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Basic Algebra Cheat Sheet, Cheat Sheet of Algebra

Crib Sheet includes arithmetic and exponential operation, Hyperbola, Ellipse, Parabola and Quadratic Functions along with common algebraic errors

Typology: Cheat Sheet

2020/2021

Uploaded on 04/27/2021

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torley 🇺🇸

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Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations
( )
,0
bab
abacabca
cc
a
aaac
b
b
cbcb
c
acadbcacadbc
bdbdbdbd
abbaabab
cddcccc
a
abacad
b
bca c
abc
d
æö
+=+=
ç÷
èø
æö
ç÷
èø
==
æö
ç÷
èø
+-
+=-=
--+
==+
--
æö
ç÷
+èø
=+¹=
æö
ç÷
èø
Exponent Properties
( )
( )
( )
( )
1
1
0
1
1, 0
11
n
m
mm
n
nmnmnm
mmn
m
nnm
nn
nnn
n
nn
nn
nn
nn
n
n
a
aaaa
aa
aaaa
aa
abab bb
aa
aa
abb
baa
+-
-
-
-
-
===
=
æö
==
ç÷
èø
==
æöæö
====
ç÷ç÷
èøèø
Properties of Radicals
1
,if is odd
,if is even
n
nnnn
n
m
nnm n
n
nn
nn
aaabab
aa
aa b
b
aan
aan
==
==
=
=
Properties of Inequalities
If thenand
If and 0 then and
If and 0 then and
abacbcacbc
ab
abcacbc
cc
ab
abcacbc
cc
<+<+-<-
<><<
<<>>
Properties of Absolute Value
if 0
if 0
aa
a
aa
³
ì
=í
-<
î
0
Triangle Inequality
aaa
a
a
abab bb
abab
³-=
==
+£+
Distance Formula
If
(
)
111
,
Pxy
= and
(
)
222
,
Pxy
= are two
points the distance between them is
( ) ( ) ( )
22
122121
,
dPPxxyy
=-+-
Complex Numbers
( ) ( ) ( )
( ) ( ) ( )
()( ) ( )
( )( )
( )
( )( )
2
22
22
2
11,0
Complex Modulus
Complex Conjugate
iiaiaa
abicdiacbdi
abicdiacbdi
abicdiacbdadbci
abiabiab
abiab
abiabi
abiabiabi
=-=--
+++=+++
+-+=-+-
++=-++
+-=+
+=+
+=-
++=+
pf3
pf4

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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 ¯