Download Basic Algebra Cheat Sheet and more Cheat Sheet Algebra in PDF only on Docsity!
Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations
b ab ab ac a b c a c c a b a^ a^ ac c bc b b c a c ad bc a c ad bc b d bd b d bd a b b a a b a b c d d c c c c a ab ac (^) b c a b ad a c bc d
+ = + Ê^ ˆ=
ÁË ˜¯
Ê ˆ
ÁË ˜¯
Ê ˆ
ÁË ˜¯
+ = +^ - = -
Ê ˆ
+ Á^ ˜
= + π Ë^ ¯= Ê ˆ ÁË ˜¯
Exponent Properties
( ) (^ )^
1 1
0
m^ n^ m m
n m n m n n m m m n n m nm n (^) n n (^) n n n
n n n n n n (^) n (^) n n n
a a a a a a a a a a a a a ab a b b b
a a a a a b b a a a b a a
= = π
= Ê^ ˆ = Á ˜ Ë ¯ = =
Ê ˆ (^) = Ê ˆ = = = Á ˜ Á ˜ Ë ¯ Ë ¯
Properties of Radicals
1
, if is odd , if is even
n (^) n n n n
m n nm n n n n n n n
a a ab a b
a a a^ a b (^) b a a n a a n
Properties of Inequalities If then and
If and 0 then and
If and 0 then and
a b a c b c a c b c a b a b c ac bc c c a b c ac bc a^ b c c
Properties of Absolute Value if 0 if 0
a a a a a
Ï^ ≥
= Ì
Ó-^ <
Triangle Inequality
a a a a a ab a b b b a b a b
Distance Formula
If P 1 = ( x 1 , y 1 )and P 2 = ( x 2 , y 2 )are two
points the distance between them is
2 2 d P P 1 , 2 (^) = x 2 (^) - x 1 (^) + y 2 (^) - y 1
Complex Numbers
2
2 2 2 2
2
Complex Modulus Complex Conjugate
i i a i a a 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 b a bi a b a bi a bi a bi a bi a bi
Logarithms and Log Properties Definition y = log (^) bx is equivalent to x = by
Example 3 log 125 5 = 3 because 5 = 125
Special Logarithms
10
ln log natural log log log common log
x (^) ex x x
where e = 2.718281828K
Logarithm Properties
( ) ( )
log
log 1 log 1 0 log log log log log log
log log log
b
b b x x b r b b b b b
b b b
b b x b x x r x xy x y x (^) x y y
Ê ˆ
Á ˜=^ -
Ë ¯
The domain of log b x is x > 0
Factoring and Solving
Factoring Formulas ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2 2 2 2 2 2 2 2
x a x a x a x ax a x a x ax a x a x a b x ab x a x b
( ) ( ) ( )( ) ( ) (^) ( )
3 2 2 3 3 3 2 2 3 3 3 3 2 2 3 3 2 2
x ax a x a x a x ax a x a x a x a x a x ax a x a x a x ax a
x^2^ n^ - a^2 n^ = (^) ( xn - an (^) )( xn + an )
If n is odd then,
( ) ( )
( )( )
1 2 1
1 2 2 3 1
n n n n n n n n n n n
x a x a x ax a x a x a x ax a x a
L
L
Quadratic Formula Solve ax^2 + bx + c = 0 , a π 0 (^2 ) 2
x b^ b^ ac a
=^ -^ ±^ -
If b^2 - 4 ac > 0 - Two real unequal solns. If b^2 - 4 ac = 0 - Repeated real solution. If b^2 - 4 ac < 0 - Two complex solutions.
Square Root Property If x^2 = p then x = ± p
Absolute Value Equations/Inequalities If b is a positive number or
or
p b p b p b p b b p b p b p b p b
= fi = - = < fi - < <
fi < - >
Completing the Square Solve 2 x^2 - 6 x - 10 = 0
(1) Divide by the coefficient of the x^2 x^2 - 3 x - 5 = 0 (2) Move the constant to the other side. x^2 - 3 x = 5 (3) Take half the coefficient of x , square it and add it to both sides 2 2 2 3 3 5 3 5 9 29 2 2 4 4
x - x + ÊÁ^ - ˆ˜^ = + ÊÁ^ - ˆ˜ = + = Ë ¯ Ë ¯
(4) Factor the left side 3 2 29 2 4
ÊÁ (^) x - ˆ˜ = Ë ¯ (5) Use Square Root Property 3 29 29 2 4 2
x - = ± = ± (6) Solve for x 3 29 2 2
x = ±
Common Algebraic Errors
Error Reason/Correct/Justification/Example (^2 ) 0
π and^2 0
π (^) Division by zero is undefined!
- 3 2 π 9 -^3 2 = -^9 ,^ ( -^3 )^2 =^9 Watch parenthesis!
( ) 2 3 5 x π x ( ) 2 3 2 2 2 6 x = x x x = x a a a b c b c
π +
= π + =
2 3 2 3
(^1) x x x x
π -^ + -
A more complex version of the previous error. a bx a
π 1 + bx^1
a bx a bx bx a a a a
Beware of incorrect canceling!
- a x ( - (^1) )π - ax - a (^ )
- a x - 1 = - ax + a Make sure you distribute the “-“!
( ) (^2 2 ) x + a π x + a ( ) ( ) ( ) (^2 2 ) x + a = x + a x + a = x + 2 ax + a x^2^ + a^2 π x + a 5 = 25 = 32 + 42 π 32 + 42 = 3 + 4 = 7 x + a π x + a See previous error.
( ) x + a n^ π xn + an and n^ x + a π n^ x + na More general versions of previous three errors.
( ) ( ) 2 2 2 x + 1 π 2 x + 2
( ) (^) ( ) 2 x + 1 2 = 2 x^2^ + 2 x + 1 = 2 x^2 + 4 x + 2 ( ) 2 x + 2 2 = 4 x^2 + 8 x + 4 Square first then distribute!
( ) ( ) 2 x + 2 2 π 2 x + 12 See the previous example. You can notfactor out a constant if there is a power on the parenthesis!
- x 2^ + a^2^ π - x^2 + a^2 (^ )
1
- x^2 + a^2 = - x^2 + a^2 Now see the previous error. a ab b c c
π Ê ˆ Á ˜ Ë ¯
a a a c ac b b (^) b b c c
Ê ˆ
ÁË ˜¯ Ê ˆÊ ˆ
= = Á ˜Á ˜=
Ê ˆ Ê ˆ Ë ¯Ë ¯
ÁË ˜¯ ÁË ˜¯
a b ac c b
Ê ˆ
Á ˜
Ë ¯ π
a a b b a a c c b c bc
Ê ˆ Ê ˆ
Á ˜ Á ˜ Ê ˆÊ ˆ
Ë ¯ = Ë ¯= =
Ê ˆ Á^ ˜Á^ ˜
Ë ¯Ë ¯
Á ˜
Ë ¯
