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4: Solving Quadratic Equations—Which Method?

4: Solving Quadratic Equations—Which Method?

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10.4 • Solving Quadratic Equations—Which Method?



437



Quadratic Formula Method

x2 ϩ 4x Ϫ 12 ϭ 0





-4 Ϯ 242 Ϫ 41121 - 122

2112



-4 Ϯ 264

2

-4 Ϯ 8



2







-4 ϩ 8

2

xϭ2

or







-4 Ϫ 8

2

x ϭ -6







or



The solution set is 5- 6, 26.

We have also discussed the use of the property x2 ϭ a if and only if x ϭ Ϯ2a for certain types of quadratic equations. For example, we can solve x2 ϭ 4 easily by applying the

property and obtaining x ϭ 24 or x ϭ -24; thus, the solutions are 2 and Ϫ2.

Which method should you use to solve a particular quadratic equation? Let’s consider

some examples in which the different techniques are used. Keep in mind that this is a decision you must make as the need arises. So become as familiar as you can with the strengths

and weaknesses of each method.



Classroom Example

Solve 3x2 ϩ 18x Ϫ 120 ϭ 0.



EXAMPLE 1



Solve 2x2 ϩ 12x Ϫ 54 ϭ 0.



Solution

First, it is very helpful to recognize a factor of 2 in each of the terms on the left side.

2x2 ϩ 12x Ϫ 54 ϭ 0

1

Multiply both sides by

x2 ϩ 6x Ϫ 27 ϭ 0

2

Now you should recognize that the left side can be factored. Thus we can proceed as follows.

1x ϩ 921x Ϫ 32 ϭ 0

xϩ9ϭ0

or

x ϭ -9

or



The solution set is 5-9, 36.

Classroom Example

Solve (8n Ϫ 11)2 ϭ 49.



EXAMPLE 2



xϪ3ϭ0

xϭ3



Solve 14x ϩ 322 ϭ 16.



Solution

The form of this equation lends itself to the use of the property x2 ϭ a if and only if x ϭ Ϯ1a.

14x ϩ 322 ϭ 16

4x ϩ 3 ϭ Ϯ216

4x ϩ 3 ϭ 4

or

4x ϭ 1

or

1

or



4



4x ϩ 3 ϭ - 4

4x ϭ -7

7

xϭ 4



7 1

The solution set is e- , f.

4 4



438



Chapter 10 • Quadratic Equations



Classroom Example

1

Solve x ϩ ϭ 3.

x



EXAMPLE 3



Solve n ϩ



1

ϭ 5.

n



Solution

First, we need to clear the equation of fractions by multiplying both sides by n.





1

ϭ 5,

n



n Z 0



1

nan ϩ b ϭ 51n2

n

n2 ϩ 1 ϭ 5n

Now we can change the equation to standard form.

n2 Ϫ 5n ϩ 1 ϭ 0

Because the left side cannot be factored using integers, we must solve the equation by using

either the method of completing the square or the quadratic formula. Using the formula, we

obtain







-1- 52 Ϯ 21 -522 Ϫ 4112112

2112

5 Ϯ 221

2



The solution set is e



Classroom Example

Solve x2 ϭ 27x.



5 Ϫ 221 5 ϩ 221

,

f.

2

2



Solve t2 ϭ 22t.



EXAMPLE 4

Solution



A quadratic equation without a constant term can be solved easily by the factoring method.

t 2 ϭ 22t

t 2 Ϫ 22t ϭ 0

t1t Ϫ 222 ϭ 0

tϭ0

or

t Ϫ 22 ϭ 0

tϭ0

or

t ϭ 22



The solution set is 50, 226. (Check each of these solutions in the given equation.)

Classroom Example

Solve x2 ϩ 9x Ϫ 162 ϭ 0.



Solve x2 Ϫ 28x ϩ 192 ϭ 0.



EXAMPLE 5

Solution



Determining whether or not the left side is factorable presents a bit of a problem because of

the size of the constant term. Therefore, let’s not concern ourselves with trying to factor;

instead we will use the quadratic formula.

x2 Ϫ 28x ϩ 192 ϭ 0



-1-282 Ϯ 21- 2822 Ϫ 411211922

2112

28 Ϯ 2784 Ϫ 768



2







10.4 • Solving Quadratic Equations—Which Method?







28 Ϯ 216

2







28 ϩ 4

2



x ϭ 16







or



or



The solution set is 512, 16}.



EXAMPLE 6



Classroom Example

Solve d 2 ϩ 16d ϭ 4.



439



28 Ϫ 4

2



x ϭ 12



Solve x2 ϩ 12x ϭ 17.



Solution

The form of this equation, and the fact that the coefficient of x is even, makes the method of

completing the square a reasonable approach.

x2 ϩ 12x ϭ 17

x ϩ 12x ϩ 36 ϭ 17 ϩ 36

1x ϩ 622 ϭ 53

x ϩ 6 ϭ Ϯ253

x ϭ - 6 Ϯ 253

2



The solution set is 5-6 Ϫ 253, - 6 ϩ 253}.



Concept Quiz 10.4

For Problems 1–7, choose the method that you think is most appropriate for solving the

given equation.

1.

2.

3.

4.

5.

6.

7.



2x2 ϩ 6x Ϫ 3 ϭ 0

(x ϩ 1)2 ϭ 36

x2 Ϫ 3x ϩ 2 ϭ 0

x2 ϩ 6x ϭ 19

4x2 ϩ 2x Ϫ 5 ϭ 0

4x2 ϭ 3

x2 Ϫ 4x Ϫ 12 ϭ 0



A.

B.

C.

D.



Factoring

Square root property (Property 10.1)

Completing the square

Quadratic formula



Problem Set 10.4

Solve each of the following quadratic equations using the

method that seems most appropriate to you. (Objective 1)

1. x ϩ 4x ϭ 45

2



3. 15n ϩ 62 ϭ 49

2



2. x ϩ 4x ϭ 60

2



4. (3n Ϫ 1)2 ϭ 25



5. t Ϫ t Ϫ 2 ϭ 0



6. t ϩ 2t Ϫ 3 ϭ 0



7. 8x ϭ 3x



8. 5x ϭ 7x



2



2



9. 9x Ϫ 6x ϩ 1 ϭ 0

2



2



2



10. 4x ϩ 36x ϩ 81 ϭ 0

2



11. 5n ϭ 28n



12. 23n ϭ 2n2



13. n2 Ϫ 14n ϭ 19



14. n2 Ϫ 10n ϭ 14



15. 5x2 Ϫ 2x Ϫ 7 ϭ 0



16. 3x2 Ϫ 4x Ϫ 2 ϭ 0



2



17. 15x2 ϩ 28x ϩ 5 ϭ 0

18. 20y2 Ϫ 7y Ϫ 6 ϭ 0

19. x2 Ϫ 28x Ϫ 7 ϭ 0

20. x2 ϩ 25x Ϫ 5 ϭ 0

21. y2 ϩ 5y ϭ 84



22. y2 ϩ 7y ϭ 60



3

n

25. 3x2 Ϫ 9x Ϫ 12 ϭ 0



24. n ϩ



23. 2n ϭ 3 ϩ



26. 2x2 ϩ 10x Ϫ 28 ϭ 0

27. 2x2 Ϫ 3x ϩ 7 ϭ 0



1

ϭ7

n



440



Chapter 10 • Quadratic Equations



28. 3x2 Ϫ 2x ϩ 5 ϭ 0



38. x2 Ϫ 33x ϩ 266 ϭ 0



29. n1n Ϫ 462 ϭ -480



39.



x2

1

Ϫxϭ 3

2



30. n1n ϩ 422 ϭ -432

31. n Ϫ



3

ϭ -1

n



40.



2x Ϫ 1

5

ϭ

3

xϩ2



32. n Ϫ



2

3

ϭ

n

4



41.



2

1

Ϫ ϭ3

x

xϩ2



1

25

33. x ϩ ϭ

x

12



42.



3

2

3

ϩ ϭ

x

xϪ1

2



1

65

34. x ϩ ϭ

x

8



43.



2

nϩ2

ϭ

3n Ϫ 1

6



35. t ϩ 12t ϩ 36 ϭ 49



44.



x2

1

ϭxϩ

2

4



36. t2 Ϫ 10t ϩ 25 ϭ 16



45. 1n Ϫ 221n ϩ 42 ϭ 7



2



46. 1n ϩ 321n Ϫ 82 ϭ - 30



37. x Ϫ 28x ϩ 187 ϭ 0

2



Thoughts Into Words

49. How can you tell by inspection that the equation

x2 ϩ x ϩ 4 ϭ 0 has no real number solutions?



47. Which method would you use to solve the equation

x2 ϩ 30x ϭ Ϫ216? Explain your reasons for making this

choice.

48. Explain how you

0 ϭ -x2 Ϫ x ϩ 6.



would



solve



Answers to the Concept Quiz

Answers for these questions may vary.

1. D

2. B

3. A



10.5



the



equation



4. C



5. D



6. B



7. A



Solving Problems Using Quadratic Equations



OBJECTIVE



1



Use quadratic equations to solve a variety of word problems



The following diagram indicates our approach in this text.

Develop

skills



Use skills

to solve

equations



Use equations

to solve word

problems



Now you should be ready to use your skills relative to solving systems of equations (Chapter

8) and quadratic equations to help with additional types of word problems. Before you consider such problems, let’s review and update the problem-solving suggestions we offered in

Chapter 3.



10.5 • Solving Problems Using Quadratic Equations



441



Suggestions for Solving Word Problems

1. Read the problem carefully and make certain that you understand the meanings of all the

words. Be especially alert for any technical terms used in the statement of the problem.

2. Read the problem a second time (perhaps even a third time) to get an overview of the

situation being described and to determine the known facts as well as what is to be found.

3. Sketch any figure, diagram, or chart that might be helpful in analyzing the problem.

*4. Choose meaningful variables to represent the unknown quantities. Use one or two

variables, whichever seems easiest. The term “meaningful” refers to the choice of

letters to use as variables. Choose letters that have some significance for the problem

under consideration. For example, if the problem deals with the length and width of

a rectangle, then l and w are natural choices for the variables.

*5. Look for guidelines that you can use to help set up equations. A guideline might be

a formula such as area of a rectangular region equals length times width, or a

statement of a relationship such as the product of the two numbers is 98.

*6. (a) Form an equation containing the variable, which translates the conditions of the

guideline from English into algebra; or

(b) Form two equations containing the two variables, which translate the guidelines

from English into algebra.

*7. Solve the equation (system of equations) and use the solution (solutions) to

determine all facts requested in the problem.

8. Check all answers back in the original statement of the problem.



The asterisks indicate those suggestions that have been revised to include using systems of

equations to solve problems. Keep these suggestions in mind as you study the examples and

work the problems in this section.

Classroom Example

The length of a rectangular region is

8 inches more than its width. The

area of the region is 48 square inches.

Find the length and width of the

rectangle.



EXAMPLE 1

The length of a rectangular region is 2 centimeters more than its width. The area of the

region is 35 square centimeters. Find the length and width of the rectangle.



Solution

We let l represent the length, and we let w represent the width (see Figure 10.10). We can

use the area formula for a rectangle, A ϭ lw, and the statement “the length of a rectangular

region is 2 centimeters greater than its width” as guidelines to form a system of equations.



Area is 35 cm2.

l

Figure 10.10



w



a



lw ϭ 35

b

lϭwϩ2



The second equation indicates that we can substitute w ϩ 2 for l. Making this substitution in

the first equation yields

1w ϩ 221w2 ϭ 35

Solving this quadratic equation by factoring, we get

w2 ϩ 2w ϭ 35

w2 ϩ 2w Ϫ 35 ϭ 0

1w ϩ 721w Ϫ 52 ϭ 0

wϩ7ϭ0

or

wϪ5ϭ0

w ϭ -7

or

wϭ5



 



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