Tải bản đầy đủ - 0 (trang)

# 5: Quadratic Equations: Complex Solutions

Tải bản đầy đủ - 0trang

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

2 Ϯ 21-222 Ϫ 4112122

2

2 Ϯ 24 Ϫ 8

2

2 Ϯ 2- 4

2

2 Ϯ 2i

2

2> 11 Ϯ i2

2>

xϭ1 Ϯ i

-b Ϯ 2b2 Ϫ 4ac

2a

476

✔ 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

1Ϫ1՘0

1Ϫ1՘0

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

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.

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

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

(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