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 = log 10 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 - 2 ax + a 2 = ( x - a )
2 2
Logarithm Properties log b b = 1 log b 1 = 0 log b b x = x log b ( x
r
æ b ö ab aç ÷ = ècø c a ac = æbö b çc÷ è ø a c ad - bc - = b d bd a+b a b = + c c c æaö ç b ÷ ad è ø= æ c ö bc çd ÷ è ø
Properties of Inequalities If a < b then a + c < b + c and a - c < b - c a b < c c a b If a < b and c < 0 then ac > bc and > c c If a < b and c > 0 then ac < bc and Properties of Absolute Value if a ³ 0 ìa a =í if a < 0 î-a a ³0 -a = a ab = a b a +b £ a + b a a = b b Triangle Inequality
b logb x = x
b
) = r log
x
log b ( xy ) = log b x + logb y æxö log b ç ÷ = log b x - logb y è yø The domain of log b x is x > 0
Factoring and Solving
Quadratic Formula Solve ax 2 + bx + c = 0 , a ¹ 0 -b ± b 2 - 4 ac 2a If b 2 - 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 If x2 = p then x = ± p Absolute Value Equations/Inequalities If b is a positive number p =b Þ p = - b or p = b p b Þ Þ -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 - 3ax2 + 3a 2 x - a 3 = ( x - a )
3 3
Distance Formula If P = ( x1, y1 ) and P2 = ( x2 , y2 ) are two 1 points the distance between them is d ( P , P2 ) = 1
(a )
n m
x3 + a 3 = ( x + a ) ( x2 - ax + a 2 ) x3 - a 3 = ( x - a ) ( x2 + ax + a 2 ) x 2 n - a...
