Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations ab + ac = a ( b + c ) a a b = c bc a c ad + bc + = b d bd a −b b−a = c−d d −c ab + ac = b + c, a ≠ 0 a Exponent Properties a n a m = a n+m an 1 = a n−m = m−n m a a a 0 = 1, a ≠ 0 a a = n b b 1 = an −n a
n n n
b ab a = c c a ac = b b c a c ad − bc − = b d bd a+b a b = + c c c a ad b = c bc d
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 Properties of Absolute Value if a ≥ 0 a a = if a < 0 −a a ≥0 −a = a ab = a b a+b ≤ a + b a a = b b Triangle Inequality
Distance Formula If P = ( x1 , y1 ) and P2 = ( x2 , y2 ) are two 1 points the distance between them is d ( P , P2 ) = 1
(a )
n m
= a nm = a nb n 1 an bn b = = n a a
( x2 − x1 ) + ( y2 − y1 )
2
2
( ab )
n
Complex Numbers i = −1
n
1 m
a −n = a b
−n
i 2 = −1
−a = i a , a ≥ 0
a = a
n m
( ) = (a )
1 m
n
Properties of Radicals
n
( 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 = an a = nm a
1
n
ab = n a n b a na = b nb
m n
n
n n
a n = a, if n is odd a n = a , if n is even
( a + bi ) = a − bi Complex Conjugate 2 ( a + bi )( a + bi ) = a + bi
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 Example log 5 125 = 3 because 53 = 125 Special Logarithms ln x = log e x natural log log x = log10 x common log where e = 2.718281828K Factoring Formulas x 2 − a 2 = ( x + a )( x − a ) x 2 + 2ax + a 2 = ( x + a ) x 2 − 2ax + a 2 = ( x − a )
2 2
Logarithm Properties log b b = 1 log b 1 = 0 log b b x = x log b ( x r ) = r log b x b logb x = x
log b ( xy ) = log b x + log b y x log b = log b x − log b y y The domain of log b x is x > 0
Factoring and Solving
Quadratic Formula Solve ax 2 + bx + c = 0 , a ≠ 0 −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= Square Root Property If x 2 = p then x = ± p Absolute Value Equations/Inequalities If b is a positive number p =b ⇒ p = −b or p = b p b ⇒ ⇒ −b < p < b p < −b or p>b
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
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 = ( x + a ) ( x n −1 − ax n − 2 + a 2 x n −3 − L + a n −1 )
2
Solve 2 x − 6 x − 10 = 0
Completing the Square (4) Factor the left side 3 29 x− = 2 4 (5) Use Square Root Property 3 29 29 x− = ± =± 2 4 2 (6) Solve for x 3 29 x= ± 2 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 9 29 3 3 x 2 − 3x + − = 5 + − = 5 + = 4 4 2 2
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 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 b b at g − , − . 2a 2 a Circle 2 2 ( x − h) + ( y − k ) = r 2 Graph is a circle with radius r and center ( h, k ) . Ellipse =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 =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 =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
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 )
( x − h)
2
( y −k) +
2
( x − h)
2
( y −k) −
2
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 b b at − , f − . 2a 2 a
(y −k)
2
−
( x − h)
2
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
Common Algebraic Errors
Error 2 2 ≠ 0 and ≠ 2 0 0 −32 ≠ 9 Reason/Correct/Justification/Example Division by zero is undefined! −32 = −9 ,
( −3 )
2
= 9 Watch parenthesis!
(x )
2 3
≠ x5
(x )
2 3
= x2 x2 x2 = x6
a a a ≠ + b+c b c 1 ≠ x −2 + x −3 2 3 x +x a + bx ≠ 1 + bx a − a ( x − 1) ≠ − ax − a
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 “-“!
( x + a)
2
≠ x2 + a2
( x + a)
2
= ( x + a )( x + a ) = x 2 + 2ax + a 2
x2 + a2 ≠ x + a x+a ≠ x + a
( x + a)
n
≠ x n + a n and
2 2
n
x+a ≠ n x + n a
5 = 25 = 32 + 42 ≠ 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
2 ( x + 1) ≠ ( 2 x + 2 )
( 2 x + 2)
2
≠ 2 ( x + 1)
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 parethesis! − x2 + a2 = ( − x2 + a 2 ) 2 Now see the previous error. a a 1 a c ac = = = b b 1 b b c c a a b = b = a 1 = a c c b c bc 1
1
( 2 x + 2)
− x2 + a2 ≠ − x2 + a2 a ab ≠ b c c a ac b ≠ c b
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins