Algebra Cheat Sheet Basic Properties & Facts Arithmetic Operations
b = ab c c
ab + ac = a ( b + c )
a b = a b
a
a
c
c
ad + bc
a
d
bd
b
a−b c−d
=
ab + ac
If a < b and c < 0 then ac > bc and
c
ad ad − bc
d
bd
− =
b−a
a+ b
d −c
c a
= b + c,
a
If a < b and c > 0 then ac < bc and
a ac = b b c
bc
+ =
Properties of Inequalities If a < b then a + c < b + c and a − c < b − c
a
b
c
c
= +
b = ad c bc d
a≠0
( an )
a
= an +m
a n am m
a
= a nm
= an −m =
m
a
a0
= 1,
≥0
ab
=
a+b
−a = a
a b
≤
a
=
b
+b
c
>
b c b c
a a b
Triangle Inequality
Distance Formula If P1 = ( x1 , y1 ) and P2
1 a
c a
<
Properties of Absolute Value if a ≥ 0 a a = if a < 0 −a
Exponent Properties n
a
= ( x2 , y2 ) are two
points the distance between them is
m−n
a≠0
d ( P1 , P2 )
= ( x2 − x1 ) 2 + ( y2 − y1 ) 2
n
n
( ab ) = a b a
−n
n
a = a n b bn
n
1
=
a
1
n
−n
a =b b a
a n
= bn an
−n n
am
n
a
m n
=a a
n
an
n
an
1 n
= nm a
n
n
ab a b
= n an b
=
= an
= (a
Properties of Radicals
Complex Numbers
1 m
i=
)
n
= ( an )
1 m
−1
= −1 −a = i a , a ≥ 0 di) = a + c + ( b + d ) i ( a + bi ) + ( c + di ( a + bi ) − ( c + di) = a − c + ( b − d ) i b d + ( ad + bc) i ( a + bi ) ( c + di ) = ac − bd ( a + bi ) ( a − bi ) = a2 + b2 a + bi
=
i2
a 2 + b2
Complex Modulus
n
a
bi ) = a − bi bi ( a + bi
n
b
bi ) ( a + bi b i ) = a + bi bi ( a + bi
= a, if n is odd = a , if n is even even
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu http://tutorial.math.lamar.edu..
Complex Conjugate 2
© 2005 Paul Dawkins
Logarithms and Log Properties Definition y = log b x is equivalent to x = b y
Example log 5 125 = 3 because 53
Logarithm Properties logb b = 1 logb 1 = 0
= x blog x = x logb ( x r ) = r logb x logb ( y ) = log b x + logb y x logb = logb − logb y y logb b x
= 125
Special Logarithms ln x = log e x natural log log x = log10 x
common log
where e = 2.718281828 K
b
is x > 0
The domain of logb
Factoring and Solving Factoring Formulas x 2 − a 2 = ( x + a )( x − a)
x 2 + 2 ax + a2
Quadratic Formula Solve ax 2 + bx + c = 0 , a ≠ 0
= ( x + a) 2
x =
b 2 − 4 ac
2a 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 − a)2 x 2 + ( a + b ) x + ab = ( x + a)( x + b) 3 x3 + 3ax 2 + 3a 2 x + a3 = ( x + a) x 2 − 2ax + a2
2
= ( x − a )3 x3 + a3 = ( x + a ) ( x2 − ax + a2 ) x3 − 3ax 2 + 3a2 x − a3
Square Root Property
If x 2
= ( x − a ) ( x2 + ax + a2 ) x 2 n − a 2 n = ( xn − an )( xn + an ) x3 − a3
=
p then x = ± p
Absolute Value Equations/Inequalities If b is a positive number p = b p = − b or p = b ⇒
If n is odd then, x n − an
−b ±
= ( x − a ) ( xn−1 + axn −2 + L + an−1 )
p p
x n + a n
b
⇒ ⇒
−b < p < b p < − b or
p> b
= ( x + a ) ( xn−1 − axn−2 + a2 xn−3 − L + an−1 ) Solve 2 x
2
Completing the Square (4) Factor the left side
− 6 x − 10 = 0
2
(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
x
2
3 3 − 3 x + − = 5 + − 2 2
2
9 = 5 + = 4
x − 3 = 29 2 4 (5) Use Square Root Property x −
3
2 (6) Solve for x
29 4
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
29
=±
x =
4 3 2
±
=±
29 2
29 2
© 2005 Paul Dawkins
Functions and Graphs Constant Function y = a or f ( x) = a
Parabola/Quadratic Function x = ay 2 + by + c g ( y ) = ay 2 + by + c
Graph is a horizontal line passing through the point ( 0, a ) .
The graph is a parabola that opens right if a > 0 or left if a < 0 and has a vertex
Line/Linear Function y = mx + b or f ( x ) = mx + b
at g −
Graph is a line with point ( 0,b ) and
Circle
slope m.
2
2
( − h) + ( y − k ) = r2 Graph is a circle with radius r and center ( h, k ) .
Slope Slope of the line containing the two points ( 1 , y1 ) and ( x2 , y2 ) is m=
y2 − y1 x2 − x1
=
rise
Ellipse
run
( x − h )
= mx + b Point – Slope form The equation of the line with slope m and passing through the point ( 1 , y1 ) is y = y1 + m ( x − x1 ) Parabola/Quadratic Function
y = a ( x − h )
2
+k
f ( x) = a ( x − h)
2
+ 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 = ax 2 + bx + c f ( x ) = ax2 + bx + c
The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex b b . − 2a , f − 2a
2
( y − k )
2
+ =1 a2 b2 Graph is an ellipse with center ( h, k )
Slope – intercept form The equation of the line with slope m and y-intercept ( 0,b ) is
at
, − b . 2a 2a b
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
2 0
≠ 0 and
2 0
≠2
Division by zero is undefined!
−32 = −9 , ( −3) 2 = 9 Watch parenthesis!
−32 ≠ 9 3
( ) ≠ x5 2
a b+c 1
3
= x 2 x2 x2 = x6
a
1
b
c
2 1+ 1 1 1 A more complex version of the previous error.
≠ x −2 + x −3
3
a
( x2 )
a
≠ +
+ x a + bx 2
Reason/Correct/Justification/Example
1
=
a + bx
≠ 1 + bx
a
= +
bx
Make sure you distribute the “-“!
2
2
( x + a ) = ( x + a)( x + a) = x2 + 2 ax + a2
x 2 + a 2
≠ x+ a x + a ≠ x + a
5 = 25 = 32 + 42
n
( + a ) ≠ x n + an and n x + a ≠ n x + n a
≠ ( 2 x + 2) 2
( 2 x + 2 ) ≠ 2 ( x + 1)
2
≠
32
+
42
= 3+ 4 = 7
See previous error. More general versions of previous three errors. 2 ( x + 1)
2
= 1+
−a ( x − 1) = − ax + a
( + a ) ≠ x2 + a2
2
bx
a a a a Beware of incorrect canceling!
−a ( x − 1) ≠ − ax − a
2 ( x + 1)
1 1
≠ + =2
2
= 2 ( x2 + 2 x + 1) = 2 x2 + 4 x + 2
2
( 2 x + 2) = 4 x2 + 8 x + 4 Square first then distribute! See the previous example. You can not factor out a constant if there is a power on the parethesis! 1
− x 2 + a 2 ≠ − a ab ≠ b c c
a b ac ≠ c
b
x 2 + a2
− + a = (− x + a )2 2
2
2
2
Now see the previous error. a 1 a c ac a = = = b b 1 b b
c c a a b b a 1 a = = c = bc c c b 1
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
