of 65
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MATHEMATICS FOR
BUSINESS
Handouts for University PreparatoryStudents
By
SURESH KUMAR, S.T.,M.SI
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To be familiar with sets, real numbers, real-number line. To relate properties of real numbers in terms of their
operations.
To review the procedure of rationalizing the denominator.
To perform operations of algebraic expressions. To state basic rules for factoring.
To rationalize the denominator of a fraction.
To solve linear equations.
To solve quadratic equations.
Chapter 0: Review of Algebra
Chapter ObjectivesChapter Objectives
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Sets of Real NumbersSome Properties of Real Numbers
Exponents and Radicals
Operations with Algebraic Expressions
Factoring
Fractions
Equations, in Particular Linear Equations
Quadratic Equations
Chapter 0: Review of Algebra
Chapter OutlineChapter Outline
0.1)0.2)
0.3)
0.4)0.5)
0.6)
0.7)
0.8)
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A set is a collection of objects.
An object in a set is called an element of thatset.
Different type of integers:
The real-number line is shown as
Chapter 0: Review of Algebra
0.1 Sets of Real Numbers0.1 Sets of Real Numbers
_ a...,3,2,1!integerspositiveofSet
_ a1,2,3..., !i t rsatift
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The set ofrational numbers consists of
numbers that can be written as aquotient of two integers.
Exe.
Numbers represented by nonterminatingnonrepeating decimals are called
irrational numbers. Exe. (pi) and .
Together rational and irrational numbersform the set of real numbers
2
%60,5.0,2
6,
7
2,
20
19
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Important properties of real numbers
1. The Transitive Property of Equality
2. The Closure Properties of Addition and
Multiplication
3. The Commutative Properties of Addition
and Multiplication
Chapter 0: Review of Algebra
0.2 Some Properties of Real Numbers0.2 Some Properties of Real Numbers
.th,a dIf cacbba !!!
.a d
umb rsr alu iquarth rumb rs,r alallF r
abba
baababba !! a d
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4. The Commutative Properties of Additionand Multiplication
5. The Identity Properties
6. The Inverse Properties
7. The Distributive Properties
Chapter 0: Review of Algebra
0.2 Some Properties of Real Numbers
cabbcacbacba !! a d
aaaa !! 1n0
0! aa 11 ! aa
cabaacbacabcba !! a d
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Chapter 0: Review of Algebra
0.2 Some Properties of Real Numbers
Example 1 Applying Properties of Real Numbers
a. The commutative property of multiplication
b. The associative property of multiplication
Example 3 Applying Properties of RealNumbers
354543.
2323.!
! xwzywzyxSol tion:
a. Show that
Sol tion:
.0for {
! c
ca
c
ab
!
!!
c
ba
cba
cab
c
ab 11
By t e ssoci tive property
By t e efinition of ivision
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Chapter 0: Review of Algebra
0.2 Some Properties of Real Numbers
Example 3 Applying Properties of Real Numbers
b. Show that
Sol tion:
T e efinition of ivision n t e istri tive property.
.0forc
{! cca
cba
c
bc
ac
bac
ba 111!!
c
b
c
a
c
ba
c
b
c
a
cbca
!
!
11
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Properties:
Chapter 0: Review of Algebra
0.3 Exponents and Radicals0.3 Exponents and Radicals
14.
13.
0for11
2.
1.
0
!
!
{
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!
x
xx
xxxxxx
x
xxxxx
n
n
factorsn
n
n
factorsn
n
.
.
nxexponent
se
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Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 1 Exponents
xx
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!!
!!
!
!
1
000
55-
55-
4
e.
1)5(,1,12.
243331
c.
243
1
3
13.
161
21
21
21
21
21.
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Chapter 0: Review of Algebra
0.3 Exponents and Radicals
T e symbol is c lle radical.
n is t e index, xis t e radicand, n is t e
radical sign.
n x
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Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 3 Rationalizing Denominators
Sol tion:
Example 5 Exponents
xxxxxxx 3323323 23 232b.
5
52
5
52
55
52
5
2
5
2.
6 55
6 566 5
1
61
65
61
21
21
21
21
21
!!!
!
!
!
!!
a. Eliminate negative exponents in andsimplify.
Sol tion:
11 yx
xy
xy
yx
yx
!! 1111
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Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 5 Exponents
b. Simplify by using the distributive law.Sol tion:
c. Eliminate negative exponents in
Sol tion:
12/12/12/3 ! xxxx
2/12/3 xx
. xx
2222
22
49
17
7
1777
xxxx
xx !!
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Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 5 Exponents
d. Eliminate negative exponents in
Sol tion:
e. Apply the distributive law toSol tion:
.211
yx
2222
22
211
11
xy
yx
xy
xy
xy
xy
yxyx
!
!
!
!
.2 5
62
15
2
xyx
582152562152 22 xyxxyx !
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Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 7 Radicals
a. Simplify
Sol tion:
b. Simplify
Sol tion:
3233 33 323 46 )( yyxyyxyx !!
7
14
77
72
7
2!
!
.3 46
yx
.2
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Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 7 Radicals
c. Simplify
Sol tion:
d. If x is any real number, simplify
Sol tion:
T s, n
210105
2152510521550250
!
!
.21550250
.2x
u!
0if
0if2
xx
xxx
222! .33
2!
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Simplify and express all answers in terms ofpositive exponents
1.
2.
3.
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Problems 1 Exponents
23
22
59
53
yy
xx
43
2332
x
xx
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Evaluate the expressions.
1.
2.
3.
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Problems 1 Exponents
3
27
8
32
27
64
54
32
1
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Simplify the expressions
1.
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Problems 1 Exponents
3 12827582
43
12
32
3
256.3
8
27.2
x
t
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Write the expressions in terms ofpositive exponents only. Avoid all
radicals in the final form. For example,
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Problems 1 Exponents
y
xxy
211
!
4 322
5 1032
2
35
.3
.2.1
zxyx
zyxc
ba
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Rationalize the denominators
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Problems 1 Exponents
4 2
5
3 2
3
2.3
3
2.2
3
1.1
ba
y
x
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Express all answers in terms of positiveexponents. Rationalize the denominator
where necessary to avoid fractional
exponents in the denominator.
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Problems 1 Exponents
432233 23 32
2
3
2
23.4.3
16
2
1.2
3
243.1
z
zyyxxyyzx
x
x
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If symbols are combined by any or all of theoperations, the resulting expression is called
an algebraic expression.
A polynomia
l in x is an algebraic expressionof the form:
where n=
non-negative integercn = constants
Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions0.4 Operations with Algebraic Expressions
011
1 cxcxcxcn
n
n
n
-
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Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 1 Algebraic Expressions
a. is an algebraic expression in t e
variablex.
b. is an algebraic expression in t e
variable y.
c. is an algebraic expression in t e
variablesxan y.
3
3
10253
xxx
2
3
y
xyyx
27
5310
yy
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Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 3 Subtracting Algebraic Expressions
Simplify
Sol tion: .364123
22
xyxxyx
48
316243)364()123(
364123
2
2
22
22
!
!
!
xyx
xyx
xyxxyx
xyxxyx
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A list of products may be obtained from thedistributive property:
Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
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Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 5 Special Products
a. By R le 2,
b. By R le 3,
103
5252
52
2
2
!
!
xx
xx
xx
204721
45754373
4753
2
2
!
!
zz
zz
zz
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Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 5 Special Products
c. By R le 5,
. By R le 6,
e. By R le 7,
168
4424
2
22
2
!
!
xx
xx
x
8
31
3131
2
22
2
22
!
!
y
y
yy
8
365
427
23233233
23
23
3223
3
!
!
xxx
xxx
x
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Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 7 Dividing a Multinomial by a Monomial
z
zz
z
zzz
xx
xx
3
2
342
2
6384b.
33
a.
2
23
23
!
!
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Perform the indicated operations andsimplify
Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 1 Algebraic Expressions
? A_ a
2
35
23
22
22
22
2
2
146.7
2323.61234.5
54.4
522332.372
435
.2
422106.1
x
xx
xxxxxxx
xx
xxyxyxyxyx
xyzxyx
z
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If two or more expressions are multipliedtogether, the expressions are called thefactors of the product.
Chapter 0: Review of Algebra
0.5 Factoring0.5 Factoring
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Chapter 0: Review of Algebra
0.5 Factoring
Example 1 Common Factors
a. Factor
completely.
Sol tion:
b. Factor completely.
Sol tion:
xkxk 322 93
kxxkxkxk 3393 2322 !
224432325268 zxybayzbayxa
24232232224432325
342
268
xyzbazbyxayazxy
bayz
bayx
a
!
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Chapter 0: Review of Algebra
0.5 Factoring
Example 3 Factoring
zzzz
xxx
yyyyyy
xxxxxxx
!
!
!
!!
1e.
396.
23231836c.
2313299b. 4168
a
.
4/14/54/1
22
23
2
42
2222
333366
23
2222
3/13/13/13/2
24
j.
2428i.bbaah.
4145.
1111f.
yxyxyxyxyxyx
yxyxyx
xxxx
bayxyxyxyx
xxxx
xxxx
!
!
!
!
!
!
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Simplifying Fractions
Allows us to multiply/divide the numerator anddenominator by the same nonzero quantity.
Multiplication and Division of Fractions
The rule for multiplying and dividing is
Chapter 0: Review of Algebra
0.6 Fractions0.6 Fractions
bd
ac
d
c
b
a
!
bc
ad
d
c
b
a
!z
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Factor the following expressions completely.
Chapter 0: Review of Algebra
0.5 Factoring
Problems Factoring
yyxyx
xxxx
xx
pp
dcbcdabbca
24
223
2
2
22433
2.51313
.4
30255.3
34.2
4128.1
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Rationalizing the Denominator
For a denominator with square roots, it maybe rationalized by multiplying an expressionthat makes the denominator a difference oftwo squares.
Addition and Subtraction of Fractions
If we add two fractions having the same
denominator, we get a fraction whosedenominator is the common denominator.
Chapter 0: Review of Algebra
0.6 Fractions
Ch t 0 R i f Al b
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Chapter 0: Review of Algebra
0.6 Fractions
Example 1 Simplifying Fractions
a. SimplifySol tion:
b. Simplify
Sol tion:
.127
62
2
xx
xx
4
2
43
23
127
6
2
2
!
!
x
x
xx
xx
xx
xx
.448
862
2
2
xx
xx
22
4214
412448
8622
2
!
!
x
x
xx
xx
xx
xx
Ch t 0 R i f Al b
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Chapter 0: Review of Algebra
0.6 Fractions
Example 3 Dividing Fractions
412
82
1
1
4
1
821
4
c.
32
5
2
1
3
5
2
35
b.
325
35
253
2a.
222
2
!
!
!
!
!
!
z
xxxx
x
x
x
x
xxx
x
xx
x
xx
x
x
x
x
xx
xx
x
x
x
x
x
x
x
x
Ch t 0 R i f Al b
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Chapter 0: Review of Algebra
0.6 Fractions
Example 5 Adding and Subtracting Fractions
233
2
235
22325a.
2
2
2
!
!
p
pp
p
pp
pp
pp
34
32
2
31
41
65
2
324
5b. 2
2
2
2
!
!
x
xx
xx
xx
xx
xxxx
xxxx
17
7
7
425
149
84
7
2
7
5c.
22
2
22
!
!
!
x
x
x
xxx
xx
x
x
x
x
xx
Chapter 0 Re ie of Algebra
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Chapter 0: Review of Algebra
0.6 Fractions
Example 7 Subtracting Fractions
332615
332
6512102
3323
2
322
92
2
96
2
2
2
2
22
222
!
!
!
xxxx
xx
xxxx
xxxxxx
xx
xxx
Chapter 0: Review of Algebra
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Perform the operations and simplify as muchas possible
Chapter 0: Review of Algebra
0.6 Fractions
Problems Fractions
63
2.8
2
2
1
3.7
45
33
1
4.6
3
65
3.5
45
1
82
22.4
.31
1.22
209.1
2
2
2
2
2
2
3
3
2
2
z
y
x
xx
x
xx
x
x
x
x
x
x
xx
x
xx
x
bax
xc
cx
bax
x
x
xx
xx
Chapter 0: Review of Algebra
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Equations
An equation is a statement that twoexpressions are equal.
The two expressions that make up anequation are called its sides.
They are separated by the equality sign, =.
Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations0.7 Equations, in Particular Linear Equations
Chapter 0: Review of Algebra
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Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Example 1 Examples of Equations
zw
y
y
xx
x
!
!
!
!
7.
64
c.
023b.
32a.2
Avariable (e.g.x, y) is a symbol t at can bereplace byany one ofa set of ifferent
n mbers.
Ch t 0 R i f Al b
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Equivalent Equations
Two equations are said to be equivalent ifthey have exactly the same solutions.
There are three operations that guarantee
equivalence:1. Adding/subtracting the same polynomial
to/from both sides of an equation.
2. Multiplying/dividing both sides of an equation
by the same nonzero constant.
3. Replacing either side of an equation by an equal
expression.
Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Chapter 0: Review of Algebra
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Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Operations That May Not Produce Equivalent
Equations4. Multiplying both sides of an equation by an
expression involving the variable.
5. Dividing both sides of an equation by anexpression involving the variable.
6. Raising both sides of an equation to equal
powers.
Chapter 0: Review of Algebra
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Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Linear Equations
A linear equation in the variable xcan bewritten in the form
where a and b are constants and .
A linear equation is also called a first-degreeequation or an equation of degree one.
0! bax
0{a
Chapter 0: Review of Algebra
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Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Example 3 Solving a Linear Equation
Solve
Sol tion:
.365 xx !
3
26
22
62
60662
062
33365
365
!
!
!
!
!
!
!
x
x
x
x
x
xxxx
xx
Chapter 0: Review of Algebra
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Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Example 5 Solving a Linear Equations
Solve
Sol tion:
.64
89
2
37!
xx
2
105
24145
2489372
64
4
89
2
374
!
!
!
!
!
x
x
x
xx
xx
Chapter 0: Review of Algebra
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Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Literal Equations
Equations where constants are notspecified, but are represented as a, b, c, d,
etc. are called literal equations.
The letters are called literal constants.
Chapter 0: Review of Algebra
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Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Example 7 Solving a Literal Equation
Solve for x.
Sol tion:
ac
ax
aacxaaxxxcxax
axxxca
!
!!
!
2
2
222
22
2
2
2 axxxca !
Chapter 0: Review of Algebra
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Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Example 9 Solving a Fractional Equation
Solve
Sol tion:
Fractional Equations
A fractional equation is an equation in whichan unknown is in a denominator.
.3
6
4
5
!
xx
x
xx
xxx
xxx
!
!
!
9
4635
3634
4534
Chapter 0: Review of Algebra
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Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Example 11 Literal Equation
If express u in terms of the remainingletters; that is, solve for u.
Sol tion:
,vau
u
s
!
sa
svu
svsau
usvsau
uvausvau
us
!
!
!
!
!
1
1
Chapter 0: Review of Algebra
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Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Radical Equations
Aradical equation is one in w ic an nknownocc rs in a radicand.
Example 13 Solving a Radical Equation
Solve
Sol tion:
.33 ! yy
4
2
126
963
33
!
!
!
!
!
y
y
y
yyy
yy
Chapter 0: Review of Algebra
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Solve the equations
g
0.7 Equations, in Particular Linear Equations
Problems Equation
025
21.803
7.4
32.729
47.3
32.6343275.2
2
2
1
1.5935.1
2
!
!
!!
!!
!
!
wwx
yyxx
zzzppp
ppx
Chapter 0: Review of Algebra
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A quadratic equation in the variable xis anequation that can be written in the form
where a, b, and c are constants and
A quadratic equation is also called a second-degreeequation or an equation of degree
two.
0.8 Quadratic Equations0.8 Quadratic Equations
02 ! cbxax
.0{a
Chapter 0: Review of Algebra
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p g
0.8 Quadratic Equations
Example 1 Solving a Quadratic Equation by Factoring
a. Solve
Sol tion:
actor t e left side factor:
Whenever the product of two or more quantitiesis zero, at least one of the quantities must be
zero.
.0122 !xx
043 ! xx
43
04or03
!!
!!
xx
xx
Chapter 0: Review of Algebra
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p g
0.8 Quadratic Equations
Example 1 Solving a Quadratic Equation by Factoring
b. Solve
Sol tion:
.56 2 ww !
6
5r0
056
562
!!
!
!
ww
ww
ww
Chapter 0: Review of Algebra
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p g
0.8 Quadratic Equations
Example 3 Solving a Higher-Degree Equation by
Factoring
a. Solve
Sol tion:
b. Solve
Sol tion:
.044 3 ! xx
1or1or0
0114
014
0442
3
!!!
!
!
!
xxx
xxx
xx
xx
.0252 32 ! xxxxx
? A
7/2r2r0
0722
0252
0252
2
2
32
!!!
!
!
!
xxx
xxx
xxxx
xxxxx
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0.8 Quadratic Equations
Example 5 Solution by Factoring
Solve
Sol tion:
.32 !x
3T s,
3or3033
03
32
2
s!
!!
!
!
!
x
xx
xx
x
x
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0.8 Quadratic Equations
Example 7 A Quadratic Equation with One Real Root
Solve by the quadratic formula.Sol tion:
Here a = 9, b = 62, andc= 2. T e roots are
092622
! yy
3
2
18
026r
3
2
18
026
92026
!
!!
!
s!
yy
y
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Quadratic-Form Equation
When a non-quadratic equation can betransformed into a quadratic equation by an
appropriate substitution, the given equation
is said to have qua
dra
tic-for
m.
0.8 Quadratic Equations
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0.8 Quadratic Equations
Example 9 Solving a Quadratic-Form Equation
Solve
Sol tion:
T is eation c
an be written
asS bstit ting w=1/x3, we ave
T s, t e roots are
.08
9136 !xx
081
91
3
2
3
!
xx
1r8
018
0892
!!
!
!
ww
ww
ww
1or2
1
11
or81
33
!!
!!
xx
xx
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Solve by factoring
Find all roots by using the quadratic
Solve the given quadratic-form equation.
0.8 Quadratic Equations
Problems Quadratic Equation
049.404.33613.20158.13222 !!!! ttxuutt
22224.309124.20242.1 nnxxxx !!
035
2
12
2
1.2
0209.1
2
24
!
!
xx
xx