Algebra Cheat Sheet Basic Properties & Facts Arithmetic Operations

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

æ b ö ab aç ÷ = ècø c

ab + ac = a ( b + c ) æaö ç ÷ a èbø = c bc

a ac = æbö b ç ÷ ècø

a c ad + bc + = b d bd

a c ad - bc - = b d bd

a -b b-a = c-d d -c

a+b a b = + c c c æaö ç ÷ ad èbø = æ c ö bc ç ÷ èdø

ab + ac = b + c, a ¹ 0 a

Properties of Absolute Value if a ³ 0 ìa a =í if a < 0 î -a a ³0 -a = a

a+b £ a + b

a n a m = a n+m

an 1 = a n-m = m-n m a a

(a )

a 0 = 1, a ¹ 0

( ab )

n

a -n = æaö ç ÷ èbø

-n

= a nm

n

1 an n

bn æbö =ç ÷ = n a èaø

n m

1

a = an

m n

a = nm a

( x2 - x1 ) + ( y2 - y1 ) 2

2

n

Complex Numbers i = -1

( ) = (a )

a = a

Properties of Radicals n

d ( P1 , P2 ) =

a æaö ç ÷ = n b èbø 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 ³ 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

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Complex Modulus

© 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

æxö log b ç ÷ = log b x - log b y è yø

log x = log10 x common log where e = 2.718281828K

The domain of log b x is x > 0

Factoring and Solving Factoring Formulas x 2 - a 2 = ( x + a )( x - a )

Quadratic Formula Solve ax 2 + bx + c = 0 , a ¹ 0

x 2 + 2ax + a 2 = ( x + a )

2

x 2 - 2ax + a 2 = ( x - a )

2

-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=

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 = ± 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 Þ p = -b or p = b p b

Þ

p < -b or

p>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 æ 3ö æ 3ö x 2 - 3x + ç - ÷ = 5 + ç - ÷ = 5 + = 4 4 è 2ø è 2ø

3ö 29 æ çx- ÷ = 2ø 4 è (5) Use Square Root Property 3 29 29 x- = ± =± 2 4 2 (6) Solve for x 3 29 x= ± 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 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 ) Parabola/Quadratic Function 2 2 y = a ( x - h) + k f ( x) = a ( x - h) + k The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at ( h, k ) . Parabola/Quadratic Function y = ax 2 + bx + c f ( x ) = ax 2 + bx + c The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex æ b æ b öö at ç - , f ç - ÷ ÷ . è 2a è 2 a ø ø

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 æ æ b ö b ö at ç g ç - ÷ , - ÷ . è è 2a ø 2 a ø 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 ± . 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 ± . a

For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.

-

© 2005 Paul Dawkins

Common Algebraic Errors Error

Reason/Correct/Justification/Example

2 2 ¹ 0 and ¹ 2 0 0

Division by zero is undefined!

-32 ¹ 9

-32 = -9 ,

(x )

(x )

2 3

2 3

¹ x5

a a a ¹ + b+c b c 1 ¹ x -2 + x -3 2 3 x +x

- a ( x - 1) ¹ - ax - a 2

x+a ¹ x + a n

= 9 Watch parenthesis!

= x2 x2 x2 = x6

( x + a)

¹ x2 + a2

x2 + a2 ¹ x + a

( x + a)

2

1 1 1 1 = ¹ + =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 ¹ 1 + bx a

( x + a)

( -3 )

¹ x n + a n and

n

x+a ¹ n x + n a

= ( x + a )( x + a ) = x 2 + 2ax + a 2

2

5 = 25 = 32 + 42 ¹ 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) ¹ ( 2 x + 2 )

2

( 2 x + 2)

2

2

2

¹ 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 ¹ - x2 + a2 a ab ¹ æbö c ç ÷ ècø æaö ç ÷ ac èbø ¹ c b

- x2 + a2 = ( - x2 + a 2 ) 2 Now see the previous error. æaö ç ÷ a 1 æ a öæ c ö ac = è ø = ç ÷ç ÷ = æ b ö æ b ö è 1 øè b ø b ç ÷ ç ÷ ècø ècø æaö æaö ç ÷ ç ÷ è b ø = è b ø = æ a öæ 1 ö = a ç ÷ç ÷ c æ c ö è b øè c ø bc ç ÷ è1ø

For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.

© 2005 Paul Dawkins