Algebra Cheat Sheet 2

8/6/2019 Algebra Cheat Sheet 2

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Algebra Cheat Sheet

Basic Properties & Facts

Arithmetic Operations

ab + ac = a (b + c)

a

bc bc

a c ad + bcb d bd

a − b b − a

c − d d − c

ab + aca

Exponent Properties

a na m = a n+ m

n m

b abc c

a acb b

c

a c ad − bcb d bd

a + b a b

c c ca

bc bc

d

a n 1m

a 0 = 1, a ≠ 0

Properties of InequalitiesIf a < b then a + c < b + c and a − c < b − c

a bc ca bc c

Properties of Absolute Valuea if a ≥ 0− a if a < 0

a ≥ 0 − a = a

a ab b

a + b ≤ a + b Triangle Inequality

Distance FormulaIf P = ( x1, y1 ) an d P = ( x2 , y2 ) a re tw o

points the distance between them is

2 2

(a b)n n

b b

n

Complex Numbers

1a n

− n n

b a a

Properties of Radicals

1− n

n 1m m

1m

i = − 1 i 2 = − 1 − a = i a , a ≥ 0

( a + b i ) + ( c + d i) = a + c + ( b + d ) i

( a + b i ) − ( c + d i) = a − c + ( b − d ) i

( a + b i )( c + d i ) = a c− b d + ( a d + b c) i

( a + b i )( a − b i ) = a2 + b2

n 1 n n a + bi = a 2 + b2 Complex Modulus

n

b b

n

( a +b i) = a − b i C o m p le x C o n j u g a t e2

a n = a , if n is even



= a

a

=

If a < b and c > 0 then ac < bc and <

If a < b and c < 0 then ac > bc and >

+ =

=

= b + c, a ≠ 0

(a ) = a nm

− =

= +

=

= a n− m = m− na a

a =

ab = a b =

d (P, P)=(x2 − x1)+(y2 − y1)

1 2

= a b n na a = n

a − n

a b bn

=

= a n

a

( )= ( a )n

a = a n

a = a n ab = n a b

a aa = n mm n

a n = a, if n is odd( a + bi )( a + bi) = a + bi

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

© 2005 Paul Dawkins

http://tutorial.math.lamar.edu/







Logarithms and Log Properties

Definition y = log b x is equivalent to x = b y

Example

log 5 125 = 3 because 5 3 = 125

Special Logarithmsln x = log e x natural log

log x = log 10 x common log

where e = 2.718281828

Logarithm Propertieslog b b = 1 log b 1 = 0

log b b x = x b logb x = x

log b (xr

) =r l o g

b x

log b ( x y) = lo gb x + lo gb y

x

y

The domain of log b x is x > 0

Factoring and Solving

Factoring Formulas x 2 − a 2 = ( x + a)( x − a)

2

2

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

3

x3 +

a3 =

( x + a )(x2

− a x

+a2

) x3 − a 3 = ( x − a) ( x2 + a x+ a2 )

2n 2n n n n n

If n is odd then,

x n − a n = ( x − a) ( xn− 1 + a xn− 2 + + an− 1 ) x n + a n

= ( x + a )( x n− 1 − ax n− 2 + a 2 x n− 3 − + a

n− 1 )

Quadratic FormulaSolve ax 2 + bx + c = 0 , a ≠ 0

− b ± b 2 − 4 ac

2a2

If b2 − 4ac = 0 - Repeated real solution.If b2 − 4ac < 0 - Two complex solutions.

Square Root PropertyIf x 2 = p then x = ± p

Absolute Value Equations/InequalitiesIf b is a positive number p = b ⇒ p = − b or p = b

p < b ⇒ − b < p < b

p > b ⇒ p < − b or p > b

Completing the Square

2

(1) Divide by the coefficient of the x 2

x 2 − 3 x − 5 = 0

(4) Factor the left side2

3 292 4

(5) Use Square Root Property

(2) Move the constant to the other side. x 2 − 3 x = 5

(3) Take half the coefficient of x, squareit and add it to both sides (6) Solve for x

= ±29 294 2

2 2

9 29



log b = log b b y x − log

x 2+

2ax+

a 2=

( x+

a) x 2 − 2ax + a 2 = ( x − a)

x3 + 3ax 2 + 3a 2 x + a3 = ( x + a)

x

If b − 4ac > 0 - Two real unequal solns.

x3 − 3ax 2 + 3a 2 x − a3 = ( x − a )

x − a=(x− a)(x+ a)

Solve 2 x − 6 x − 10 = 0 = x

x

−

= ±

x 2 − 3 x + − = 5 + =

x = ±

2 2 4 4

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

3 292 2









Functions and Graphs

Constant Function

y = a or f ( x) = aGraph is a horizontal line passing

through the point ( 0 ,a) .

Line/Linear Function y = mx + b or f ( x) = m x+ b

Graph is a line with point ( 0 ,b) a n dslope m.

SlopeSlope of the line containing the two

points ( x1, y1 ) an d( x2 , y2 ) is

y2 − y1 rise x2 − x1 run

Slope – intercept formThe equation of the line with slope m

and y-intercept ( 0 ,b) is y = mx + b

Point – Slope form

Parabola/Quadratic Function

x = ay 2 + by + c g ( y ) = a y2 + b y+ c

The graph is a parabola that opens rightif a > 0 or left if a < 0 and has a vertex

b b .

2a

Circle2 2

Graph is a circle with radius r and center ( h, k ) .

Ellipse2 2

+ = 12

Graph is an ellipse with center ( h, k )with vertices a units right/left from thecenter and vertices b units up/down fromthe center.

The equation of the line with slope m

and passing through the point ( x1, y1 ) is

y = y1 + m ( x − x1 )

Hyperbola2

−2

= 1

Parabola/Quadratic Function2 2

The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertexat ( h, k ) .

Parabola/Quadratic Function y = ax 2 + bx + c f ( x ) = a x2 + b x+ c

The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex

b b

2a

Graph is a hyperbola that opens left and

right, has a center at ( h, k ) , ver t icesaunits left/right of center and asymptotes

ba

Hyperbola2 2

− = 12

Graph is a hyperbola that opens up and

down, has a center at ( h, k ) , ver t icesbunits up/down from the center andasymptotes that pass through center with

ba

For a completonline Algebrvisit http://tutorialmar.edu .











at

−

, −2a

( x − h) + ( y − k ) = r 2

m = = (x− h) (y− k )a b 2

(x− h) (y− k )

a b2

y=a(x− h)+k f (x)=a(x− h)+k

that pass through center with slope ± .

(y− k ) (x− h)b a 2

at − −

,

2aslope ± .




Common Algebraic Errors

Error Reason/Correct/Justification/Example

≠ 0 and ≠ 2 Division by zero is undefined!

− 32 ≠ 9 − 32 = − 9

,

( −3) = 9 Watch parenthesis!

2 3 ≠ x52 3

= x 2 x 2 x 2 = x 6

a

b + c

a ab c

1 1 1

1 + 1 1 1

12 3

a + bxa

− a ( x − 1) ≠ − a x− a

A more complex version of the previouserror.a + bx a bx bx

a a a aBeware of incorrect canceling!− a ( x − 1) = − a x+ aMake sure you distribute the “-“!

( x + a) ≠ x 2 + a 2 ( x + a) = ( x + a)( x + a) = x2 + 2a x+ a2

x 2 + a 2 ≠ x + a

x + a ≠ x + a

5 = 25 = 32 + 42 ≠ 32 + 42 = 3 + 4 = 7See previous error.

2 2

x + a ≠ n x + n a More general versions of previous threeerrors.

2

2

Square first then distribute!

( 2 x + 2) ≠ 2( x + 1)2

See the previous example. You can notfactor out a constant if there is a power on

the parethesis!

− x 2 + a 2 ≠ − x 2 + a 2

1

Now see the previous error.

a

b

≠ab

ca

b



(x ) (x )≠ + ≠ + = 2

≠ x − 2 + x − 3

x + x

≠ 1 + bx= + = 1 +

(x+ a) ≠ xn+ an a n d

2 ( x + 1) ≠ ( 2 x + 2)2 ( x + 1) = 2( x2 + 2 x + 1) = 2 x2 + 4 x + 2

(2 x+ 2)= 4 x2+ 8 x+ 4

− x2 + a 2 = ( − x2 + a 2 )2

c

a c = = =

b ≠ aca 1

b = c b c

1 ac

a

c b

a a

c b c1

For a complete set of online Algebra notes visi t http://tutorial.math.lamar.edu . © 2005 Paul Dawkins






Algebra Cheat Sheet 2

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