5: Quadratic Equations: Complex Solutions
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11.5 • Quadratic Equations: Complex Solutions
✔ Check
1x Ϫ 222 ϭ
12 ϩ i27 Ϫ 222
1i2722
7i2
71 -12
-7 ϭ
475
1x Ϫ 22 ϭ -7
-7 12 Ϫ i27 Ϫ 22 - 7
1 -i272 - 7
-7
7i - 7
-7
-7
71- 12 - 7
- 7
- 7 ϭ -7
2
- 7
2
2
2
The solution set is 52 Ϫ i27, 2 ϩ i276.
Classroom Example
Solve a2 ϩ 6a ϭ -15 .
EXAMPLE 3
Solve x2 ϩ 2x ϭ -10 .
Solution
The form of the equation lends itself to completing the square, so we proceed as follows:
x2 ϩ 2x ϭ -10
x2 ϩ 2x ϩ 1 ϭ -10 ϩ 1
1x ϩ 122 ϭ -9
x ϩ 1 ϭ Ϯ 3i
x ϭ - 1 Ϯ 3i
✔ Check
x2 ϩ 2x ϭ
1-1 ϩ 3i22 ϩ 21- 1 ϩ 3i2
1 Ϫ 6i ϩ 9i2 Ϫ 2 ϩ 6i
-1 ϩ 9i2
- 1 ϩ 91 -12
-10 ϭ
x ϩ 2x ϭ -10
1 -1 Ϫ 3i2 ϩ 21-1 Ϫ 3i2 -10
1 ϩ 6i ϩ 9i Ϫ 2 Ϫ 6i -10
- 1 ϩ 9i -10
-1 ϩ 91 -12 -10
-10 ϭ -10
2
-10
-10
-10
-10
-10
-10
2
2
2
The solution set is 5-1 Ϫ 3i, - 1 ϩ 3i6.
Classroom Example
Solve x2 Ϫ x ϩ 1 ϭ 0.
EXAMPLE 4
Solve x2 Ϫ 2x ϩ 2 ϭ 0 .
Solution
We use the quadratic formula to obtain the solutions.
x2 Ϫ 2x ϩ 2 ϭ 0
xϭ
2 Ϯ 21-222 Ϫ 4112122
2
2 Ϯ 24 Ϫ 8
xϭ
2
2 Ϯ 2- 4
xϭ
2
2 Ϯ 2i
xϭ
2
xϭ
2> 11 Ϯ i2
2>
xϭ1 Ϯ i
xϭ
-b Ϯ 2b2 Ϫ 4ac
2a
476
Chapter 11 • Additional Topics
✔ Check
x2 Ϫ 2x ϩ 2 ϭ 0
11 ϩ i22 Ϫ 211 ϩ i2 ϩ 2 0
x2 Ϫ 2x ϩ 2 ϭ 0
11 Ϫ i22 Ϫ 211 Ϫ i2 ϩ 2 0
1 ϩ 2i ϩ i2 Ϫ 2 Ϫ 2i ϩ 2 ϭ 0 1 Ϫ 2i ϩ i2 Ϫ 2 ϩ 2i ϩ 2 0
1 ϩ i2 0
1 ϩ i2 0
1Ϫ10
1Ϫ10
0 ϭ 0
0ϭ0
The solution set is 51 Ϫ i, 1 ϩ i6.
EXAMPLE 5
Classroom Example
Solve x2 Ϫ 6x Ϫ 7 ϭ 0.
Solve x2 ϩ 3x Ϫ 10 ϭ 0 .
Solution
We can factor x2 ϩ 3x Ϫ 10 and proceed as follows:
x2 ϩ 3x Ϫ 10 ϭ 0
1x ϩ 521x Ϫ 22 ϭ 0
or x Ϫ 2 ϭ 0
xϭ2
x ϭ - 5 or
x ϩ 5 ϭ 0
The solution set is {Ϫ5, 2}. (Don’t forget that all real numbers are complex numbers; that is,
Ϫ5 and 2 can be written as Ϫ5 ϩ 0i and 2 ϩ 0i.)
To summarize our work with quadratic equations in Chapter 10 and this section, we
suggest the following approach to solve a quadratic equation.
1. If the equation is in a form that the property, if x2 ϭ a, then x ϭ ; 2a applies, use it.
(See Examples 1 and 2.)
2. If the quadratic expression can be factored using integers, factor it and apply the
property, if ab ϭ 0, then a ϭ 0 or b ϭ 0. (See Example 5.)
3. If numbers 1 and 2 don’t apply, use either the quadratic formula or the process of completing the square. (See Examples 3 and 4.)
Concept Quiz 11.5
For Problems 1–10, answer true or false.
1. The solution set for x2 ϭ Ϫ8 is [- 2i12, 2i12]
2. The solution set for (x Ϫ 3)2 ϭ Ϫ18 consists of two real numbers.
3. The equation x2 ϩ 16x Ϫ 14 ϭ 0 could be solved by completing the square.
4. The equation 3x2 ϩ 2x ϩ 1 ϭ 0 has two nonreal complex solutions.
5. The equation 5x2 Ϫ x ϩ 4 ϭ 0 has two real number solutions.
6. The equation 12x2 Ϫ 7x Ϫ 12 ϭ 0 can be solved by factoring.
7. The quadratic formula always yields nonreal complex solutions.
8. The equation 4x2 ϩ 7x ϭ 14 is in proper form to apply the quadratic formula.
9. Completing the square always yields real number solutions.
10. The equation (3x Ϫ 4)2 ϩ 6 ϭ 4 has two nonreal complex solutions.
11.6 • Pie, Bar, and Line Graphs
477
Problem Set 11.5
19. 3x2 Ϫ 2x ϩ 1 ϭ 0
Solve each of the following quadratic equations, and check
your solutions. (Objective 2)
1. x2 ϭ -64
20. 2x2 ϩ x ϩ 1 ϭ 0
2. x2 ϭ -49
3. 1x Ϫ 222 ϭ -1
5. 1x ϩ 522 ϭ -13
7. 1x Ϫ 322 ϭ -18
9. 15x Ϫ 122 ϭ 9
4. 1x ϩ 322 ϭ - 16
21. 2x2 Ϫ 3x Ϫ 5 ϭ 0
8. 1x ϩ 422 ϭ - 28
23. y2 Ϫ 2y ϭ -19
6. 1x Ϫ 722 ϭ - 21
22. 3x2 Ϫ 5x Ϫ 2 ϭ 0
10. 17x ϩ 322 ϭ 1
24. y2 ϩ 8y ϭ -24
11. a2 Ϫ 3a Ϫ 4 ϭ 0
12. a2 ϩ 2a Ϫ 35 ϭ 0
25. x2 Ϫ 4x ϩ 7 ϭ 0
13. t2 ϩ 6t ϭ -12
14. t2 Ϫ 4t ϭ - 9
26. x2 Ϫ 2x ϩ 3 ϭ 0
15. n2 Ϫ 6n ϩ 13 ϭ 0
27. 4x2 Ϫ x ϩ 2 ϭ 0
16. n2 Ϫ 4n ϩ 5 ϭ 0
28. 5x2 ϩ 2x ϩ 1 ϭ 0
17. x2 Ϫ 4x ϩ 20 ϭ 0
29. 6x2 ϩ 2x ϩ 1 ϭ 0
18. x2 ϩ 2x ϩ 5 ϭ 0
30. 7x2 ϩ 3x ϩ 3 ϭ 0
Thoughts Into Words
32. Explain why the expression b2 Ϫ 4ac from the quadratic formula will determine whether or not the solutions of a particular quadratic equation are imaginary.
31. Which method would you use to solve the equation
x2 ϩ 4x ϭ Ϫ5? Explain your reasons for making that
choice.
Answers to the Concept Quiz
1. True
2. False
3. True
4. True
11.6
5. False
6. True
7. False
8. False
9. False
10. True
Pie, Bar, and Line Graphs
OBJECTIVES
1
Construct pie charts, bar graphs, and line graphs
2
Interpret pie charts, bar graphs, and line graphs
3
Learn the strengths and weaknesses of pie charts, bar graphs, and line graphs
The saying “a picture is worth a thousand words” is also true for mathematics. In mathematics, a graph or chart is often used to present information. In this section, you will see various
types of graphs and how they are used.
Pie Chart
A pie chart or circle graph is used to illustrate the parts of a whole. The pie or circle represents the whole, and the sectors of the pie or circle represent the parts of the whole. The parts
478
Chapter 11 • Additional Topics
are usually expressed in percents. For example, a student activities group is planning a ski trip
and knows that the expenses, per person, are as follows:
Expenses for a Ski Trip, per person
Category
Food
Transportation
Equipment rental
Lift tickets
Lodging
Total
Expense
Percent
$155
360
200
85
200
$1000
15.5%
36%
20%
8.5%
20%
The pie chart in Figure 11.13 is a useful graph to show the ski club how the expenses of
the trip are distributed.
Ski trip expenses
Transportation
36%
Lift tickets
8.5%
Voter preference on a proposition
No
Food
15.5%
Lodging
20%
Equipment rental
20%
Figure 11.13
Yes
Figure 11.14
A glance at a pie chart can often tell the story and eliminate the need to compare numbers. Consider the pie chart in Figure 11.14. Without any numerical values presented, the
graph gives us the information that a majority of voters are in favor of the proposition,
because more than half the circle is shaded for yes.
Classroom Example
Use the pie chart shown in Figure
11.15 to answer these questions.
(a) What percent of the bagels sold
were garlic, onion, or blueberry?
(b) What percent of the bagels sold
were not poppy seed or sesame
seed?
(c) Were less than one-fourth of the
bagels sold either blueberry or
cinnamon?
EXAMPLE 1
Use the pie chart in Figure 11.15 to answer the questions.
Bagels sold
Garlic
7%
Blueberry
10%
Whole
wheat
12%
Onion
5%
Plain
25%
Cinnamon
14%
Poppy seed
13%
Sesame seed
14%
Figure 11.15
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