Algebra Cheat Sheet PDF

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Algebra Cheat Sheet Basic Properties & Facts 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

Arithmetic Operations b ab a æç ö÷ = ècø c

ab + ac = a ( b + c ) æ aö ç ÷ a è bø = 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 b ÷ ad è ø= æ c ö bc çd ÷ è ø

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

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

(a )

a 0 = 1, a ¹ 0

n m

=a

nm

n

n æaö = æbö = b çb÷ ça÷ an è ø è ø

n m

1

a = an

m n

a=

nm

a

n

ab = n a n b

n

a = b

n

an = a , if n is odd

n

an = a , if n is even

( ) 1

a = am

Properties of Radicals n

( x 2 - x1 ) + ( y 2 - y1 ) 2

2

n

1 an

-n

d ( P1, P2 ) =

n æa ö = a ç ÷ n èb ø b 1 = an a- n

(ab )n = an bn a -n =

Triangle Inequality

Distance Formula If P1 = ( x1 , y1 ) and P2 = ( x 2 , y 2 ) are two points the distance between them is

Exponent Properties a n a m = a n+ m

a a = b b

ab = a b

n n

a b

n

Complex Numbers i = -1 = ( an ) m 1

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 a + bi = a 2 + b 2

Complex Modulus

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

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

Logarithm Properties log b b = 1 log b 1 = 0 log b b x = x

Example log5 125 = 3 because 53 = 125

b logb x = x

log b ( xr ) = r log b x log b ( xy ) = log b x + log b y

Special Logarithms ln x = loge x natural log

æ log b ç è

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

xö = log b x - log b y y ÷ø

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 + 2 ax + 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 ) x 3 + 3ax 2 + 3a 2x + a 3 = ( x + a )

3

x 3 - 3ax 2 + 3a 2x - a 3 = ( x - a )

3

Square Root Property If x 2 = p then x = ± p

x 3 + a 3 = ( x + a ) (x 2 - ax + a 2 ) x 3 - a 3 = ( x - a ) ( x 2 + ax + a 2 ) x 2n - a 2n = (x n - a 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 +an

Absolute Value Equations/Inequalities If b is a positive number p=b p = - b or p = b Þ pb

Þ

p < - b or

p> b

= ( x + a ) ( x n -1 - ax n -2 + a 2 x n-3 -L + a n-1 ) Solve 2 x 2 - 6 x -10 = 0

Completing the Square (4) Factor the left side 2

2

(1) Divide by the coefficient of the x x2 - 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 9 29 æ 3ö æ 3ö 2 x - 3x + ç - ÷ = 5 + ç - ÷ = 5 + = 4 4 è 2ø è 2ø

æ x - 3 ö = 29 ç ÷ 2ø 4 è (5) Use Square Root Property 3 29 29 =± =± 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.

x-

© 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 ( x 2 , y 2 ) is y 2 - y1 rise = x 2 - 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 = y 1 +m (x -x 1 ) 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 = a x 2 + b x + 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 2 a è 2 a øø

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 (x -h )2 + ( y - k )2 = r 2 Graph is a circle with radius r and center ( h, k ) . Ellipse 2 ( x - h)

( 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 =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 - ( x - h )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 , (- 3 ) = 9 Watch parenthesis!

(x )

(x )

2

3

2

¹ x5

2

a + bx ¹ 1 + bx a - a ( x -1 ) ¹ - ax - a

(x + a )

¹ x +a 2

Make sure you distribute the “-“! ( x + a )2 = ( x + a )( x + a ) = x 2 + 2ax + a 2

2

x2 +a2 ¹ x +a x +a ¹ x + a

(x + a ) n ¹ x n + a n

= x2 x2x 2 = x 6

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

a a a ¹ + b+c b c 1 ¹ x -2 + x -3 2 x + x3

2

3

and n x + a ¹ n x + n a

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

(2 x + 2 ) 2 ¹ 2 ( x + 1 ) 2 -x 2 + a 2 ¹ - x 2 + a 2 a ab ¹ æbö c ç c÷ è ø æ aö ç b ÷ ac è ø¹ c b

( 2x + 2 )2 = 4x 2 + 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! -x + a = ( - x + a 2

2

2

1 2 2

)

Now see the previous error. æ aö 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 ÷ è cø è cø æ aö æ aö ç b÷ ç b÷ è ø = è ø = æ a öæ 1 ö = a ÷ç c ÷ bc c æ c ö èç b øè ø ç 1÷ è ø

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

© 2005 Paul Dawkins...