6 ()()2x−5 ()( )72 ()( )3 2 ()( )2x +1 3 ()( )6 5 ism aar.pdf · section 6.9 209 exercise set 6.9...

SECTION 6.9 209

Exercise Set 6.9 1. A binomial is an expression that contains two terms in which each exponent that appears on the variable is a whole

number. Examples: 22 3, 7, 9x x x+ − −

2. A trinomial is an expression containing three terms in which each exponent that appears on the variable is a whole number. Examples: 2 42, 2 1, 1x y x y x y+ + + + − −

3. The foil method is a method that obtains the products of the First, Outer, Inner, and Last terms of the binomials. 4. If the product of two factors is 0, then one or both of the factors must have a value of 0.

5. 0,02 ≠=++ acbxax

6. a

acbbx

2

42 −±−=

7. ( )( )2 9 18 6 3x x x x+ + = + + 8. ( ) ( )2 5 4 4 1x x x x+ + = + +

9. ( ) ( )2 6 3 2x x x x− − = − + 10. ( )( )2 6 3 2x x x x+ − = + −

11. ( )( )462422 −+=−+ xxxx 12. ( )( )2 6 8 4 2x x x x− + = − −

13. ( )( )2 2 3 1 3x x x x− − = + − 14. ( )( )16652 +−=−− xxxx

15. ( ) ( )2 10 21 7 3x x x x− + = − − 16. ( )( )2 81 9 9x x x− = − +

17. ( )( )2 25 5 5x x x− = − + 18. ( )( )2 20 5 4x x x x− − = − +

19. ( )( )472832 −+=−+ xxxx 20. ( )( )2 4 32 4 8x x x x+ − = − +

21. ( )( )796322 −+=−+ xxxx 22. ( )( )2 2 48 6 8x x x x− − = + −

23. ( )( )22 10 2 5 2x x x x− − = − + 24. ( )( )23 2 5 3 5 1x x x x− − = − +

25. ( )( )24 13 3 4 1 3x x x x+ + = + + 26. ( ) ( )22 11 21 2 3 7x x x x− − = + −

27. ( )( )2254125 2 ++=++ xxxx 28. ( )( )2521092 2 −−=+− xxxx

29. ( )( )24 11 6 4 3 2x x x x+ + = + + 30. ( )( )327221204 2 ++=++ xxxx

31. ( )( )24 11 6 4 3 2x x x x− + = − − 32. ( )( )12434116 2 −−=+− xxxx

33. ( )( )64324143 2 −+=−− xxxx 34. ( )( )1312156 2 ++=++ xxxx

( )( )35. 1 2 0

1 0 or 2 0

1 2

x x

x x

x x

− + =− = + == = −

( )( )36. 2 5 1 0

2 5 0 or 1 0

2 5 1

5

2

x x

x x

x x

x

+ − =+ = − == − =

= −

( )( )37. 3 4 2 1 0

3 4 0 or 2 1 0

3 4 2 1

4 1

3 2

x x

x x

x x

x x

+ − =+ = − == − =

= − =

38. ( )( ) 0456 =−− xx

06 =−x or 045 =−x

6=x 45 =x

5

4=x

210 CHAPTER 6 Algebra, Graphs, and Functions

( )( )239. 10 21 0

7 3 0

7 0 or 3 0

7 3

x x

x x

x x

x x

+ + =+ + =

+ = + == − = −

( )( )240. 4 5 0

5 1 0

5 0 or 1 0

5 1

x x

x x

x x

x x

+ − =+ − =

+ = − == − =

( )( )241. 4 3 0

3 1 0

3 0 or 1 0

3 1

x x

x x

x x

x x

− + =− − =

− = − == =

( )( )242. 5 24 0

8 3 0

8 0 or 3 0

8 3

x x

x x

x x

x x

− − =− + =

− = + == = −

43. xx 2152 =−

01522 =−− xx

( )( ) 035 =+− xx

05 =−x or 03 =+x

5=x 3−=x

44. 672 −=− xx

0672 =+− xx

( )( )6 1 0x x− − =

6 0x − = or 1 0x − = 6x = 1x =

45. 342 −= xx

0342 =+− xx

( )( )3 1 0x x− − =

3 0x − = or 1 0x − =

3x = 1x =

46. 040132 =+− xx

( )( )5 8 0x x− − =

5 0x − = or 8 0x − = 5x = 8x =

47. 0812 =−x

( ) ( )9 9 0x x− + =

9 0x − = or 9 0x + = 9x = 9x = −

48. 0642 =−x

( )( )8 8 0x x− + =

8 0x − = or 8 0x + = 8x = 8x = −

49. 03652 =−+ xx

( )( ) 049 =−+ xx

09 =+x or 04 =−x 9−=x 4=x

50. 020122 =++ xx

( )( )10 2 0x x+ + =

10 0x + = or 2 0x + = 10x = − 2x = −

( ) ( )

2

2

51. 3 10 8

3 10 8 0

3 2 4 0

3 2 0 or 4 0

3 2 4

2

3

x x

x x

x x

x x

x x

x

+ =+ − =− + =

− = + == = −

=

( )( )

2

2

52. 3 5 2

3 5 2 0

3 1 2 0

3 1 0 or 2 0

3 1 2

1

3

x x

x x

x x

x x

x x

x

− =− − =+ − =

+ = − == − =

= −

SECTION 6.9 211

53. 2115 2 −=+ xx

02115 2 =++ xx

( )( ) 0215 =++ xx

015 =+x or 02 =+x 15 −=x 2−=x

5

1−=x

54. 352 2 +−= xx

0352 2 =−+ xx

( )( ) 0312 =+− xx

012 =−x or 03 =+x 12 =x 3−=x

2

1=x

55. 143 2 −=− xx

0143 2 =+− xx

( )( ) 0113 =−− xx

013 =−x or 01 =−x

13 =x 1=x

3

1=x

56. 012165 2 =++ xx

( )( ) 0265 =++ xx

065 =+x or 02 =+x 65 −=x 2−=x

5

6−=x

57. 0294 2 =+− xx

( )( ) 0214 =−− xx

014 =−x or 02 =−x

14 =x 2=x

4

1=x

58. 026 2 =−+ xx

( ) ( )2 1 3 2 0x x− + =

2 1 0x − = or 3 2 0x + = 2 1x = 3 2x = −

1

2x =

2

3x = −

59. 2 2 15 0x x+ − =

1, 2, 15a b c= = = −

( ) ( )( )

( )

22 2 4 1 15

2 1x

− ± − −=

2 4 60 2 64 2 8

2 2 2x

− ± + − ± − ±= = =

6

32

x = = or 52

10 −=−=x

60. 2 12 27 0x x+ + =

1, 12, 27a b c= = =

( ) ( )( )

( )

212 12 4 1 27

2 1x

− ± −=

12 144 108 12 36 12 6

2 2 2x

− ± − − ± − ±= = =

6

32

x−= = − or

189

2x

−= = −

61. 2 3 18 0x x− − =

1, 3, 18a b c= = − = −

( ) ( ) ( )( )

( )

23 3 4 1 18

2 1x

− − ± − − −=

3 9 72 3 81 3 9

2 2 2x

± + ± ±= = =

12

62

x = = or 6

32

x−= = −

62. 2 6 16 0x x− − =

1, 6, 16a b c= = − = −

( ) ( ) ( )( )

( )

26 6 4 1 16

2 1x

− − ± − − −=

6 36 64 6 100 6 10

2 2 2x

± + ± ±= = =

16

82

x = = or 22

4 −=−=x


63. 982 =− xx

0982 =−− xx 9,8,1 −=−== cba

( ) ( ) ( )( )

( )12

91488 2 −−−±−−=x

2

108

2

1008

2

36648 ±=±=+±=x

92

18 ==x or 12

2 −=−=x

64. 1582 +−= xx

01582 =−+ xx

15,8,1 −=== cba

( ) ( )( )

( )12

151488 2 −−±−=x

2

3128

2

1248

2

60648 ±−=±−=+±−=x

314 ±−=x

65. 0322 =+− xx 3,2,1 =−== cba

( ) ( ) ( )( )

( )12

31422 2 −−±−−=x

2

82

2

1242 −±=−±=x

No real solution

66. 032 2 =−− xx 3,1,2 −=−== cba

( ) ( ) ( )( )

( )22

32411 2 −−−±−−=x

4

51

4

251

4

2411 ±=±=+±=x

2

3

4

6 ==x or 14

4 −=−=x

67. 0242 =+− xx 2,4,1 =−== cba

( ) ( ) ( )( )

( )12

21444 2 −−±−−=x

2

224

2

84

2

8164 ±=±=−±=x

22 ±=x

68. 0252 2 =−− xx

2,5,2 −=−== cba

( ) ( ) ( )( )

( )22

22455 2 −−−±−−=x

4

415

4

16255 ±=+±=x

69. 23 8 1 0x x− + =

3, 8, 1a b c= = − =

( ) ( ) ( )( )

( )

28 8 4 3 1

2 3x

− − ± − −=

8 64 12 8 52 8 2 13

6 6 6x

± − ± ±= = =

4 13

3x

±=

70. 22 4 1 0x x+ + =

2, 4, 1a b c= = =

( ) ( ) ( )

( )

24 4 4 2 1

2 2x

− ± −=

4 16 8 4 8 4 2 2

4 4 4x

− ± − − ± − ±= = =

2 2

2x

− ±=

SECTION 6.9 213

71. 014 2 =−− xx

1,1,4 −=−== cba

( ) ( ) ( )( )

( )42

14411 2 −−−±−−=x

8

171

8

1611 ±=+±=x

72. 0354 2 =−− xx

3,5,4 −=−== cba

( ) ( ) ( )( )

( )42

34455 2 −−−±−−=x

8

735

8

48255 ±=+±=x

73. 0572 2 =++ xx

5,7,2 === cba

( ) ( )( )

( )22

52477 2 −±−=x

4

37

4

97

4

40497 ±−=±−=−±−=x

14

4 −=−=x or 2

5

4

10 −=−=x

74. 593 2 −= xx

0593 2 =+− xx

5,9,3 =−== cba

( ) ( ) ( )( )

( )32

53499 2 −−±−−=x

6

219

6

60819 ±=−±=x

75. 07103 2 =+− xx

7,10,3 =−== cba

( ) ( ) ( )( )

( )32

7341010 2 −−±−−=x

6

410

6

1610

6

8410010 ±=±=−±=x

3

7

6

14 ==x or 16

6 ==x

76. 0174 2 =−+ xx 1,7,4 −=== cba

( ) ( )( )

( )42

14477 2 −−±−=x

8

657

8

16497 ±−=+±−=x

77. 013114 2 =+− xx

13,11,4 =−== cba

( ) ( ) ( )( )

( )42

13441111 2 −−±−−=x

8

8711

8

20812111 −±=−±=x

No real solution

78. 0295 2 =−+ xx

2,9,5 −=== cba

( ) ( )( )

( )52

25499 2 −−±−=x

10

119

10

1219

10

40819 ±−=±−=+±−=x

5

1

10

2 ==x or 210

20 −=−=x


79. Area of backyard ( ) 230 20 600 mlw= = =

Let x = width of grass around all sides of the flower garden

Width of flower garden 20 2x= −

Length of flower garden 30 2x= −

Area of flower garden ( )( )30 2 20 2lw x x= = − −

Area of grass ( )( )600 30 2 20 2x x= − − −

( )( )( )

( )( )

2

2

2

2

2

600 30 2 20 2 336

600 600 100 4 336

600 600 100 4 336

4 100 336

4 100 336 0

25 84 0

21 4 0

21 0 or 4 0

21 4

x x

x x

x x

x x

x x

x x

x x

x x

x x

− − − =

− − + =

− + − =− + =

− + =− + =− − =

− = − == =

21x ≠ since the width of the backyard is 20 m.

Width of grass = 4 m Width of flower garden

( )20 2 20 2 4 20 8 12 mx= − = − = − =

Length of flower garden

( )30 2 30 2 4 30 8 22 mx= − = − = − =

80. 10015000,45 2 −+= xx

0100,45152 =−+ xx

( )( ) 0205220 =−+ xx

0220 =+x or 0205 =−x 220−=x 205=x

Cannot produce a negative number of air conditioners. Thus, 205=x air conditioners.

81. a) Since the equation is equal to 6 and not 0, the zero-factor property cannot be used.

b) ( )( ) 674 =−− xx

628112 =+− xx

022112 =+− xx 22,11,1 =−== cba

( ) ( ) ( )( )

( )12

22141111 2 −−±−−=x

2

3311

2

8812111 ±=−±=x

37.8≈x or 63.2≈x

SECTION 6.10 215

82. a

acbbx

2

42 −±−=

The acb 42 − is the radicand in the quadratic formula, the part under the square root sign.

a) If 042 >− acb , then you are taking the square root of a positive number and there are two solutions.

These solutions are a

acbbx

2

42 −+−= and a

acbbx

2

42 −−−= .

b) If 042 =− acb , then you are taking the square root of zero and there is one solution. This solution

is a

b

a

bx

22

0 −=±−= .

c) If 042 <− acb , then you are taking the square root of a negative number and there is no real solution.

83. ( )( )1 3 0x x+ − =

2 2 3 0x x− − =

84.

Exercise Set 6.10 1. A function is a special type of relation where each value of the independent variable corresponds to a unique value of the dependent variable. 2. A relation is any set of ordered pairs. 3. The domain of a function is the set of values that can be used for the independent variable. 4. The range of a function is the set of values obtained for the dependent variable. 5. The vertical line test can be used to determine if a graph represents a function. If a vertical line can be drawn so that it intersects the graph at more than one point, then each value of x does not have a

unique value of y and the graph does not represent a function. If a vertical line cannot be made to

intersect the graph in at least two different places, then the graph represents a function. 6. The area of a square is a function of the length of a side, the average stopping distance of a car is a function of its speed, the cost of apples is a function of the number of apples

7. Not a function since x = 2 is not paired with a unique value of y.

8. Function since each value of x is paired with a unique value of y.

D: x = -2, -1, 1, 2, 3 R: y = -1, 1, 2, 3


9. Function since each vertical line intersects the graph at only one point.

D: all real numbers R: all real numbers




D: all real numbers R: y = 2

12. Not a function since x = -1 is not paired with a unique value of y.


D: all real numbers R: y ≥ -4


D: all real numbers R: y ≤ 10

15. Not a function since it is possible to draw a vertical line that intersects the graph at more than one point.


D: 0 ≤ x ≤ 8 R: -1 ≤ y ≤ 1


D: 0 ≤ x < 12 R: y = 1, 2, 3







D: all real numbers R: y > 0


D: 0 ≤ x ≤ 10 R: -1 ≤ y ≤ 3



D: all real numbers R: y ≥ 0



27. Not a function since x = 4 is paired with two

different values of y. 28. Not a function since x = 3 is paired with three

different values of y.


30. Not a function since x = 1 is paired with three different values of y.

31. ( ) 3, 2f x x x= + =

( )2 2 3 5f = + =

32. ( ) 2 5, 4f x x x= + =

( ) ( )4 2 4 5 8 5 13f = + = + =

33. ( ) 4,72 −=−−= xxxf

( ) ( ) 1787424 =−=−−−=−f

34. ( ) 1,35 −=+−= xxxf

( ) ( ) 8353151 =+=+−−=−f

35. ( ) 0,610 =−= xxxf

( ) ( ) 66060100 −=−=−=f

36. ( ) 4,67 =−= xxxf

( ) ( ) 226286474 =−=−=f

SECTION 6.10 217

37. ( ) 2 3 1, 4f x x x x= − + =

( ) ( ) ( )24 4 3 4 1 16 12 1 5f = − + = − + =

38. ( ) 2 5, 7f x x x= − =

( ) ( )27 7 5 49 5 44f = − = − =

39. ( ) 2,822 2 −=−−= xxxxf

( ) ( ) ( ) 4848822222 2 =−+=−−−−=−f

40. ( ) 2,732 =++−= xxxxf

( ) ( ) ( ) 976472322 2 =++−=++−=f

41. ( ) 3,453 2 −=++−= xxxxf

( ) ( ) ( ) 3841527435333 2 −=+−−=+−+−−=−f

42. ( ) 4,525 2 =++= xxxxf

( ) ( ) ( ) 935880542454 2 =++=++=f

43. ( ) 25 3 9, 1f x x x x= − + − = −

( ) ( ) ( )21 5 1 3 1 9 5 3 9 17f − = − − + − − = − − − = −

44. ( ) 23 6 10, 2f x x x x= − − + = −

( ) ( ) ( )22 3 2 6 2 10 12 12 10 10f − = − − − − + = − + + =

45.

46.

47.

48.

49.

50.


51. 2 16y x= −

a) 1 0, opens upwarda = >

b) ( ) ( )0 c) 0, 16 d) 0, 16x = − −

e) ( ) ( )4,0 , 4,0−

f)

g) D: all real numbers R: 16y ≥ −

52. 2 9y x= −


b) ( ) ( )0 c) 0, 9 d) 0, 9x = − −

e) ( ) ( )3,0 , 3,0−

f)


53. 2 4y x= − +

a) 1 0, opens downwarda = − <

b) ( ) ( )0 c) 0,4 d) 0,4x =

e) ( ) ( )2,0 , 2,0−

f)

g) D: all real numbers R: 4y ≤

54. 2 16y x= − +


b) ( ) ( )0 c) 0,16 d) 0,16x =

e) ( ) ( )4,0 , 4,0−

f)


55. ( ) 2 4f x x= − −


b) ( ) ( )0 c) 0, 4 d) 0, 4x = − −

e) no x-intercepts f)

g) D: all real numbers R: 4y ≤ −

56. 22 8y x= − −


b) ( ) ( )0 c) 0, 8 d) 0, 8x = − −



SECTION 6.10 219

57. 22 3y x= −


b) ( ) ( )0 c) 0, 3 d) 0, 3x = − −

e) ( ) ( )1.22,0 , 1.22,0−

f)


58. ( ) 23 6f x x= − −


b) ( ) ( )0 c) 0, 6 d) 0, 6x = − −



59. ( ) 2 2 6f x x x= + +


b) ( ) ( )1 c) 1,5 d) 0,6x = − −


g) D: all real numbers R: 5y ≥

60. 2 8 1y x x= − +


b) ( ) ( )4 c) 4, 15 d) 0,1x = −

e) ( ) ( )7.87,0 , 0.13,0

f)



61. 2 5 6y x x= + +


b) ( ) ( )5c) 2.5, 0.25 d) 0,6

2x = − − −

e) ( ) ( )3,0 , 2,0− −

f)

g) D: all real numbers R: 0.25y ≥ −

62. 2 7 8y x x= − −


b) ( )7 7 81c) , d) 0, 8

2 2 4x

= − −

e) ( ) ( )1,0 , 8,0−

f)

g) D: all real numbers R: 81

4y ≥ −

63. 2 4 6y x x= − + −


b) ( ) ( )2 c) 2, 2 d) 0, 6x = − −



64. 2 8 8y x x= − + −


b) ( ) ( )4 c) 4,8 d) 0, 8x = −

e) ( ) ( )1.17,0 , 6.83,0

f)


SECTION 6.10 221

65. 23 14 8y x x= − + −


b) ( )7 7 25c) , d) 0, 8

3 3 3x

= −

e) ( )2,0 , 4,0

3

f)

g) D: all real numbers R: 25

3y ≤

66. 22 6y x x= − −


b) ( )18

1 1c) , 6 d) 0, 6

4 4x

= − −

e) ( ) 32,0 , ,0

2 −

f)


67.

D: all real numbers R: 0>y

68.


69.


70.



71.


72.

D: all real numbers R: 1−>y

73.


74.

D: all real numbers R: 1−>y

75.


76.


77.


78.


SECTION 6.10 223

79. ( ) 300 0.10m s s= +

( ) ( )20,000 300 0.10 20,000

300 2000 $2300

m = += + =

80. ( ) ttd 60=

a) ( ) ( ) 1803603,3 === dt mi

b) ( ) ( ) 4207607,7 === dt mi

81. a) 2000 8x→ =

( ) ( ) ( )28 0.56 8 5.43 8 59.83

35.84 43.44 59.83 52.23%

f = − += − + =

b) 1997

c) ( )( )

5.43 5.434.848214286

2 2 0.56 1.12

4.85

bx

a

− −−= = = =

≈

( ) ( ) ( )24.85 0.56 4.85 5.43 4.85 59.83

13.1726 26.3355 59.83

46.6671 46.67%

f = − += − += ≈

82. a) 1999 5x→ =

( ) ( ) ( )25 4.25 5 30.32 5 150.14

106.25 151.6 150.14 195.49

195,490 free lunches

l = − + += − + + =≈

b) 1998

c) ( )30.32 30.32

3.5670588242 2 4.25 8.5

3.57

bx

a

− − −= = = =− −

≈

( ) ( ) ( )( )

23.57 4.25 3.57 30.32 3.57 150.14

4.25 12.7449 108.2424 150.14

54.165825 108.2424 150.14

204.216575 204,216.575

204,217 free lunches

l = − + +

= − + += − + += =≈

83. ( ) ( ) xxP 1.03.14000=

a) ( ) ( ) ( )101.03.1400010,10 == Px

( ) 52003.14000 == people

b) ( ) ( ) ( )501.03.1400050,50 == Px

( )71293.34000=

852,1472.851,14 ≈= people

84. tePP 00003.00

−=

( )5000003.02000 −= eP

0015.02000 −= e

( )9985011244.02000=

1997002249.1997 ≈= g

85. a) Yes

b) 6500 scooter injuries≈

86. a) No, the average cost is increasing and then decreasing.

b) 1200$≈


87. ( )( )( )12

2029.21

xd

−=

a) ( )( )( )12

192029.21,19

−== dx

( )( ) 2.23059463094.19.21 ≈= cm

b) ( )( )( )12

42029.21,4

−== dx

( )( ) 2.55519842099.29.21 ≈= cm

c) ( )( )( )12

02029.21,0

−== dx

( )( ) 5.69174802105.39.21 ≈= cm

88. ( ) ( )nnf 04.1000,85=

a) ( ) ( )804.1000,858 =f

( ) 328,11636856905.1000,85 ≈= ;

The value of the house after 8 years is

about .328,116$

b) 15 years since ( ) 50.192,147$14 ≈f and

( ) 20.080,153$15 ≈f

89. ( ) 18785.0 +−= xxf

a) ( ) ( ) 1701872085.020 =+−=f beats per minute

b) ( ) ( ) 1625.1611873085.030 ≈=+−=f beats per minute

c) ( ) ( ) 1455.1441875085.050 ≈=+−=f beats per minute

d) ( ) ( ) 1361876085.060 =+−=f beats per minute

e) 8518785.0 =+− x

10285.0 −=− x

120=x years of age

90. ( ) ttd 000,186=

a) ( ) ( ) 800,2413.1000,1863.1,3.1 === dt mi

b) m sec m

186,000 60 11,160,000sec min min

× =

( ) ttd 000,160,11=

c) ( ) ( ) 000,628,923.8000,160,113.8 ==d mi

6 ()()2x−5 ()( )72 ()( )3 2 ()( )2x +1 3 ()( )6 5 ism aar.pdf · section 6.9 209 exercise set 6.9...

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