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Mathematics LibreTexts

6.5E: Exercises

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    30869
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    Practice Makes Perfect

    Use the Zero Product Property

    In the following exercises, solve.

    1. \((3a−10)(2a−7)=0\)

    Answer

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

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

    3. \(6m(12m−5)=0\)

    Answer

    \(m=0,\; m=\frac{5}{12}\)

    4. \(2x(6x−3)=0\)

    5. \((2x−1)^2=0\)

    Answer

    \(x=\frac{1}{2}\)

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

    Solve Quadratic Equations by Factoring

    In the following exercises, solve.

    7. \(5a^2−26a=24\)

    Answer

    \(a=−\frac{5}{4},\; a=6\)

    8. \(4b^2+7b=−3\)

    9. \(4m^2=17m−15\)

    Answer

    \(m=\frac{5}{4},\; m=3\)

    10. \(n^2=5−6n\)

    11. \(7a^2+14a=7a\)

    Answer

    \(a=−1,\; a=0\)

    12. \(12b^2−15b=−9b\)

    13. \(49m^2=144\)

    Answer

    \(m=\frac{12}{7},\; m=−\frac{12}{7}\)

    14. \(625=x^2\)

    15. \(16y^2=81\)

    Answer

    \(y=−\frac{9}{4},\; y=\frac{9}{4}\)

    16. \(64p^2=225\)

    17. \(121n^2=36\)

    Answer

    \(n=−\frac{6}{11},\; n=\frac{6}{11}\)

    18. \(100y^2=9\)

    19. \((x+6)(x−3)=−8\)

    Answer

    \(x=2,\; x=−5\)

    20. \((p−5)(p+3)=−7\)

    21. \((2x+1)(x−3)=−4x\)

    Answer

    \(x=\frac{3}{2},\; x=−1\)

    22. \((y−3)(y+2)=4y\)

    23. \((3x−2)(x+4)=12x\)

    Answer

    \(x=\frac{3}{2},\; x=−1\)

    24. \((2y−3)(3y−1)=8y\)

    25. \(20x^2−60x=−45\)

    Answer

    \(x=−\frac{2}{3}\)

    26. \(3y^2−18y=−27\)

    27. \(15x^2−10x=40\)

    Answer

    \(x=2,\; x=−\frac{4}{3}\)

    28. \(14y^2−77y=−35\)

    29. \(18x^2−9=−21x\)

    Answer

    \(x=−\frac{3}{2},\; x=\frac{1}{3}\)

    30. \(16y^2+12=−32y\)

    31. \(16p^3=24p^2+9p\)

    Answer

    \(p=0,\; p=\frac{3}{4}\)

    32. \(m^3−2m^2=−m\)

    33. \(2x^3+72x=24x^2\)

    Answer

    \(x=0,\space x=6\)

    34. \(3y^3+48y=24y^2\)

    35. \(36x^3+24x^2=−4x\)

    Answer

    \(x=0,\space x=\frac{1}{3}\)

    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.

    Answer

    ⓐ \(x=2\) 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.

    Answer

    ⓐ \(x=\frac{3}{2}\) 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. \(f(x)=9x^2−4\)

    Answer

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

    42. \(f(x)=25x^2−49\)

    43. \(f(x)=6x^2−7x−5\)

    Answer

    ⓐ \(x=\frac{5}{3}\) 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.

    Answer

    \(−13,\space −11\) 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.

    Answer

    \(−14,\space −12\) 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.

    Answer

    \(−4\) 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.

    Answer

    \(5,\space 11\)

    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.

    Answer

    \(6,\space 8\)

    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.

    Answer

    \(8,\space 15,\space 17\)

    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.

    Answer

    ⓐ 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?

    Answer

    Answers will vary.

    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.

    This table has 4 columns, 3 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don’t get it. The first column has the following statements: solve quadratic equations by using the zero product property, solve quadratic equations by factoring and solve applications modeled by quadratic equations.

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

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