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
 b  ab
a  =
c c
ab + ac = a ( b + c )
a
  a
b =
c
bc
a
ac
=
b b
 
c
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
a
  ad
b =
 c  bc
 
d
ab + ac
= b + c, a ≠ 0
a
Exponent Properties
n
m
a a =a
n m
(a )
( ab )
n
a −n =
a
 
b
−n
= a nm
n
1
an
n
bn
b
=  = n
a
a
n
m
1
m n
a=
nm
a
Triangle Inequality
Distance Formula
If P1 = ( x1 , y1 ) and P2 = ( x2 , y2 ) are two
points the distance between them is
d ( P1 , P2 ) =
2
( x2 − x1 ) + ( y2 − y1 )
2
n
a
a
=
 
bn
b
1
= an
−n
a
=a b
a = an
a+b ≤ a + b
a 0 = 1, a ≠ 0
n n
a
a
=
b
b
ab = a b
an
1
= a n−m = m−n
m
a
a
n+m
n
Complex Numbers
i = −1
( ) = (a )
a = a
Properties of Radicals
n
Properties of Absolute Value
if a ≥ 0
a
a =
if a < 0
 −a
−a = a
a ≥0
1
m
n
1
m
i 2 = −1
−a = i a , a ≥ 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
n
a na
=
b nb
( a + bi ) = a − bi Complex Conjugate
2
( a + bi )( a + bi ) = a + bi
n
a n = a, if n is odd
n
a n = a , if n is even
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
Complex Modulus
© 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
b logb x = x
log b b x = x
Example
log 5 125 = 3 because 53 = 125
log b ( x r ) = r log b x
log b ( xy ) = log b x + logb y
Special Logarithms
ln x = log e x
natural log
x
log b   = log b x − log b y
 y
log x = log10 x common log
where e = 2.718281828K
The domain of log b x is x > 0
Factoring and Solving
Factoring Formulas
x 2 − a 2 = ( x + a )( x − a )
x 2 + 2ax + a 2 = ( x + a )
2
x 2 − 2ax + a 2 = ( x − a )
2
Quadratic Formula
Solve ax 2 + bx + c = 0 , a ≠ 0
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
−b ± b 2 − 4ac
x=
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.
Square Root Property
If x 2 = p then x = ± p
x3 + a3 = ( x + a ) ( x 2 − ax + a 2 )
x3 − a 3 = ( x − a ) ( x 2 + ax + a 2 )
x 2 n − a 2 n = ( 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 + a n
Absolute Value Equations/Inequalities
If b is a positive number
p =b
⇒
p = −b or p = b
p <b
⇒
−b < p < b
p >b
⇒
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
2
Solve 2 x − 6 x − 10 = 0
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
2
2
9 29
 3
 3
x − 3x +  −  = 5 +  −  = 5 + =
4 4
 2
 2
2
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
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
 b
 b 
at  − , f  −   .
 2a  2 a  
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 ) = r2
Graph is a circle with radius r and center
( h, k ) .
Ellipse
( x − h)
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.
Hyperbola
( x − h)
2
( y −k)
−
2
( x − h)
−
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
=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
≠ 0 and ≠ 2
0
0
Division by zero is undefined!
−32 ≠ 9
−32 = −9 ,
3
( −3 )
2
= 9 Watch parenthesis!
3
( x2 ) ≠ x5
( x2 ) = x2 x2 x2 = x6
a
a a
≠ +
b+c b c
1
≠ x −2 + x −3
2
3
x +x
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 + bx
a
− a ( x − 1) ≠ − ax − a
( x + a)
2
≠ x2 + a2
( x + a)
x2 + a2 ≠ x + a
x+a ≠ x + a
( x + a)
n
≠ x n + a n and
n
x+a ≠ n x + n a
2
= ( x + a )( x + a ) = x 2 + 2ax + a 2
5 = 25 = 32 + 42 ≠ 32 + 42 = 3 + 4 = 7
See previous error.
More general versions of previous three
errors.
2
2
2 ( x + 1) ≠ ( 2 x + 2 )
( 2 x + 2)
2
≠ 2 ( x + 1)
2
2
− x2 + a2 ≠ − x2 + a2
a
ab
≠
b c
 
c
a
  ac
b ≠
c
b
2 ( x + 1) = 2 ( x 2 + 2 x + 1) = 2 x 2 + 4 x + 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 parethesis!
( 2 x + 2)
−x + a = (−x + a
2
2
2
1
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
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
ACT Math Test COORDINATE GEOMETRY CHEAT SHEET
www.MathOnTime.com
Expect NINE Coordinate Geometry problems on your ACT Math Test.
Disclaimers: Even though it is called a “cheat” sheet, do not use this sheet during your actual test! Keep in mind that this page does not cover every concept you may need on the ACT Math Test. Also, this page may contain errors. If you find an error, we’d love to hear about it: service@MathOnTime.com
Geometry – Things to Remember!
3-D Figures:
Regular Solids:
Prism: V = Bh
Tetrahedron – 4 faces
Cube – 6 faces
Octahedron – 8 faces
Dodecahedron – 12 faces
Icosahedron – 20 faces
1
Pyramid: V = Bh
3
Cylinder: V = π r 2 h ; SA = 2π rh + 2π r 2
1
Cone: V = π r 2 h ; SA = sπ r + π r 2
3
4
Sphere: V = π r 3 ; SA = 4π r 2 = π d 2
3
Polygon Interior/Exterior Angles:
Sum of int. angles = 180(n − 2)
180(n − 2)
Each int. angle (regular) =
n
Sum of ext. angles = 360
360
Each ext. angle (regular) =
n
Related Conditionals:
Converse: switch if and then
Inverse: negate if and then
Contrapositive: inverse of the converse
(contrapositive has the same truth value
as the original statement)
Pythagorean Theorem:
c2 = a 2 + b2
Converse: If the sides of a triangle
satisfy c 2 = a 2 + b 2 then the triangle is a
right triangle.
Locus Theorems:
Fixed distance from point.
Fixed distance from a line.
Equidistant from 2 points.
Equidistant 2 parallel lines.
Triangles:
By Sides:
Equidistant from 2
Scalene – no congruent sides
intersecting lines
Isosceles – 2 congruent sides
Equilateral – 3 congruent sides
Congruent Triangles
By Angles:
NO donkey theorem
SSS
Acute – all acute angles
(SSA or ASS)
SAS
Right – one right angle
ASA
Obtuse – one obtuse angle
Equiangular – 3 congruent angles(60º) AAS
HL (right triangles only)
Equilateral ↔ Equiangular
Exterior angle of a triangle equals the
sum of the 2 non-adjacent interior
angles.
Mid-segment of a triangle is parallel
to the third side and half the length of
the third side.
Similar Triangles:
AA
SSS for similarity
SAS for similarity
Corresponding sides of similar
triangles are in proportion.
CPCTC (use after the triangles are congruent)
Inequalities:
--Sum of the lengths of any two sides of a triangle is greater
than the length of the third side.
--Longest side of a triangle is opposite the largest angle.
--Exterior angle of a triangle is greater than either of the
two non-adjacent interior angles.
Mean Proportional in Right Triangle:
Altitude Rule:
Leg Rule:
part hyp
altitude
=
altitude other part hyp
Copyright © Regents Exam Prep Center
hyp
leg
=
leg projection
Parallels: If lines are parallel …
Corresponding angles are equal.
m<1=m<5, m<2=m<6, m<3=m<7, m<4=m<8
Alternate Interior angles are equal.
m<3=m<6, m<4=m<5
Alternate Exterior angles are equal.
m<1=m<8, m<2=m<7
Same side interior angles are supp.
m<3 + m<5=180, m<4 + m<6=180
Circle Segments
In a circle, a radius perpendicular to a chord
bisects the chord.
Intersecting Chords Rule:
(segment part)•(segment part) =
(segment part)•(segment part)
Secant-Secant Rule:
(whole secant)•(external part) =
(whole secant)•(external part)
Secant-Tangent Rule:
(whole secant)•(external part) = (tangent)2
Hat Rule: Two tangents are equal.
Slopes and Equations:
vertical change
y −y
m=
= 2 1.
horizontal change x2 − x1
y = mx + b slope-intercept
y − y1 = m( x − x1 ) point-slope
Quadrilaterals:
Parallelogram:
opp. sides parallel
opp sides =
opp angles =
consec. angles supp
diag bis each other
Rectangle: add 4 rt
angles, diag. =
Rhombus: add 4 =
sides, diag. perp,
diag bisect angles.
Square: All from
above.
Circle Angles:
Central angle = arc
Transformations:
Trapezoid:
Only one set
parallel sides.
Median of trap is
parallel to both
bases and = ½
sum bases.
Isosceles Trap:
legs =
base angles =
diagonals =
opp angles supp
rx −axis ( x, y ) = ( x, − y )
ry −axis ( x, y ) = (− x, y )
ry = x ( x, y ) = ( y, x)
ry =− x ( x, y ) = (− y, − x)
rorigin ( x, y ) = (− x, − y )
Ta ,b ( x, y ) = ( x + a, y + b)
Dk ( x, y ) = (kx, ky )
Inscribed angle = half arc
Angle formed by 2 chords
= half the sum of arcs
Glide
reflection is
composition
of a reflection
and a
translation.
Isometry –
keeps length.
Orientation –
label order
Angle by tangent/chord = half arc
Angle formed by 2 tangents, or 2 secants, or a tangent/secant
= half the difference of arcs
Coordinate Geometry Formulas:
Distance Formula:
d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
Midpoint Formula:
⎛ x + x y + y2 ⎞
( x, y ) = ⎜ 1 2 , 1
⎟
2 ⎠
⎝ 2
Copyright © Regents Exam Prep Center
Circles:
Equation of circle center at origin:
x 2 + y 2 = r 2 where r is the radius.
Equation of circle not at origin:
( x − h) 2 + ( y − k ) 2 = r 2 where (h,k) is the
center and r is the radius.