Download Section 4.8 Practice Exercises

S ection 4 .8
c. Set 11(/)
~
1152 and so lve [or
Solving Equations by Using the Zero Pro duct Rule
I.
11(/) ~ - 161" + 2881
1152 ~ - I 61" + 2881
Substitute 1152 [o r 11 (/).
161" - 2881 + 11 52 ~ 0
Se t th e equatio n equ al to ze ro.
16(1" - 181 + 72) ~ 0
16(1 - 6)(1 - 12)
I ~
6
or
~
I ~
FaclOr o ut th e GeF.
0
FaclOr.
12
The rocke t will reach a he ight
o f 11 52 fl afte r 6 sec (o n
th e wa y up) and a fl e r 12 sec
(on th e way do wn ). (See
Figure 4-8.)
-::
::::.
~
't:j
I:
Hcight of Rockct Versus TIme
After Launch
[, (1)
t500
1250
1000
750
500
250
I) = - 16? + 2881
(6, 11 52) ~_--,I"":.!,
( 12. 1152)
0~--~--~~~~~~7-~ 1
- 250
3
6
9
12
15
18
Time (sec)
Figure 4-8
Section 4.8
Practice Exercises
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Study Skills Exercise
1. De fin e th e key terms.
a. Quadratic equation
c. Quadnltic function
b. Zero product rule
d. Parabola
Review Exercises
2. Write th e fac to red form fo r each bino mia l, if possible.
a.
x' - y'
b.
x' + y'
c.
x' - y'
d.
x' + y'
For Exe rcises 3-8. [acto r complete ly.
3. lOx'
+ 3x
6. 3q 2 _ 4q - 4
4. 7x2
7.
-
5. 2p' - 9p - 5
28
i' - 1
8.
Z2 -
liz
+ 30
Objective 1: Solving Equations by Using the Zero Product Rule
9. What condit io ns are necessa ry to solve an
equ atio n by using th e zero product rul e?
10. State th e zero product rul e.
341
342
Chapter 4
Polynomials
For Exercises 11 - 16. de termine which of th e equations are written in the correct form to apply the zero product
rule directly. If an equation is not in the correct form . explai n what is wrong.
11. 2x(x - 3) = 0
12. (/I + 1)(1/ - 3) = 10
l3. 3p' - 7p + 4 = 0
14. (1 - 1 - 12 = 0
15. a(a + 3)'
16.
For Exercises 17- 52, solve the equation.
17. (x + 3)(x + 5) = 0
o = 5(y -
021.
0.4 )(y + 2.) )
24.
(~x -
5 )(x +
D
=0
Examples Hi.)
18. (x + 7)(x - 4)
20. (3a + 1)( 4a - 5) = 0
23.
(See
=5
x (x + 4)( )Ox - 3)
o = - 4(:
19. (2.v + 9)(51V - 1)
=0
=0
22. 1(1 - 6)(31 - II )
- 7.5 )(z - 9.3)
=0
=0
25 . .<'+ 6x - 27 = 0
26. 2x' + x - 15 = 0
27. 2x2 + 5x = 3
28. - Il x = 3x' - 4
29. lOx2 = 15x
30. 5x' = 7x
31. 6(y - 2) - 3(y + 1)
32. 4x + 3(x - 9) = 6x + I
33. - 9 = y(y + 6)
34. - 62
36. 6y' + 2y = 48
37. (x + 1)(2r - I)(x - 3)
35. 9p' - 15p - 6
=0
38. 2x(x - 4)'(4x + 3)
=0
039. (y - 3)(y + 4) = 8
= 1(1 - 16) + 2
43. p' + (p + 7)'
44. x' + (x + 2)' = 100
45. 31(1 + 5) - (1 = 2(1 + 41 - 1
46. a' - 4a - 2
47. 2r' - 8x' - 24x = 0
48. 2p 3 + 20p' + 42p
49. 1V3 = 16w
50. 12.-' = 27x
51. 0 = 2x3 + 5x' - 18x - 45
=6
=0
40. (I + 10)(1 + 5) = 6
42. 1V(61V + I) = 2
41. (2a - 1)(a - I)
=8
=0
= 169
= (a + 3)(a - 5)
52. 0 = 3y' + y' - 48y - 16
Objective 2: Applications of Quadratic Equations
53. If 5 is added to the square of a number, the result is 30. Find a ll such numbers.
54. Four less than the square of a number is 77 . Find a ll such numbers.
55. The square of a numbe r is eq ual to 12 more tha n the number. Find all such numbe rs.
56. The square of a num ber is equal
to
20 more than the number. Find a ll such numbe rs.
57. The product of two consecutive integers is 42. Find the integers.
58. 111e product of two consecutive intege rs is 110. Find the int egers.
59. The product of two consecutive odd integers is 63. Find the integers.
(See
Example 7.)
60. The product of two consecuti ve even integers is 120. Find the integers.
61. A rectangular pe n is to contain 35 ft' of area. If the widt h is 2 ft less th an the le ngth , find the dimensio ns
of the pen. (See Example 8.)
\
Section 4.8
343
Solving Equations by Using the Zero Product Rule
0 62. The length of a rectangular photograph is 7 in. more than the width. If the area is 78 in .'. what are the
dimensions of the photograph?
63. The length of a rectangular room is 5 yd more than the width. If the area is 300 yd' . find the length and
the width of the room .
64. 111e top of a rectangular dining room table is twice as long as it is wide. Find the dimensions of the table if
the area is 18 ft '.
.
65. 111e he ight of a triangle is I in . more than the base. If the height is increased by 2 in . while the base
rem ains the same. th e new area becomes 20 in .2
a. Find the base a nd heigh t of the origi nal tri angle.
b. Find the area of the original tria ngle.
66. The base of a tria ngle is 2 cm more tha n the he ight. If the base is increased by 4 cm while the he ight
remains the same, the new area is 56 cm2 .
a. Find the base and height of the ori gi na l tri angle.
h. Find the area of the original triangle.
67. The area of a triangular garde n is 25 ft' . The base is twice the height. Find the base and the height of the
triangle.
68. The height of a tria ngle is I in. more tha n twice the base. If the area is 18 in.' . find the base a nd height of
the triangle.
69. The sum of the squares of two consecutive positive integers is 41. Find the integers.
70. The sum of the sq ua res of two consecutive. positive even integers is 164. Find the integers.
71. Justi n must travel from Summe rsville to Clayton. He can drive 10 mi through
the mountains at 40 mph. Or he can drive east and then north on
superhighways at 60 mph . The a lt ernati ve rou te form s a right a ngle as shown
in the diagram. The easte rn leg is 2 mi less than the northe rn leg. (See Exampte 9.1
Clayton
a. Find the total dista nce Justin would travel in goi ng the al te rn ati ve route.
b. I r Justin wants to minimize th e time or th e trip, which route should he take ?
72. A 17-ft ladder is standing up against a wall. The distance between the base of
the ladder and the wall is 7 ft less than the distance between the top of the
ladder and the base of the wall. Find the distance between the base of the
ladde r and the wall.
73. A right tria ngle has side lengths represented by three consecutive even
integers. Find the le ngths of the three sides, measured in meters.
Summ ersvill e
74. The hypote nuse of a ri ght triangle is 3 m more than twice the short leg. The lo nger leg is 2 m more than
twice the short er leg. Find the le ngths of the sides.
75. Dete rmine the length of the radius of a circle whose area is nume rica lly eq ual to its circum ference.
76. De te rmine the length of the rad ius of a circle whose area is numerically twice its circumference.