Algebra cheat sheet

Algebra cheat sheet

1. For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu . © 2005 Paul Dawkins
Algebra Cheat Sheet

Basic Properties & Facts
Arithmetic Operations
()
,0
bab
abacabca
cc
a
aaacb
bcbcb
c
acadbcacadbc
bdbdbdbd
abbaabab
cddcccc
a
abacad b
bca
cabc
d
æö
+=+=
ç÷
èø
æö
ç÷
èø
==
æö
ç÷
èø
+-
+=-=
--+
==+
--
æö
ç÷
+ èø
=+¹=
æö
ç÷
èø
Exponent Properties
()
()
()()
1
1
0
1
1, 0
11
n
m
mm
n
nmnmnm
mmn
m
nnm
n
n
n
nn
n
nn
nn
nn
n
n
n
n
a
aaaa
aa
aaaa
aa
abab
bb
aa
aa
abb
aaa
baa
+-
-
-
-
-
===
==¹
æö
==
ç÷
èø
==
æöæö
====
ç÷ç÷
èøèø

Properties of Radicals

1
,if is odd
,if is even
nnnnn
n
mnnm
n
n
nn
nn
aaabab
aa
aa
bb
aan
aan
==
==
=
=




Properties of Inequalities
If thenand
If and 0 then and
If and 0 then and
abacbcacbc
ab
abcacbc
cc
ab
abcacbc
cc
<+<+-<-
<><<
<<>>

Properties of Absolute Value
if 0
if 0
aa
a
aa
³ì
=í
-<î

0
Triangle Inequality
aaa
aa
abab
bb
abab
³-=
==
+£+


Distance Formula
If ()
111
,Pxy= and ( )
222
,Pxy= are two
points the distance between them is

( )( )( )
22
122121
,dPPxxyy =-+-

Complex Numbers

( )( ) ( )
( )( ) ()
( )( ) ( )
( )( )
( )
( )( )
2
22
22
2
11,0
Complex Modulus
Complex Conjugate
iiaiaa
abicdiacbdi
abicdiacbdi
abicdiacbdadbci
abiabiab
abiab
abiabi
abiabiabi
=-=--=³
+++=+++
+-+=-+-
++=-++
+-=+
+=+
+=-
++=+

2. For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu . © 2005 Paul Dawkins
Logarithms and Log Properties
Definition
log is equivalent to
y
b
yxxb==

Example
3
5
log1253 because 5125==

Special Logarithms
10
lnlognatural log
loglogcommon log
e
xx
xx
=
=

where 2.718281828e= K
Logarithm Properties
()
()
log
log1log10
log
loglog
logloglog
logloglog
b
bb
xx
b
r
bb
bbb
bbb
b
bxbx
xrx
xyxy
x
xy
y
==
==
=
=+
æö
=-
ç÷
èø


The domain of log
b
x is 0x>
Factoring and Solving
Factoring Formulas
()()
()
()
() ()()
22
2
22
2
22
2
2
2
xaxaxa
xaxaxa
xaxaxa
xabxabxaxb
-=+-
++=+
-+=-
+++=++

()
()
()( )
()( )
3
3223
3
3223
3322
3322
33
33
xaxaxaxa
xaxaxaxa
xaxaxaxa
xaxaxaxa
+++=+
-+-=-
+=+-+
-=-++

( )( )
22nnnnnn
xaxaxa-=-+
If n is odd then,
()( )
()( )
121
12231
nnnnn
nn
nnnn
xaxaxaxa
xa
xaxaxaxa
---
----
-=-+++
+
=+-+-+
L
L
Quadratic Formula
Solve
2
0axbxc++= , 0a¹
2
4
2
bbac
x
a
-±-
=
If
2
40bac-> - Two real unequal solns.
If
2
40bac-= - Repeated real solution.
If
2
40bac-< - Two complex solutions.

Square Root Property
If
2
xp= then xp=±

Absolute Value Equations/Inequalities
If b is a positive number
or
or
pbpbpb
pbbpb
pbpbpb
=Þ=-=
<Þ-<<
>Þ<->

Completing the Square
Solve
2
26100xx--=

(1) Divide by the coefficient of the
2
x
2
350xx--=
(2) Move the constant to the other side.
2
35xx-=
(3) Take half the coefficient of x, square
it and add it to both sides
22
2 33929
355
2244
xx
æöæö
-+-=+-=+=
ç÷ç÷
èøèø

(4) Factor the left side
2
329
24
x
æö
-=
ç÷
èø

(5) Use Square Root Property
32929
242
x-=±=±
(6) Solve for x
329
22
x=±

3. For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu . © 2005 Paul Dawkins
Functions and Graphs
Constant Function
()or yafxa==
Graph is a horizontal line passing
through the point ()0,a.

Line/Linear Function
()or ymxbfxmxb=+=+
Graph is a line with point ()0,b and
slope m.

Slope
Slope of the line containing the two
points ()
11
,xy and ( )
22
,xy is
21
21
rise
run
yy
m
xx
-
==
-

Slope – intercept form
The equation of the line with slope m
and y-intercept ()0,b is
ymxb=+
Point – Slope form
The equation of the line with slope m
and passing through the point ()
11
,xy is
( )
11
yymxx=+-

Parabola/Quadratic Function
() () ()
22
yaxhkfxaxhk=-+=-+

The graph is a parabola that opens up if
0a> or down if 0a< and has a vertex
at (),hk.

Parabola/Quadratic Function
()
22
yaxbxcfxaxbxc=++=++

The graph is a parabola that opens up if
0a> or down if 0a< and has a vertex
at ,
22
bb
f
aa
æö æö
--
ç÷ç÷
èøèø
.

Parabola/Quadratic Function
()
22
xaybycgyaybyc=++=++

The graph is a parabola that opens right
if 0a> or left if 0a< and has a vertex
at ,
22
bb
g
aa
æöæö
--
ç÷ç÷
èøèø
.

Circle
()( )
22
2
xhykr-+-=
Graph is a circle with radius r and center
(),hk.

Ellipse
()( )
22
22
1
xhyk
ab
--
+=
Graph is an ellipse with center (),hk
with vertices a units right/left from the
center and vertices b units up/down from
the center.

Hyperbola
()( )
22
22
1
xhyk
ab
--
-=
Graph is a hyperbola that opens left and
right, has a center at (),hk, vertices a
units left/right of center and asymptotes
that pass through center with slope
b
a
±.
Hyperbola
( )()
22
22
1
ykxh
ba
--
-=
Graph is a hyperbola that opens up and
down, has a center at (),hk, vertices b
units up/down from the center and
asymptotes that pass through center with
slope
b
a
±.

4. 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
0
0
¹ and
2
2
0
¹ Division by zero is undefined!
2
39-¹
2
39-=- , ()
2
39-= Watch parenthesis!
()
3
25
xx¹ ()
3
22226
xxxxx==
aaa
bcbc
¹+
+

1111
2
21111
=¹+=
+

23
23
1
xx
xx
--
¹+
+

A more complex version of the previous
error.
abx
a
+
1bx¹+
1
abxabxbx
aaaa
+
=+=+
Beware of incorrect canceling!
()1axaxa--¹--
()1axaxa--=-+
Make sure you distribute the “-“!
()
2
22
xaxa+¹+ ()()()
2
22
2xaxaxaxaxa+=++=++
22
xaxa+¹+
2222
5253434347==+¹+=+=
xaxa+¹+ See previous error.
()
n
nn
xaxa+¹+ and
nnn
xaxa+¹+
More general versions of previous three
errors.
()( )
22
2122xx+¹+
()( )
2
22
21221242xxxxx+=++=++
( )
2
2
22484xxx+=++
Square first then distribute!
( )()
22
2221xx+¹+
See the previous example. You can not
factor out a constant if there is a power on
the parenthesis!
2222
xaxa-+¹-+
( )
1
2222
2
xaxa-+=-+
Now see the previous error.
aab
b c
c
¹
æö
ç÷
èø
1
1
a
aacac
bb bb
cc
æö
ç÷
æöæöèø
===
ç÷ç÷
æöæö èøèø
ç÷ç÷
èøèø

a
acb
cb
æö
ç÷
èø
¹
1
1
aa
aabb
ccbcbc
æöæö
ç÷ç÷
æöæöèøèø
===
ç÷ç÷
æöèøèø
ç÷
èø