# 6.5E: Exercises

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## Practice Makes Perfect

Use the Zero Product Property

In the following exercises, solve.

1. $$(3a−10)(2a−7)=0$$

$$a=\frac{10}{3},\; a=\frac{7}{2}$$

2. $$(5b+1)(6b+1)=0$$

3. $$6m(12m−5)=0$$

$$m=0,\; m=\frac{5}{12}$$

4. $$2x(6x−3)=0$$

5. $$(2x−1)^2=0$$

$$x=\frac{1}{2}$$

6. $$(3y+5)^2=0$$

Solve Quadratic Equations by Factoring

In the following exercises, solve.

7. $$5a^2−26a=24$$

$$a=−\frac{4}{5},\; a=6$$

8. $$4b^2+7b=−3$$

9. $$4m^2=17m−15$$

$$m=\frac{5}{4},\; m=3$$

10. $$n^2=5−6n$$

11. $$7a^2+14a=7a$$

$$a=−1,\; a=0$$

12. $$12b^2−15b=−9b$$

13. $$49m^2=144$$

$$m=\frac{12}{7},\; m=−\frac{12}{7}$$

14. $$625=x^2$$

15. $$16y^2=81$$

$$y=−\frac{9}{4},\; y=\frac{9}{4}$$

16. $$64p^2=225$$

17. $$121n^2=36$$

$$n=−\frac{6}{11},\; n=\frac{6}{11}$$

18. $$100y^2=9$$

19. $$(x+6)(x−3)=−8$$

$$x=2,\; x=−5$$

20. $$(p−5)(p+3)=−7$$

21. $$(2x+1)(x−3)=−4x$$

$$x=\frac{3}{2},\; x=−1$$

22. $$(y−3)(y+2)=4y$$

23. $$(3x−2)(x+4)=12x$$

$$x=\frac{3}{2},\; x=−1$$

24. $$(2y−3)(3y−1)=8y$$

25. $$20x^2−60x=−45$$

$$x=−\frac{2}{3}$$

26. $$3y^2−18y=−27$$

27. $$15x^2−10x=40$$

$$x=2,\; x=−\frac{4}{3}$$

28. $$14y^2−77y=−35$$

29. $$18x^2−9=−21x$$

$$x=−\frac{3}{2},\; x=\frac{1}{3}$$

30. $$16y^2+12=−32y$$

31. $$16p^3=24p^2-9p$$

$$p=0,\; p=\frac{3}{4}$$

32. $$m^3−2m^2=−m$$

33. $$2x^3+72x=24x^2$$

$$x=0,\space x=6$$

34. $$3y^3+48y=24y^2$$

35. $$36x^3+24x^2=−4x$$

$$x=0,\space x=\frac{1}{3}$$

36. $$2y^3+2y^2=12y$$

Solve Equations with Polynomial Functions

In the following exercises, solve.

37. For the function, $$f(x)=x^2−8x+8$$, ⓐ find when $$f(x)=−4$$ ⓑ Use this information to find two points that lie on the graph of the function.

ⓐ $$x=2$$ or $$x=6$$ ⓑ $$(2,−4)$$ $$(6,−4)$$

38. For the function, $$f(x)=x^2+11x+20$$, ⓐ find when $$f(x)=−8$$ ⓑ Use this information to find two points that lie on the graph of the function.

39. For the function, $$f(x)=8x^2−18x+5$$, ⓐ find when $$f(x)=−4$$ ⓑ Use this information to find two points that lie on the graph of the function.

ⓐ $$x=\frac{3}{2}$$ or $$x=\frac{3}{4}$$
ⓑ $$(\frac{3}{2},−4)$$ $$(\frac{3}{4},−4)$$

40. For the function, $$f(x)=18x^2+15x−10$$, ⓐ find when $$f(x)=15$$ ⓑ Use this information to find two points that lie on the graph of the function.

In the following exercises, for each function, find: ⓐ the zeros of the function ⓑ the $$x$$-intercepts of the graph of the function ⓒ the $$y$$-intercept of the graph of the function.

41. $$f(x)=9x^2−4$$

ⓐ $$x=\frac{2}{3}$$ or $$x=−\frac{2}{3}$$
ⓑ $$(\frac{2}{3},0)$$, $$(−\frac{2}{3},0)$$
ⓒ $$(0,−4)$$

42. $$f(x)=25x^2−49$$

43. $$f(x)=6x^2−7x−5$$

ⓐ $$x=\frac{5}{3}$$ or $$x=−\frac{1}{2}$$
ⓑ $$(\frac{5}{3},0)$$, $$(−\frac{1}{2},0)$$
ⓒ $$(0,−5)$$

44. $$f(x)=12x^2−11x+2$$

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

45. The product of two consecutive odd integers is $$143$$. Find the integers.

$$−13,\space −11$$ and $$11,\space 13$$

46. The product of two consecutive odd integers is $$195$$. Find the integers.

47. The product of two consecutive even integers is $$168$$. Find the integers.

$$−14,\space −12$$ and $$12,\space 14$$

48. The product of two consecutive even integers is $$288$$. Find the integers.

49. The area of a rectangular carpet is $$28$$ square feet. The length is three feet more than the width. Find the length and the width of the carpet.

$$−4$$ and $$7$$

50. A rectangular retaining wall has area $$15$$ square feet. The height of the wall is two feet less than its length. Find the height and the length of the wall.

51. The area of a bulletin board is $$55$$ square feet. The length is four feet less than three times the width. Find the length and the width of the a bulletin board.

$$5,\space 11$$

52. A rectangular carport has area $$150$$ square feet. The height of the carport is five feet less than twice its length. Find the height and the length of the carport.

53. A pennant is shaped like a right triangle, with hypotenuse $$10$$ feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.

$$6,\space 8$$

54. A stained glass window is shaped like a right triangle. The hypotenuse is $$15$$ feet. One leg is three more than the other. Find the lengths of the legs.

55. A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is $$9$$ feet longer than the side along the building. The third side is $$7$$ feet longer than the side along the building. Find the lengths of all three sides of the reflecting pool.

$$8,\space 15,\space 17$$

56. A goat enclosure is in the shape of a right triangle. One leg of the enclosure is built against the side of the barn. The other leg is $$4$$ feet more than the leg against the barn. The hypotenuse is $$8$$ feet more than the leg along the barn. Find the three sides of the goat enclosure.

57. Juli is going to launch a model rocket in her back yard. When she launches the rocket, the function $$h(t)=−16t^2+32t$$ models the height, $$h$$, of the rocket above the ground as a function of time, $$t$$. Find:

ⓐ the zeros of this function which tells us when the rocket will hit the ground. ⓑ the time the rocket will be $$16$$ feet above the ground.

ⓐ 0, 2 ⓑ 1

58. Gianna is going to throw a ball from the top floor of her middle school. When she throws the ball from $$48$$ feet above the ground, the function $$h(t)=−16t^2+32t+48$$ models the height, $$h$$, of the ball above the ground as a function of time, $$t$$. Find:

ⓐ the zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be $$48$$ feet above the ground. ⓒ the height the ball will be at $$t=1$$ seconds which is when the ball will be at its highest point.

## Writing Exercises

59. Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?

60. Give an example of a quadratic equation that has a GCF and none of the solutions to the equation is zero.

## Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

## Additional Factoring Applications

Number problems

61. The product of two consecutive even integers is $$624$$. Find the integers.

$$-26$$ and $$−24$$, and $$24$$ and $$26$$

62. The product of two consecutive even integers is $$528$$. Find the integers.

63. The product of two consecutive positive integers is $$552$$. Find the integers.

$$23$$, $$24$$

64. The product of two consecutive positive integers is $$756$$. Find the integers..

65.

The product of two consecutive odd integers is $$483$$. Find the integers.

$$−23$$ and $$−21$$, and $$21$$ and $$23$$

66. The product of two consecutive odd integers is $$783$$. Find the integers.

67. You have two positive numbers. The second number is three more than two times the ﬁrst number. The diﬀerence of their squares is $$144$$. Find both positive numbers.

$$5$$ and $$13$$

68. You have two positive numbers. The second number is two more than three times the ﬁrst number. The diﬀerence of their squares is $$60$$. Find both positive numbers.

69.

Two numbers diﬀer by $$5$$. The sum of their squares is $$97$$. Find the two numbers.

$$4$$ and $$9$$, and $$−4$$ and $$−9$$

70. Two numbers diﬀer by $$6$$. The sum of their squares is $$146$$. Find the two numbers.

Area of Rectangles

71. A rectangular canvas picture measures $$14$$ inches by $$36$$ inches. The canvas is mounted inside a frame of uniform width, increasing the total area covered by both canvas and frame to $$720$$ square inches. Find the uniform width of the frame.

$$2$$ inches

72. A rectangular canvas picture measures $$10$$ inches by $$32$$ inches. The canvas is mounted inside a frame of uniform width, increasing the total area covered by both canvas and frame to $$504$$ square inches. Find the uniform width of the frame.

73. A rectangle has perimeter $$42$$ feet and area $$104$$ square feet. Find the dimensions of the rectangle.

$$8$$ feet by $$13$$ feet

74. A rectangle has perimeter $$32$$ feet and area $$55$$ square feet. Find the dimensions of the rectangle.

75. The length of a rectangle is three feet longer than six times its width. If the area of the rectangle is $$165$$ square feet, what is the width of the rectangle?

$$5$$ feet

76. The length of a rectangle is three feet longer than nine times its width. If the area of the rectangle is $$90$$ square feet, what is the width of the rectangle?

77. The ratio of the width to the length of a given rectangle is $$2$$ to $$3$$, or $$\dfrac {2}{3}$$. If the width and length are both increased by $$4$$ inches, the area of the resulting rectangle is $$80$$ square inches. Find the width and length of the original rectangle.

$$4$$ inches by $$6$$ inches

78. The ratio of the width to the length of a given rectangle is $$3$$ to $$4$$, or $$\dfrac {3}{4}$$. If the width is increased by $$3$$ inches and the length is increased by $$6$$ inches, the area of the resulting rectangle is $$126$$ square inches. Find the width and length of the original rectangle.

Area of Triangles

 79. Find the area of $$\triangle ABC$$ 80. Find the area of $$\triangle ABC$$ x

79. $$x=24,$$ area is $$240$$  square units

 81. Find the value of $$x$$ given the area of $$\triangle ABC$$ is 35: 82. Find the value of $$x$$ given the area of $$\triangle ABC$$ is 24.

81. $$x=5$$

 83. Find the value of $$x$$ given the area of $$\triangle ABC$$ is 12: 84. Find the value of $$x$$ given the area of $$\triangle ABC$$ is 108:

83. $$x=4$$

Area of circles

85. The radius of the outer circle is one inch longer than twice the radius of the inner circle.

If the area of the shaded region is $$40\pi$$ square inches, what is the length of the inner radius?

$$3$$ inches

86. The radius of the outer circle is two inches longer than three times the radius of the inner circle.

If the area of the shaded region is $$180\pi$$ square inches, what is the length of the inner radius?

 87. Find the area of the circle to the nearest tenth 88. Find the area of the circle to the nearest tenth

87.  $$78.5$$  square units

Projectile Problems

89. A projectile is ﬁred at an angle into the air from atop a cliﬀ overlooking the ocean. The projectile’s distance (in feet) from the base of the cliﬀ is given by the equation $$x = 180t$$ and the projectile’s height above sea level (in feet) is given by the equation $$y = −16t^2 + 352t + 1664$$ where $$t$$ is the amount of time (in seconds) that has passed since the projectile’s release. How much time passes before the projectile splashes into the ocean? At that time, how far is the projectile from the base of the cliﬀ?

$$26$$ seconds, $$4,680$$ feet
90. A projectile is ﬁred at an angle into the air from atop a cliﬀ overlooking the ocean. The projectile’s distance (in feet) from the base of the cliﬀ is given by the equation $$x = 140t$$ and the projectile’s height above sea level (in feet) is given by the equation $$y = −16t^2 + 288t + 1408$$ where $$t$$ is the amount of time (in seconds) that has passed since the projectile’s release. How much time passes before the projectile splashes into the ocean? At that time, how far is the projectile from the base of the cliﬀ?