ALGEBRA - Cheat Sheet
For those who make a di erence
Welcome to the fascinating world of algebra, where equations hold the keys to
unlocking a universe of mathematical possibilities. Algebra may seem daunting
at first, with its complex symbols and seemingly endless rules, but fear not! In
this book, we will guide you through the intricate labyrinth of algebraic
concepts, demystifying the subject and equipping you with the tools to
conquer its challenges. "Mastering Algebra: Unlocking the Secrets with Cheat
Sheets" is designed to be your ultimate companion on this mathematical
journey. Whether you are a student struggling to grasp the fundamentals, a
teacher seeking innovative teaching aids, or a curious mind eager to explore
the beauty of algebra, this book has something for everyone. Our primary goal
is to make algebra accessible and enjoyable for learners of all levels. To
achieve this, we have incorporated a powerful tool into this book: cheat
sheets. These cheat sheets are designed to provide concise summaries of key
algebraic principles, formulas, and strategies. They will serve as your handy
reference, allowing you to quickly refresh your memory and find solutions to
common algebraic problems. But don't let the term "cheat sheets" mislead
you. This book is not about shortcuts or avoiding the hard work necessary to
master algebra. Instead, it is a strategic approach to help you navigate the
intricate web of algebraic concepts more efficiently. By condensing complex
ideas into easily digestible snippets, we aim to enhance your understanding
and reinforce your knowledge. In "Mastering Algebra: Unlocking the Secrets
with Cheat Sheets," we will guide you through a comprehensive exploration of
algebraic topics, starting from the fundamental principles and building up to
advanced concepts. You will gain a solid foundation in essential algebraic
techniques, including solving equations, manipulating variables, simplifying
expressions, and graphing functions. The cheat sheets, strategically placed
within each chapter, will serve as quick references and study aids, saving you
time and effort when you need them most. Whether you are a student
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"Mastering Algebra: Unlocking the Secrets with Cheat Sheets" is your key to
success. Embrace this journey, and watch as the intricate world of algebra
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transforms into a playground of intellectual discovery.
Algebra Cheat Sheet
Basic Properties and Facts
Arithmetic Operations
ab + ac = a (b + c)
a
b
c
=
Properties of Inequalities
b
ab
a
=
c
c
a
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
ab + ac
= b + c, a 6= 0
a
ad
cb =
bc
d
an am = an+m
(ab)n = an bn
(an )m
a0
anm
an
1
= an−m = m−n
m
a
a
n
1
n
1
am = am
= (an ) m
a −n
b
If a < b and c > 0 then ac < bc and
a
b
<
c
c
If a < b and c < 0 then ac > bc and
a
b
>
c
c
Properties
( of Absolute Value
a if a ≥ 0
|a| =
−a if a < 0
|a| ≥ 0
|−a| = |a|
|ab| = |a| |b|
a
|a|
=
b
|b|
|a + b| 6 |a| + |b|
Exponent Properties
=
If a < b then a + c < b + c and a − c < b − c
n
b
bn
=
= n
a
a
= 1 , a 6= 0
a n
b
1
a−n
a−n
=
an
bn
Triangle Inequality
Distance Formula
If P1 = (x1 , y1 ) and P2 = (x2 , y2 ) are two
points the distance between them is
q
d (P1 , P2 ) = (x2 − x1 )2 + (y2 − y1 )2
= an
1
= n
a
Properties of Radicals
√
√
√ √
1
n
n
a = an
ab = n a n b
r
√
n
p√
√
a
a
m n
n
nm
a=
a
= √
n
b
b
√
n n
a = a if n is odd
√
n n
a = |a| if n is even
Complex Numbers
√
i = −1
i2 = −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) = a2 + b2
√
|a + bi| = a2 + b2
Complex Modulus
(a + bi) = a − bi
Complex Conjugate
(a + bi) (a + bi) = |a + bi|2
© Paul Dawkins - https://tutorial.math.lamar.edu
Algebra Cheat Sheet
Logarithms and Log Properties
Definition
y = logb (x) is equivalent to x = by
Logarithm Properties
logb (b) = 1
logb (1) = 0
Example
log5 (125) = 3 because 53 = 125
logb (bx ) = x
blogb (x) = x
Special Logarithms
ln(x) = loge (x)
logb (xy) = logb (x) + logb (y)
x
logb
= logb (x) − logb (y)
y
logb (xr ) = r logb (x)
natural log
log(x) = log10 (x)
common log
where e = 2.718281828 . . .
The domain of logb (x) is x > 0
Factoring and Solving
Factoring Formulas
x2 − a2 = (x + a) (x − a)
Quadratic Formula
Solve ax2 + bx + c = 0, a 6= 0
√
−b ± b2 − 4ac
x=
2a
2
If b − 4ac > 0 – Two real unequal solns.
x2 + 2ax + a2 = (x + a)2
x2 − 2ax + a2 = (x − a)2
x2 + (a + b) x + ab = (x + a) (x + b)
x3
x3
+
3ax2
−
3ax2
+
3a2 x
3a2 x
+
a3
−
a3
= (x + a)
If b2 − 4ac = 0 – Repeated real solution.
3
If b2 − 4ac < 0 – Two complex solutions.
3
= (x − a)
x3 + a3 = (x + a) x2 − ax + a2
x3 − a3 = (x − a) x2 + ax + a2
+
Square Root Property
√
If x2 = p then x = ± p
Absolute Value Equations/Inequalities
x2n − a2n = (xn − an ) (xn + an )
If b is a positive number
|p| = b
⇒
p = −b or
If n is odd then,
xn
−
an
= (x − a)
xn−1
+
axn−2
+ ··· +
an−1
|p| < b
⇒
+
=
|p| > b
(x + a) xn−1 − axn−2 + a2 xn−3 − · · · + an−1
⇒
xn
an
p=b
−b < p < b
p < −b or
p>b
Completing the Square
Solve 2x2 − 6x − 10 = 0
(1) Divide by the coefficient of the x2
x2 − 3x − 5 = 0
(4) Factor the left side
3 2 29
x−
=
2
4
(2) Move the constant to the other side.
x2 − 3x = 5
(5) Use Square Root Property
r
√
3
29
29
x− =±
=±
2
4
2
(3) Take half the coefficient of x, square it and
(6) Solve for x
add it to both sides
√
2
2
3
29
3
3
9
29
x= ±
2
x − 3x + −
=5+ −
=5+ =
2
2
2
2
4
4
© Paul Dawkins - https://tutorial.math.lamar.edu
Algebra Cheat Sheet
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 The graph is a parabola that opens right if
point (0, a).
a > 0 or left if a < 0 and has a vertex at
b
b
g −
,−
.
Line/Linear Function
2a
2a
y = mx + b
or
f (x) = mx + b
Circle
Graph is a line with point (0, b) and slope m.
(x − h)2 + (y − k)2 = r2
Slope
Slope of the line containing the two points
(x1 , y1 ) and (x2 , y2 ) is
y2 − y1
rise
m=
=
x2 − x1
run
Slope – intercept form
The equation of the line with slope m and
y-intercept (0, b) is
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.
y = mx + b
Point – Slope form
The equation of the line with slope m and
passing through the point (x1 , y1 ) is
y = y1 + m (x − x1 )
Parabola/Quadratic Function
y = a (x − h)2 + k
f (x) = a(x − h)2 + k
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 that pass through
b
center with slope ± .
a
The graph is a parabola that opens up if a > 0
Hyperbola
or down if a < 0 and has a vertex at (h, k).
(y − k)2 (x − h)2
−
=1
Parabola/Quadratic Function
b2
a2
y = ax2 + bx + c
f (x) = ax2 + bx + c
Graph is a hyperbola that opens up and down,
The graph is a parabola that opens up if a > 0 has a center at (h, k), vertices b units up/down
from the center and asymptotes that pass
or down if a < 0 and has a vertex at
b
through center with slope ± .
b
b
a
− ,f −
.
2a
2a
© Paul Dawkins - https://tutorial.math.lamar.edu
Algebra Cheat Sheet
Common Algebraic Errors
Error
Reason/Correct/Justification/Example
2
2
6= 0 and 6= 2
0
0
Division by zero is undefined!
−32 6= 9
−32 = −9, (−3)2 = 9 Watch parenthesis!
x2
3
6= x5
x2
3
= x2 x2 x2 = x6
a
a a
6= +
b+c
b
c
1
6= x−2 + x−3
2
x + x3
A more complex version of the
previous error.
a + bx
6= 1 + bx
a
a + bx
a bx
bx
= +
=1+
a
a
a
a
Beware of incorrect canceling!
−a (x − 1) 6= −ax − a
−a (x − 1) = −ax + a
Make sure you distribute the “-”!
(x + a)2 6= x2 + a2
√
x2 + a2 6= x + a
(x + a)2 = (x + a) (x + a) = x2 + 2ax + a2
√
√
√
√
5 = 25 = 32 + 42 6= 32 + 42 = 3 + 4 = 7
√
x + a 6=
√
(x + a)n 6= xn + an and
x+
√
n
√
x + a 6=
(2x + 2)2 6= 2 (x + 1)2
√
−x2 + a2 6= − x2 + a2
a
ab
6=
b
c
c
a
b
c
6=
ac
b
See previous error.
a
2 (x + 1)2 6= (2x + 2)2
√
1
1
1 1
=
6= + = 2
2
1+1
1 1
√
n
x+
√
n
a
More general versions of previous
three errors.
2
2 (x + 1) = 2 x2 + 2x + 1 = 2x2 + 4x + 2
(2x + 2)2 = 4x2 + 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
−x2 + a2 = −x2 + a2 2
Now see the previous error.
a
a c ac
a
= 1 =
=
b
b
1
b
b
c
c
a
a
1
a
b = b = a
=
c
c
b
c
bc
1
© Paul Dawkins - https://tutorial.math.lamar.edu