Elementary Education Chapter 11 Solve the related quadratic equation to find

Section 11.4 Equations Quadratic in Form

53.



21

32 42 10xx





Using a graphing utility we find the solution set is





3, 5

.

Both solutions check.

54.

42

10 9 0xx

Using a graphing utility we find the solution set is



55.

26 8

26 80

xx





Using a graphing utility we find the solution set is





1.

The solution checks.

56.



2

21513

215130

xx

 

 

Using a graphing utility we find the solution set is

57.









2

22

32 3240xx xx 

Using a graphing utility we find the solution set is





1, 4 .

The solutions check.

58.

11

24

11

45

xx



59.

33

320xx

Using a graphing utility we find the solution set is





1, 8 .

The solutions check.

60. makes sense

61. makes sense

62. does not make sense; Explanations will vary.

Sample explanation: If at any point in the solution

process both sides of an equation are raise to an

even power, a check is required to see if extraneous

Chapter 11 Quadratic Equations and Functions

66. false; Changes to make the statement true will vary.

68.

42

520xx

Let

2

.ux

42

520

xx



2

533

2

533 5332

222

x







  

69.

63

518

5180

xx





Let

3.ux

63

5180

xx



5

Substitute

3for .xu

Rationalize the denominator.

33

22

33

3

333

23

995 95 225

55

555



  

3

3225





1

2

445

112

5

510

2

xx

uu

















5

1 2.5 1

2

ab c 

Section 11.4 Equations Quadratic in Form

Substitute

4 for .

1

xu

x





41 42

xx





71.



22

32 2

221

51

10 2 2 5 1

xx

x

xx xx







72.



2

22

22122

111 1

23113 13

11 2 22

i i i iii

iii i

ii

i





 

 





Chapter 11 Quadratic Equations and Functions

75.

32

44

440

xx x





76.

112(3)

2

333

xxx



 



11.5 Check Points

1.

2

20

20 0

xx



 

Solve the related quadratic equation to find the boundary points.

The boundary points are

4

and 5.

Interval Test Value Test Conclusion





,4 

5

2

20

(5) (5) 20

30 20, true

xx









,4 

belongs to the solution set.

Section 11.5 Polynomial and Rational Inequalities

2.

2

261

2610

xx

 



Solve the related quadratic equation to find the boundary points.

2

2610xx

37 37

or

22

0.2 2.8

xx

 



 



2

Interval Test Value Test Conclusion

37 37

, 10 2 10 6 10 1 , is not part of the solution set

22

200 59, false

 

 

       

 

 



3.

32

33

330

xxx





Solve the related quadratic equation to find the boundary points.

Chapter 11 Quadratic Equations and Functions

30

3

x





or

10

1

x





or

10

1

x









32

Interval Test Value Test Conclusion

, 3 5 ( 5) 3( 5) ( 5) 3 , 3 is part of the solution set

50 2, true

          



4. 50

2

x





Find the values of x that make the numerator and denominator zero.

50 20

52

xx

 



The boundary points are –2 and 5.

Interval Test Value Test Conclusion





,2  −3

35 0

32

80, false

 









,2 

does not belong to the

solution set.





















Section 11.5 Polynomial and Rational Inequalities

5.

21

1

210

1

x









The boundary points are

1

and 1.

Interval Test Value Test Conclusion





,1 

2

2( 2) 1

21

41, true











,1 

belongs to the

solution set.

Exclude

1

from the solution set because it would make the denominator zero. The solution set is









,1 1,   

.

6.





2

16 80

st t t

 

To find when the object will be more than 64 feet above the ground, solve the inequality

2

16 80 64.

tt



Chapter 11 Quadratic Equations and Functions

The boundary points are 1 and 4.

Interval Test Value Test Conclusion





0,1

0.5

2

16(0.5) 80(0.5) 64

36 64, false









0,1

does not belong to the solution set.





















11.5 Concept and Vocabulary Check

1.

2

8150

xx



; boundary





11.5 Exercise Set

1.









420

xx



Solve the related quadratic equation.









420

xx



40or 20

42

xx

 



The boundary points are –2 and 4.

Interval Test Value Test Conclusion





,2 

−3









34 32 0

70, true

  







,2 

belongs to the solution set.

























Section 11.5 Polynomial and Rational Inequalities

2.









350

xx



Solve the related quadratic equation.









350

xx



30 or 50

35

xx

 

 

Interval Test Value Test Conclusion





,3 

−4









43 45 0

true

  





,3 

belongs to the solution set.

























3.









730

xx



Solve the related quadratic equation.









730

xx



70or 30

73

xx

 



























Chapter 11 Quadratic Equations and Functions

4.









170

xx



Solve the related quadratic equation.









170

xx



10 or 70

17

xx

  

 

Interval Test Value Test Conclusion





,1 

−2









21 27 0

false

  





,1 

does not belong to the solution set.



5.

2

540

xx



Solve the related quadratic equation.

2

540

410

xx





6.

430

xx



Solve the related quadratic equation.



2

430

310

xx





30or 10

31

xx

 



Interval Test Value Test Conclusion





,1

0





2

0 4 0 3 0 false





,1

does not belong to the solution set.









7.

2

540

xx



Solve the related quadratic equation.



2

540

410

xx







Chapter 11 Quadratic Equations and Functions

8.

2

60

xx



Solve the related quadratic equation.

2

60

xx



9.

2

680

xx



Solve the related quadratic equation.



2

680

420

xx





The solution set is





2, 4

.

Section 11.5 Polynomial and Rational Inequalities

10.

2

230

xx



Solve the related quadratic equation.



2

230

310

xx













11.

2

31080

xx



Solve the related quadratic equation.

2

31080

xx



Interval Test Value Test Conclusion





,4 

−5



2

35 105 80

17 0, false









,4 

does not belong to the solution set.

Chapter 11 Quadratic Equations and Functions

12.

2

9320

xx



Solve the related quadratic equation.

2

Interval Test Value Test Conclusion

2

,3



 





−1



2

9 1 3 1 2 0 true  

2

,3





 





belongs to the solution set.













13.

2

215

2150

xx



 

Solve the related quadratic equation.

The boundary points are –3 and

5

2

.

Interval Test Value Test Conclusion





,3 

−4



2

24 4 15

28 15, false









,3 

does not belong to the solution set.

Section 11.5 Polynomial and Rational Inequalities

14.

2

61

610

xx





Solve the related quadratic equation.

2

610

xx



Interval Test Value Test Conclusion

1

,2



 





−1



2

6 1 1 1 true

1

,2





 









belongs to the solution set.













15.

2

47 3

4730

xx





Solve the related quadratic equation.

The boundary points are –1 and

3

4



.

Interval Test Value Test Conclusion





,1 

-2



2

42 72 3

23, false









,1 

does not belong to the solution set.













Chapter 11 Quadratic Equations and Functions

16.

2

316 5

31650

xx





Solve the related quadratic equation.

2

Interval Test Value Test Conclusion





,5 

−6



2

3 6 16 6 5 false





,5 

does not belong to the solution set.

17.

2

40xx

Solve the related quadratic equation.



2

40

xx



0or 4 0

4

xx

x





The boundary points are 0 and 4.

Interval Test Value Test Conclusion





,0

−1



2

1410

50, true









,0

belongs to the solution set.





















Section 11.5 Polynomial and Rational Inequalities

18.

2

20xx

Solve the related quadratic equation.

























2

20

xx



The solution set is





2, 0

19.

2

230xx

Solve the related quadratic equation.

2

230

xx



Interval Test Value Test Conclusion

3

,2



 





−2



2

22 32 0

20, true

 



3

,2





 









belongs to the solution set.















Chapter 11 Quadratic Equations and Functions

20.

2

350xx

Solve the related quadratic equation.

2

350

xx



Interval Test Value Test Conclusion





,0

−1



2

3 1 5 1 0 false





,0

does not belong to the solution set.

21.

2

0xx

Solve the related quadratic equation.



2

0

10

xx





The boundary points are 0 and 1.

Interval Test Value Test Conclusion





,0

−1



2

110

20, false

  







,0

does not belong to the solution set.

Section 11.5 Polynomial and Rational Inequalities

22.

2

20xx 

Solve the related quadratic equation.



2

20

xx

 



0or 2 0

02

xx

 



Interval Test Value Test Conclusion

23.

2

42

420

xx





Solve the related quadratic equation, using the quadratic formula.

2

420xx

The boundary points are

22 and 22.



2

Interval Test Value Test Conclusion

00402

,2 2 ,2 2 does not

02, false

belong to the solution set.



   



Chapter 11 Quadratic Equations and Functions

24.

2

22

220

xx





Solve the related quadratic equation.

2

220

xx



25.

2

342

3420

xx





Solve the related quadratic equation.

2

3420xx

Interval Test

Value Test Conclusion

210

,3











−1

2

3( 1) 4( 1) 2

32, true



 210

,3















 belongs to the solution set.











