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10
10.1
Factoring and
Completing the Square
10.2
The Quadratic Formula
10.3
Graphing Parabolas
10.4
More on Quadratic
Equations
10.5
Quadratic and Rational
Inequalities
Chapter
Q
uadratic Equations
and Inequalities
Is it possible to measure beauty? For thousands of years artists and philosophers
have been challenged to answer this question. The seventeenthcentury philoso
pher John Locke said, “Beauty consists of a certain composition of color and ±gure
causing delight in the beholder.” Over the centuries many architects, sculptors,
and painters have searched for beauty in their work by exploring numerical pat
terns in various art forms.
Today many artists and architects still use the concepts of beauty given to us
by the ancient Greeks. One principle,
called the Golden Rectangle, concerns
the most pleasing proportions of a rec
tangle. The Golden Rectangle appears
in nature as well as in many cultures.
Examples of it can be seen in Leonardo
da Vinci’s
Proportions of the Human
Figure
as well as in Indonesian temples
and Chinese pagodas. Perhaps one
of the bestknown examples of the
Golden Rectangle is in the façade and
²oor plan of the Parthenon, built in
Athens in the ±fth century
B
.
C
.
In Exercise 91 of Section 10.4 we
will see that the principle of
the Golden Rectangle is
based on a proportion that
we can solve using the
quadratic formula.
L
W
W
W
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Chapter 10
Quadratic Equations and Inequalities
102
In this Section
•R
eview of Factoring
eview of the EvenRoot
Property
•Completing the Square
•Miscellaneous Equations
• Imaginary Solutions
10.1
Factoring and Completing the Square
Factoring and the evenroot property were used to solve quadratic equations in
Chapters 5, 6, and 9. In this section we ﬁrst review those methods. Then you will
learn the method of completing the square, which can be used to solve any
quadratic equation.
Review of Factoring
A quadratic equation has the form
ax
2
±
bx
±
c
²
0, where
a
,
b
, and
c
are real num
bers with
a
³
0. In Section 5.6 we solved quadratic equations by factoring and then
applying the zero factor property.
Of course we can only use the factoring method when we can factor the quadratic poly
nomial. To solve a quadratic equation by factoring we use the following strategy.
Zero Factor Property
The equation
ab
²
0 is equivalent to the compound equation
a
²
0o
r
b
²
0.
Strategy for Solving Quadratic Equations by Factoring
1.
Write the equation with 0 on one side.
2.
Factor the other side.
3.
Use the zero factor property to set each factor equal to zero.
4.
Solve the simpler equations.
5.
Check the answers in the original equation.
EXAMPLE
1
Solving a quadratic equation by factoring
Solve 3
x
2
´
4
x
²
15 by factoring.
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 Spring '09
 ALKOFAHI
 Algebra, Equations, Inequalities

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