
6.5E: Exercises


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

In the following exercises, solve.

7. $$5a^2−26a=24$$

$$a=−\frac{5}{4},\; a=6$$

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?