C
H
A
P
T
E
R
9
Quadratic Equations
and Quadratic
Functions
hroughout time, humans have been building bridges over waterways. Primitive people threw logs across streams or attached
ropes to branches to cross the waters. Later, the Romans built
stone structures to span rivers and chasms. Throughout the centuries, bridges have been made of wood and stone and later from cast
iron, concrete, and steel. Today’s bridges are among the most beautiful
and complex creations of modern engineering. Whether the bridge spans
a small creek or a four-mile-wide stretch of water, mathematics is a
part of its very foundation.
The function of a bridge, the length it must span, and the load it
must carry often determine the type of bridge that is built. Some common types designed by civil engineers are cantilevered, arch, cablestayed, and suspension bridges. The military is known for building
trestle bridges and floating or pontoon bridges.
New technology has enabled engineers to build bridges that
are stronger, lighter, and less expensive than in the past, as well
as being esthetically pleasing. Currently, some engineers are
working on making bridges earthquake resistant. Another
idea that is being explored is incorporating carbon fibers
in cement to warn of small cracks through electronic
signals.
In Exercise 45 of Section 9.6 you will see how
quadratic equations and functions are used in
designing suspension bridges.
T
y
20
10
z
x
0
5 10 15
25 30 35 40
454
(9-2)
Chapter 9
Quadratic Equations and Quadratic Functions
9.1
In this
section
●
Definition
●
Using the Square Root
Property
●
Solving Equations by
Factoring
THE SQUARE ROOT PROPERTY
AND FACTORING
We solved some quadratic equations in Chapters 5 and 6 by factoring. In Chapter 8
we solved some quadratic equations using the square root property. In this section we will review the types that you have already learned to solve. In the next
section you will learn a method by which you can solve any quadratic equation.
Definition
We saw the definition of a quadratic equation in Chapter 5, but we will repeat it here.
Quadratic Equation
A quadratic equation is an equation of the form
ax 2 bx c 0,
where a, b, and c are real numbers with a 0.
Equations that can be written in the form of the definition may also be called
quadratic equations. In Chapters 5, 6, and 8 we solved quadratic equations such as
x 2 10,
5(x 2)2 20,
and
x 2 5x 6.
Using the Square Root Property
If b 0 in ax 2 bx c 0, then the quadratic equation can be solved by using
the square root property.
E X A M P L E
1
Using the square root property
Solve the equations.
b) 2x 2 3 0
a) x 2 9 0
c) 3(x 1)2 6
Solution
a) Solve the equation for x 2, and then use the square root property:
x2 9 0
x2 9
Add 9 to each side.
x 3 Square root property
Check 3 and 3 in the original equation. Both 3 and 3 are solutions to
x 2 9 0.
b) 2x 2 3 0
2x 2 3
3
x2 2
2
x
3
Square root property
3
2
x 2
2
6
x 2
Rationalize the denominator.
Check. The solutions to 2x 2 3 0 are
6
2
6
and .
2
9.1
study
tip
Effective studying involves
actively digging into the subject. Make sure that you are
making steady progress. At
the end of each week, take
note of the progress that you
have made. What do you
know on Friday that you didn’t
know on Monday?
The Square Root Property and Factoring
(9-3)
455
c) This equation is a bit different from the previous two equations. If we actually
squared the quantity (x 1), then we would get a term involving x. Then b
would not be equal to zero as it is in the other equations. However, we can solve
this equation like the others if we do not square x 1.
3(x 1)2 6
(x 1)2 2
x 1 2
x 1 2
Divide each side by 3.
Square root property
Subtract 1 from each side.
Check x 1 2
in the original equation:
3(1 2 1)2 3(2)2 3(2) 6
The solutions are 1 2
and 1 2.
E X A M P L E
2
■
A quadratic equation with no real solution
Solve x 2 12 0.
Solution
The equation x 2 12 0 is equivalent to x 2 12. Because the square of any
■
real number is nonnegative, this equation has no real solution.
Solving Equations by Factoring
In Chapter 5 you learned to factor trinomials and to use factoring to solve some quadratic equations. Recall that quadratic equations are solved by factoring as follows.
Strategy for Solving Quadratic Equations by Factoring
1.
2.
3.
4.
5.
E X A M P L E
helpful
3
hint
After you have factored the
quadratic polynomial, use
FOIL to check that you have
factored correctly before proceeding to the next step.
Write the equation with 0 on one side of the equal sign.
Factor the other side.
Use the zero factor property. (Set each factor equal to 0.)
Solve the two linear equations.
Check the answers in the original quadratic equation.
Solving a quadratic equation by factoring
Solve by factoring.
b) 3x 2 13x 10 0
a) x 2 2x 8
1
1
c) x 2 x 3
6
2
Solution
a)
x 2 2x 8
2
x 2x 8 0
Get 0 on the right-hand side.
(x 4)(x 2) 0
Factor.
x40
or
x 2 0 Zero factor property
x 4
or
x 2 Solve the linear equations.
Check in the original equation:
(4)2 2(4) 16 8 8
22 2 2 4 4 8
Both 4 and 2 are solutions to the equation.
456
(9-4)
Chapter 9
Quadratic Equations and Quadratic Functions
b) 3x 2 13x 10 0
(3x 2)(x 5) 0
3x 2 0
3x 2
2
x
3
Factor.
or
or
or
x 5 0 Zero factor property
x 5
x 5
2
Check in the original equation. Both 5 and are solutions to the equation.
3
1 2 1
c)
x x 3
6
2
x 2 3x 18
Multiply each side by 6.
2
Get 0 on the right-hand side.
x 3x 18 0
Factor.
(x 6)(x 3) 0
Zero factor property
x60
or
x30
x6
or
x 3
Check in the original equation. The solutions are 3 and 6.
■
CAUTION
You can set each factor equal to zero only when the product of
the factors is zero. Note that x 2 3x 18 is equivalent to x(x 3) 18, but you
can make no conclusion about two factors that have a product of 18.
M A T H
A T
W O R K
Even as a child, Gregory Brown was building tunnels
and bridges in the sand. Now as a structural engineer
for Stone and Webster, he is designing and analyzing
real-life bridges and buildings. Many factors must be
considered in analyzing a new or existing structure.
For example, in the northern locations the effects of
the weight of snowfall on a roof must be considered,
while in the southern areas the strength of hurricane
winds must be taken into account. Of course, earthquakes pose yet another consideration.
At the present time, Mr. Brown is working on bridge ratings. To rate a bridge, he
first studies the design and construction of the bridge. Then he examines the structure for any kind of deterioration such as rust, cracks, or holes. In addition to the
traffic load a bridge carries, the weight and size of trucks using the bridge are evaluated. When all this information is collected, collated, and analyzed, a bridge rating
report is submitted. This report notes the extent of the deterioration, provides recommendations regarding vehicle weight limitations, and provides suggested repairs
to strengthen the structure.
In Exercises 57 and 58 of this section you will see how an engineer can use a
quadratic equation to find the length of a diagonal brace on a bridge.
STRUCTURAL
ENGINEER
9.1
WARM-UPS
(9-5)
The Square Root Property and Factoring
457
True of false? Explain your answer.
Both 4 and 4 satisfy the equation x 2 16 0. True
. False
The equation (x 3)2 8 is equivalent to x 3 22
Every quadratic equation can be solved by factoring. False
Both 5 and 4 are solutions to (x 4)(x 5) 0. True
The quadratic equation x 2 3 has no real solutions. True
The equation x 2 0 has no real solutions. False
The equation (2x 3)(4x 5) 0 is equivalent to x 23 or x 54.
False
8. The only solution to the equation (x 2)2 0 is 2. True
9. (x 3)(x 5) 4 is equivalent to x 3 2 or x 5 2. False
10. All quadratic equations have two distinct solutions. False
1.
2.
3.
4.
5.
6.
7.
9. 1
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a quadratic equation?
A quadratic equation is an equation of the form
ax 2 bx c 0, where a 0.
2. What property do we use to solve quadratic equations in
which b 0?
If b 0 a quadratic equation can be solved by the square
root property.
3. How can a quadratic equation in which b 0 fail to have a
real solution?
If b 0 we can get the square root of a negative number
and no real solution.
4. What method is discussed for solving quadratic equations
in which b 0?
If b 0, some quadratics can be solved by factoring.
5. When do you need to solve linear equations to find the
solutions to a quadratic equation?
After applying the zero factor property we will have linear
equations to solve.
6. What new material is presented in this section?
This section reviews material about quadratic equations
that was given earlier in this text.
Solve each equation. See Examples 1 and 2.
7. x 2 36 0
6, 6
9. x 2 10 0
No real solution
11. 5x 2 50
10, 10
8. x 2 81 0
9, 9
10. x 2 4 0
No real solution
12. 7x 2 14
2, 2
13. 3t 2 5 0
15 15
, 3
3
15. 3y2 8 0
26 26
, 3
3
17. (x 3)2 4
1, 5
19. (y 2)2 18
2 32, 2 32
1
21. 2(x 1)2 2
3 1
, 2 2
1
23. (x 1)2 2
2 2
2 2
, 2
2
25.
x 2 2
1
2
1
1 2 1 2
, 2
2
27. (x 11)2 0
11
14. 5y2 7 0
35 35
, 5
5
16. 5w2 12 0
215 215
, 5
5
18. (x 5)2 9
8, 2
20. (m 5)2 20
5 25, 5 25
3
22. 3(x 1)2 4
3 1
, 2 2
1
24. (y 2)2 2
4 2 4 2
, 2
2
26.
x 2 2
1
2
3
1 6
1 6
, 2
2
28. (x 45)2 0
45
Solve each equation by factoring. See Example 3.
29. x 2 2x 15 0
30. x 2 x 12 0
3, 5
3, 4
2
31. x 6x 9 0
32. x 2 10x 25 0
3
5
(9-6)
Chapter 9
33. 4x 2 4x 8
1, 2
35. 3x 2 6x 0
0, 2
37. 4t 2 6t 0
3
0, 2
39. 2x 2 11x 21 0
3
7, 2
41. x 2 10x 25 0
5
7
43. x 2 x 15
2
5
, 6
2
12
1 2
45. a a 0
5
10
4, 6
Quadratic Equations and Quadratic Functions
34. 3x 2 3x 90
6, 5
36. 5x 2 10x 0
0, 2
38. 6w2 15w 0
5
0, 2
40. 2x 2 5x 2 0
1
2, 2
2
42. x 4x 4 0
2
2
1
44. 3x 2 x 5
5
1 1
, 5 3
2 2 5
46. w w 3 0
9
3
3
9, 2
Solve each equation.
27
49. x 12 x
3, 9
1 4x
48. x 2 2x 2
2 2
, 2
2
6
50. x x1
3, 2
51. 3x8 x 2
3, 4
52. 3x4
1 x 4
5, 6
47. x 2 2x 2(3 x)
6, 6
Solve each problem.
53. Side of a square. If the diagonal of a square is 5 meters,
then what is the length of a side?
52
meters
2
54. Diagonal of a square. If the side of a square is 5 meters,
then what is the length of the diagonal? 52 meters
55. Howard’s journey. Howard walked eight blocks east and
then four blocks north to reach the public library. How far
was he then from where he started? 45
blocks
56. Side and diagonal. Each side of a square has length s, and
its diagonal has length d. Write a formula for s in terms
of d.
d2
s 2
57. Designing a bridge. Find the length d of the diagonal
brace shown in the accompanying diagram.
26
1 feet
d
River
Park
Art
Museum
4
blocks
w3
FIGURE FOR EXERCISES 57 AND 58
58. Designing a bridge. Find the length labeled w in the
accompanying diagram.
9 feet
59. Two years of interest. Tasha deposited $500 into an
account that paid interest compounded annually. At the
end of two years she had $565. Solve the equation
565 500 (1 r)2 to find the annual rate r.
6.3%
60. Rate of increase. The price of a new 1998 BMW 740iL
was $68,570 (Edmund’s, www.edmunds.com). If a new
2000 BMW 740iL costs $74,165, then the average annual
rate of increase r satisfies
74,165 68,570(1 r)2.
Solve the equation to find r.
4.0%
61. Projectile motion. If an object is projected upward with
initial velocity v0 ft/sec from an initial height of s0 feet,
then its height s (in feet) t seconds after it is projected is
given by the formula s 16t 2 v0 t s0.
a) If a baseball is hit upward at 80 ft/sec from a height of
6 feet, then for what values of t is the baseball 102 feet
above the ground?
102 ft
FIGURE FOR EXERCISE 55
102 ft
80
40
0
8 blocks
12
15
120
Public
library
10
12
w
Post Office
City Hall
d
10
Height (ft)
458
0
1
2 3 4 5
Time (sec)
6
FIGURE FOR EXERCISE 61
9.2
65. Writing. One of the following equations has no real solutions. Find it by inspecting all of the equations (without
solving). Explain your answer.
a) x 2 99 0
b) 2(v 77)2 0
c) 3( y 22)2 11 0
d) 5(w 8)2 9 0
c
66. Cooperative learning. For each of three soccer teams A, B,
and C to play the other two teams once, it takes three
games (AB, AC, and BC). Work in groups to answer the following questions.
a) How many games are required for each team of a fourteam league to play every other team once?
6
b) How many games are required in a five-team soccer
league?
10
c) Find an expression of the form an2 bn c that gives
the number of games required in a soccer league of n
teams.
1 2 1
n n
2
2
d) The Urban Soccer League has fields available for a
120-game season. If the organizers want each team to
play every other team once, then how many teams
should be in the league?
16
n2 n
63. Sum of integers. The formula S gives the sum of
2
the first n positive integers. For what value of n is this sum
equal to 45?
9
64. Serious reading. Kristy’s New Year’s resolution is to read
one page of Training Your Boa to Squeeze on January 1,
two pages on January 2, three pages on January 3, and so
section
●
Perfect Square Trinomials
●
Solving a Quadratic
Equation by Completing
the Square
●
Applications
459
GET TING MORE INVOLVED
FIGURE FOR EXERCISE 62
In this
(9-7)
on. On what date will she finish the 136-page book? See
Exercise 63.
January 16
b) For what value of t is the baseball back at a height of
6 feet?
a) 2 sec and 3 sec b) 5 sec
62. Diving time. A springboard diver can perform complicated
maneuvers in a short period of time. If a diver springs into
the air at 24 ft/sec from a board that is 16 feet above the
water, then in how many seconds will she hit the water?
Use the formula from Exercise 61.
2 sec
9.2
Completing the Square
COMPLETING THE SQUARE
The quadratic equations in Section 9.1 were solved by factoring or the square
root property, but some quadratic equations cannot be solved by either of those
methods. In this section you will learn a method that works on any quadratic
equation.
Perfect Square Trinomials
The new method for solving any quadratic equation depends on perfect square trinomials. Recall that a perfect square trinomial is the square of a binomial. Just as we
recognize the numbers
1,
4,
9,
16,
25,
36,
...
as being the squares of the positive integers, we can recognize a perfect square trinomial. The following is a list of some perfect square trinomials with a leading