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Reference Guide

# College Mathematics - Reference Guides

4 pages495 viewsFall 2015

Department
BAD - Business Administration
Course Code
BAD 200
Professor
All
Chapter
Permachart

This preview shows page 1. to view the full 4 pages of the document.
ALGEBRA
l e a r n r e f e r e n c e r e v i e w
College Mathematics
College Mathematics
ALGEBRAIC LAWS
SPECIAL PRODUCT FORMULAS
LOGARITHMS
Sign Laws
Commutative Laws Associative Laws
Distributive Laws Laws Involving Zero
Exponent Laws
• Given an = b, then the following are true:
Logarithm: n = logab = logarithm of bto the base a
Anti-logarithm: b= antilogan= anti-logarithm of nto the base a
Common logarithm (log): Logarithm to the base 10
Natural logarithm (In): Logarithm to the base e= 2.718282
Remember: Logarithms are exponents
Laws of Logarithms
Change of Base
Special Relations
SOLUTION OF EQUATIONS
Two Linear Equations in Two Variables
Algebraic Procedure
Solve either equation for one unknown in terms of the other
Substitute solution into second equation and solve to obtain
first unknown
Substitute this unknown into either of the original equations
and solve for second unknown
Graphical Procedure
Equations: x+ y= 1 and x+ y= 0
Solution: None
Reason: • Lines do not intersect
• Equations are inconsistent,
as x+ycannot be both
0 and 1
Equations: x+ y= 1 and xy= 0
Solution: One
Reason: • Lines intersect at single
point: x= 1/2 and y= 1/2
• Unique solution
Equations: x+y= 1 and 2x+ 2y= 2
Solution: Infinite number of solutions
Reason: • Lines are identical
• Equations are dependent
(i.e., multiples of each other)
Quadratic Equations in One Variable
• The roots of the quadratic equation ax2+ bx + c= 0
are given by the relation:
• The nature of the roots is determined by the discriminant
D = b24ac as follows:
−−
()
=
()
=−
()
()
=
()
=−
()
()
=
()()
=
()
=−
()
=− −
()()
=−
()
=− −
()()
=−
()()
=−
AAA ABABAB ABABAB
AAA ABABAB AB ABAB
AB BA AB BA+=+ =
AA A A A AB AB
AA A A A AB AB
AAAAB BA
AA
mnmn nmmm
mn mn m m mm m
mnmn mm
xyx
y
n
== =
()
== =
()
()
==
()
=
()
=
+
−−
1
1
1
0
ABC ABCABC ABC+ +
()
=+
()
+
()
=
()
AB A ABB AB A AB AB B
AB ABAB AB ABAABB
ABC A B C ABC BC
AB
±
()
=± + ±
()
=± + ±
−=+
()
()
±=±
()
+
()
+++
()
=+++
()
+++
()
++
()
±
(
222 332 23
22 33 2 2
222 2
233
22

))
+
()
±
()
()
+
−− −
nnn n n
AnABnn AB nn n AB
1
1
12
12
12 3
122 33
()( ) ()( )( )
AB A B
A
A
===
=∞
=
000
0
0000
means and or
or does not exist
is indeterminate
AB C AB AC
AB A B
mmm
+
()
=+
()
=
log log log log log
log log log log log
aaaa
na
aaaa
na
xy x y x n x
xy x y x nx
()
=+ =±
()
=− =
±
1
log log log log . ln ln . log
aAA
xxax xx x===0 43429 2 30259
xABi
DxBi
=
=
2Roots are real and unequal
Roots are real and equal
Roots are complex conjugates:
Roots are pure imaginary:
1,2
1,2
xbb ac
a
12
24
2
=−± −
,
Binomial Series
COMPLEX NUMBERS
Imaginary unit
Complex number
Conjugate
complex numbers
Modulus r:
Argument
q
:
Complex number
(polar form)
ii nin=−=− −=1 1
2
or Always write:
0
zabi z
a
" " 
"
Example:
If
23i
then z is pure imaginary ; 0b"If then z is real
zabi z abi
12
=+ =−and
zr i= +(cos sin )
θθ
Equality
Addition/subtraction
Multiplication
Multiplication of conjugates
Multiplication in polar form
Division
Division in polar form
Integer powers of i
Algebraic Operations
()()()()a bi c di a c b d i+±+ = ±+±
()()( )( )a bi c di ac bd ad bc i++= − + +
()()a bi a bi a b+−=+
22
zz rr i
12 12 1 2 1 2
=+++[cos( ) sin( )]
θθ θθ
abi
cdi
abicdi
cdicdi
ac bd bc ad i
cd
+
+=+−
+−
=++
+
()()
()()
()()
22
iiiiiii
2341 2
11 1=−== =− =
−−
;;; ;
© 1996-2014 Mindsource Technologies Inc.
COLLEGE MATHEMATICS • A-632-71
TM
permacharts
www.permacharts.com
Y
X
1
1
Y
X
1
1
Y
X
1
1
Db ac
Db a
=-4>0 Rootsarerealandunequal
=-4
2
2cc
Db ac
=0 Rootsarerealandequal
=-4<0 Roots
2 are complex conjugates : = ±
=-4 <0
1
Dac( ) Rootsarepureimaginary  =±
1
b=0 x Bi:
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