For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. ©
2005 Paul Dawkins. Algebra Cheat Sheet. Basic Properties & Facts. Arithmetic ...
Logarithms and Log Properties Definition y = log b x is equivalent to x = b y
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 a if a < 0 î -a = a a ³0
a+b £ a + b
Exponent Properties a n a m = a n+ m
an 1 = a n-m = m- n am a
(a )
a = 1, a ¹ 0
n
m
( ab ) a
=a
0
nm
n
n
1 = n a
-n
æaö ç ÷ èbø
-n
n
bn æbö =ç ÷ = n a èaø
n m
n
1 m
n
n
1 m
i = -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
Complex Modulus
a = nm a
n
a na = b nb
( a + bi ) = a - bi ( a + bi )( a + bi ) =
Complex Conjugate
n
a = a, if n is odd
n
a n = a , if n is even
n
x 2 + 2ax + a 2 = ( x + a )
-b ± b 2 - 4ac 2a 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.
x 2 - 2ax + a 2 = ( x - a )
2
x=
2
x 2 + ( a + b ) x + ab = ( x + a )( x + b )
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
x 3 - 3ax 2 + 3a 2 x - a 3 = ( x - a ) x + a = ( x + a ) ( x - ax + a 3
x
1 n
a =a
m n
( ) = (a )
a = a
Properties of Radicals
2
Complex Numbers i = -1
a + bi
The domain of log b x is x > 0 Quadratic Formula Solve ax 2 + bx + c = 0 , a ¹ 0
x3
2
æxö log b ç ÷ = log b x - log b y è yø
Factoring Formulas x 2 - a 2 = ( x + a )( x - a )
d ( P1 , P2 ) =
+ ( y 2 - y1 )
log b ( xy ) = log b x + log b y
log x = log 10 x common log where e = 2.718281828K
x 3 + 3ax 2 + 3a 2 x + a 3 = ( x + a )
( x2 - x1 )
b logb x = x
log b ( x r ) = r log b x
Special Logarithms ln x = log e x natural log
Distance Formula If P1 = ( x1 , y1 ) and P2 = ( x2 , y 2 ) are two points the distance between them is
n
a æaö ç ÷ = n b èbø 1 = an a- n
= a n bn
Triangle Inequality
2
log b b x = x
Example log 5 125 = 3 because 53 = 125
Factoring and Solving
a a = b b
ab = a b
Logarithm Properties log b b = 1 log b 1 = 0
2
3
3
Square Root Property If x 2 = p then x = ± p
) - a = ( x - a ) ( x + ax + a ) - a = ( x - a )( x + a ) 3
2
2
3
2
2
2n
2n
n
n
n
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
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 2 - 6 x - 10 = 0
2
2
(1) Divide by the coefficient of the x 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 x2 - 3x + ç - ÷ = 5 + ç - ÷ = 5 + = 2 2 4 4 è ø è ø
© 2005 Paul Dawkins
Absolute Value Equations/Inequalities If b is a positive number p =b Þ p = -b or p = b
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 - ÷ ÷ . 2 a è 2a ø ø è
Common Algebraic Errors
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 2a ø Circle 2 2 ( x - h) + ( y - k ) = r2 Graph is a circle with radius r and center ( h, k ) .
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.
2
-
(y -k) 2
-
(y -k)
2
=1
( x - h) 2
2
=1
b a 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.
-32 ¹ 9
-32 = -9 ,
(x )
(x )
2
3
¹ x5
2
a a a ¹ + b+c b c 1 ¹ x -2 + x - 3 x2 + x3
( x + a)
2
x+a ¹ x + a n
2
= 9 Watch parenthesis!
= x2 x2 x 2 = x 6
( x + a)
¹ x2 + a 2
x2 + a2 ¹ x + a
( x + a)
3
( -3 )
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
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 + 4 2 = 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 ( x + 1) ¹ ( 2 x + 2 )
2
( 2x + 2)
2
2
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 2
Division by zero is undefined!
2
Hyperbola
( x - h)
Reason/Correct/Justification/Example
2 2 ¹ 0 and ¹ 2 0 0
-a ( x - 1) ¹ -ax - a
Ellipse
( x - h)
Error
© 2005 Paul Dawkins
2
¹ 2 ( x + 1)
( 2x + 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
-x + a ¹ - x + a 2
2
a ab ¹ b æ ö c ç ÷ ècø æaö ç ÷ ac èbø ¹ c b
2
2
-x2 + a2 = ( -x 2 + a 2 )2 Now see the previous error. æaö ç ÷ a 1 æ a ö æ 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 ø è c ø bc ç ÷ è1ø
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
