466
(9-14)
Chapter 9
Quadratic Equations and Quadratic Functions
the members will sell 5000 200x tickets. So the total
revenue for the tickets is given by R x (5000 200x).
a) What is the revenue if the tickets are sold at $8 each?
b) For what ticket price is the revenue $30,000?
c) Use the accompanying graph to estimate the ticket price
that will produce the maximum revenue.
a) $27,200 b) $10 and $15 c) $12.50
b) Use completing the square to solve the quadratic equation formed by setting the answer to part (a) equal to
zero.
c) Write a quadratic equation (in the form
ax2 bx c 0) that has solutions
75. Exploration.
a) Find the product [x (5 3)][x (5 3)].
In this
section
●
Developing the Quadratic
Formula
●
The Discriminant
●
Which Method to Use
3 2
and .
2
d) Explain how to find a quadratic equation in the form
ax2 bx c 0 for any two given solutions.
a) x 2 10x 22
b) 5 3
7
c) x 2 3x 0
4
GET TING MORE INVOLVED
9.3
3 2
2
THE QUADRATIC FORMULA
In Section 9.2 you learned that every quadratic equation can be solved by completing the square. In this section we use completing the square to get a formula, the
quadratic formula, for solving any quadratic equation.
Developing the Quadratic Formula
To develop a formula for solving any quadratic equation, we start with the general
quadratic equation
ax 2 bx c 0
and solve it by completing the square. Assume that a is positive for now, and divide
each side by a:
ax2 bx c 0
a
a
b
c
x 2 x 0
a
a
b
c
x 2 x
a
a
Divide by a to get 1 for the coefficient of x 2.
Simplify.
Isolate the x 2- and x-terms.
b b
b 2
b2
Now complete the square on the left. One-half of is , and .
a 2a
2a
4a2
b
b2
b2
c
b2
x 2 x 2 2 Add 2 to each side.
4a
4a
4a
a
a
2
b
4ac
b 2
x 2 2
Factor on the left-hand side, and get a
4a
2a
4a
common denominator on the right-hand side.
b
b 4ac
x 2a 4a
2
2
2
b
x 2a
b2 4ac
4a 2
Square root property
b
b2
a
4c
x 2a
2a
Because a 0, 4a2 2a.
b b24ac
x 2a
Combine the two expressions.
9.3
study
tip
Try changing subjects or tasks
every hour when you study.
The brain does not easily assimilate the same material
hour after hour. You will learn
and remember more from
working on it one hour per
day than seven hours on
Saturday.
E X A M P L E
1
The Quadratic Formula
(9-15)
467
We assumed that a was positive so that 4a2 2a would be correct. If a is
negative, then 4a2 2a. Either way, the result is the same. It is called the
quadratic formula. The formula gives x in terms of the coefficients a, b, and c. The
quadratic formula is generally used instead of completing the square to solve a
quadratic equation that cannot be factored.
The Quadratic Formula
The solutions to ax2 bx c 0, where a 0, are given by
b b24ac
x .
2a
Equations with rational solutions
Use the quadratic formula to solve each equation.
b) 4x 2 9 12x
a) x 2 2x 3 0
Solution
a) To use the formula, we first identify a, b, and c. For the equation
1x 2 2x 3 0,
↑
↑
↑
a
b
c
a 1, b 2, and c 3. Now use these values in the quadratic formula:
b b2
a
4c
x 2a
2 2
2
(1
4)(
3)
x 22 4(1)(3) 4 12 16
2(1)
2 16
x 2
2 4
x 2
2 4
2 4
x or
x 2
2
x1
or
x 3
Check these answers in the original equation. The solutions are 3 and 1.
b) Write the equation in the form ax 2 bx c 0 to identify a, b, and c:
4x 2 9 12x
4x 2 12x 9 0
Now a 4, b 12, and c 9. Use these values in the formula:
(12) (
12
)2
(4
4)(
9)
x 2(4)
12 0
12 3
x 8
8
2
Check. The only solution to the equation is 3.
2
■
468
(9-16)
Chapter 9
Quadratic Equations and Quadratic Functions
The equations in Example 1 could have been solved by factoring. (Try it.) The
quadratic equation in the next example has an irrational solution and cannot be
solved by factoring.
E X A M P L E
2
An equation with an irrational solution
Solve 3x 2 6x 1 0.
Solution
For this equation, a 3, b 6, and c 1:
calculator
close-up
Check irrational solutions
using the answer key as
shown here.
(6) (
6
)2
(3
4)(
1)
x 2(3)
6 24
6
6 26
6
2(3 6)
2(3)
24 46 26
Numerator and denominator
have 2 as a common factor.
3 6
3
3 6
3 6
The two solutions are the irrational numbers and .
3
3
■
We have seen quadratic equations such as x 2 9 that do not have any real
number solutions. In general, you can conclude that a quadratic equation has no real
number solutions if you get a square root of a negative number in the quadratic
formula.
E X A M P L E
3
A quadratic equation with no real number solutions
Solve 5x 2 x 1 0.
Solution
For this equation we have a 5, b 1, and c 1:
(1
)2
4
(5
)(1)
1 x 2(5)
b 1, b 1
1 19
x 10
The equation has no real solutions because 19 is not real.
■
The Discriminant
A quadratic equation can have two real solutions, one real solution, or no real number solutions, depending on the value of b2 4ac. If b2 4ac is positive, as in
Example 1(a) and Example 2, we get two solutions. If b2 4ac is 0, we get only
one solution, as in Example 1(b). If b2 4ac is negative, there are no real number
solutions, as in Example 3. Table 9.1 summarizes these facts.
9.3
The Quadratic Formula
(9-17)
Value of b2 4ac
Number of real solutions
to ax 2 bx c 0
Positive
2
Zero
1
Negative
0
469
TABLE 9.1
The quantity b2 4ac is called the discriminant because its value determines the
number of real solutions to the quadratic equation.
E X A M P L E
4
The number of real solutions
Find the value of the discriminant, and determine the number of real solutions to
each equation.
a) 3x 2 5x 1 0
b) x 2 6x 9 0
c) 2x 2 1 x
Solution
a) For the equation 3x 2 5x 1 0 we have a 3, b 5, and c 1. Now
find the value of the discriminant:
b 2 4ac (5)2 4(3)(1) 25 12 13
Because the discriminant is positive, there are two real solutions to this
quadratic equation.
b) For the equation x 2 6x 9 0, we have a 1, b 6, and c 9:
b2 4ac (6)2 4(1)(9) 36 36 0
Since the discriminant is zero, there is only one real solution to the equation.
c) We must first rewrite the equation:
2x 2 1 x
2x 2 x 1 0
Subtract x from each side.
Now a 2, b 1, and c 1.
b2 4ac (1)2 4(2)(1) 1 8 7
Because the discriminant is negative, the equation has no real number solutions.
■
Which Method to Use
If the quadratic equation is simple enough, we can solve it by factoring or by the
square root property. These methods should be considered first. All quadratic equations can be solved by the quadratic formula. Remember that the quadratic formula
is just a shortcut to completing the square and is usually easier to use. However, you
should learn completing the square because it is used elsewhere in algebra. The
available methods are summarized as follows.
470
(9-18)
helpful
Chapter 9
Quadratic Equations and Quadratic Functions
Solving the Quadratic Equation ax2 bx c 0
hint
If our only intent is to get the
answer, then we would probably use a calculator that is
programmed with the quadratic formula. However, by
learning different methods we
gain insight into the problem
and get valuable practice with
algebra. So be sure you learn
all of the methods.
WARM-UPS
Method
Square root
property
Comments
Use when b 0.
Factoring
Quadratic
formula
Use when the polynomial
can be factored.
Use when the first two
methods do not apply.
Completing
the square
Use only when this
method is specified.
Examples
If x 2 7, then x 7.
If (x 2) 2 9, then
x 2 3
x 2 5x 6 0
(x 2)(x 3) 0
x 2 2x 6 0
2 2
2
41
(6)
x 21
x 2 4x 9 0
x 2 4x 4 9 4
(x 2)2 13
True or false? Explain your answer.
1. Completing the square is used to develop the quadratic formula. True
2. For the equation x 2 x 1 0, we have a 1, b x, and c 1.
False
3. For the equation x 2 3 5x, we have a 1, b 3, and c 5. False
b4ac
4. The quadratic formula can be expressed as x b .
2a
2
False
The quadratic equation 2x 2 6x 0 has two real solutions. True
All quadratic equations have two distinct real solutions. False
Some quadratic equations cannot be solved by the quadratic formula. False
We could solve 2x 2 6x 0 by factoring, completing the square, or the
quadratic formula. True
2
9. The equation x 2 x is equivalent to x 1 14. True
5.
6.
7.
8.
2
10. The only solution to x 2 6x 9 0 is 3.
9. 3
True
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What method presented here can be used to solve any
quadratic equation?
The quadratic formula solves any quadratic equation.
2. What is the quadratic formula?
b b2
a
4c
The quadratic formula is x .
2a
3. What is the quadratic formula used for?
The quadratic formula is used to solve ax 2 bx c 0.
4. What is the discriminant?
The discriminant is b2 4ac.
5. How can you determine whether there are no real solutions
to a quadratic equation?
If b2 4ac 0 then there are no real solutions.
6. What methods have we studied for solving quadratic
equations?
We have solved quadratic equations by factoring, the
square root property, completing the square, and the quadratic formula.
9.3
Find the value of the discriminant, and state how many real solutions there are to each quadratic equation. See Example 4.
27. 4x 2 4x 1 0
28. 9x 2 6x 1 0
0, one
0, one
29. 6x 2 7x 4 0
30. 3x 2 5x 7 0
47, none
59, none
31. 5t 2 t 9 0
32. 2w2 6w 5 0
181, two
76, two
33. 4x 2 12x 9 0
34. 9x 2 12x 4 0
0, one
0, one
35. x 2 x 4 0
36. y 2 y 2 0
15, none
7, none
37. x 5 3x 2
38. 4 3x x 2
59, none
25, two
Use the method of your choice to solve each equation.
3
7
39. x 2 x 1
40. x 2 x 2
2
2
1
1
, 4
2, 2
2
2
2
2
41. (x 1) (x 2) 5 42. x (x 3)2 29
0, 3
2, 5
(9-19)
1
5
1
43. x x 2 12
6
, 4
5
1
1
5
44. x x1 6
3
, 2
5
45. x 2 6x 8 0
46. 2x 2 5x 3 0
1
, 3
2
4, 2
47. x 2 9x 0
0, 9
48. x 2 9 0
3, 3
49. (x 5)2 9
50. (3x 1)2 0
1
3
8, 2
471
51. x(x 3) 2 3(x 4) No real solution
4
52. (x 1)(x 4) (2x 4)2 , 5
3
x x2
53. x
3
3 3
3 3 33
, 2
2
5 4
1 5 41
, 2
2
5
x2
54. x2
x
55. 2x 2 3x 0
56. x 2 5
3
0, 2
5, 5
Use a calculator to find the approximate solutions to
each quadratic equation. Round answers to two decimal places.
57. x 2 3x 3 0
0.79, 3.79
58. x 2 2x 2 0
0.73, 2.73
59. x2 x 3.2 0
1.36, 2.36
60. x 2 4.3x 3 0
0.88, 3.42
61. 5.29x 2 3.22x 0.49 0
0.30
62. 2.6x 2 3.1x 5 0
2.11, 0.91
Use a calculator to solve each problem.
63. Phasing out freon-12. The emission of CFC-12 (or
freon-12) in the U.S. can be modeled by the function
y 0.87x 2 12.25x 77.54,
Emission (thousands
of metric tons)
Solve by the quadratic formula. See Examples 1–3.
7. x 2 2x 15 0
8. x 2 3x 18 0
5, 3
6, 3
10. x 2 12x 36 0
9. x 2 10x 25 0
5
6
11. 2x 2 x 6 0
12. 2x 2 x 15 0
3
5
2, 3, 2
2
13. 4x 2 4x 3 0
14. 4x 2 8x 3 0
3 1
3 1
, , 2 2
2 2
2
15. 2y 6y 3 0
16. 3y2 6y 2 0
3 3
3 3
3 3 3 3
, , 3
2
2
3
17. 2t 2 4t 1
18. w2 2 4w
2 2 2 2
2 2, 2 2
, 2
2
19. 2x 2 2x 3 0
20. 2x 2 3x 9 0
No real solution
No real solution
21. 8x 2 4x
22. 9y2 3y 6y
1
0, 1, 0
2
23. 5w2 3 0
24. 4 7z2 0
15 15
27 27
, , 5
7
5
7
1
1
1
26. z2 6z 3 0
25. h2 7h 0
2
2
4
12 233, 12 233
7 43, 7 43
The Quadratic Formula
Emission of CFC–12
y
150
100
50
0
5
10
15
Years after 1980
x
FIGURE FOR EXERCISE 63
472
(9-20)
Chapter 9
Quadratic Equations and Quadratic Functions
where x is the number of years since 1980 and y is the
amount of emission in thousands of metric tons (Energy
Information Administration, www.eia.doe.gov).
a) In what years was the emission of CFC-12 gas 106 thousand metric tons?
1983, 1991
9.4
In this
section
●
Geometric Applications
●
Work Problems
●
Vertical Motion
b) In what year will the emission of CFC-12 gas be zero?
1999
64. Lottery tickets. The formula R 200x2 5000x was
used in Exercise 74 of Section 9.2 to predict the revenue
when lottery tickets are sold for x dollars each. For what
ticket price is the revenue $25,000? $6.91 and $18.09
APPLICATIONS OF QUADRATIC
EQUATIONS
In this section we will solve problems that involve quadratic equations.
Geometric Applications
Quadratic equations can be used to solve problems involving area.
E X A M P L E
1
Dimensions of a rectangle
The length of a rectangular flower bed is 2 feet longer than the width. If the area is
6 square feet, then what are the exact length and width? Also find the approximate
dimensions of the rectangle to the nearest tenth of a foot.
Solution
Let x represent the width, and x 2 represent the length as shown in Fig. 9.2. Write
an equation using the formula for the area of a rectangle, A LW:
x(x 2) 6
x 2x 6 0
The area is 6 square feet.
2
We use the quadratic formula to solve the equation:
8
2 2
2
(1
4)(
6) 2 2
x 2(1)
2
x + 2 ft
)
2 27 2(1 7
1 7
2
2
x ft
FIGURE 9.2
Because 1 7
is negative, it cannot be the width of a rectangle. If
,
x 1 7
then
.
x 2 1 7 2 1 7
feet. We can
So the exact width is 1 7 feet, and the exact length is 1 7
check that these dimensions give an area of 6 square feet as follows:
)(1 7
) 1 7
7
76
LW (1 7
Use a calculator to find the approximate dimensions of 1.6 and 3.6 feet.
■
Work Problems
The work problems in this section are similar to the work problems that you solved
in Chapter 5. However, you will need the quadratic formula to solve the work
problems presented in this section.