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Mathematics LibreTexts

9.1E: Exercises

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    Practice Makes Perfect

    ExerciseS 1 - 22: Solve Quadratic Equations of the Form \(ax^{2}=k\) Using the Square Root Property

    In the following exercises, solve each equation.

    1. \(a^{2}=49\)

    2. \(b^{2}=144\)

    3. \(r^{2}-24=0\)

    4. \(t^{2}-75=0\)

    5. \(u^{2}-300=0\)

    6. \(v^{2}-80=0\)

    7. \(4 m^{2}=36\)

    8. \(3 n^{2}=48\)

    9. \(\frac{4}{3} x^{2}=48\)

    10. \(\frac{5}{3} y^{2}=60\)

    11. \(x^{2}+25=0\)

    12. \(y^{2}+64=0\)

    13. \(x^{2}+63=0\)

    14. \(y^{2}+45=0\)

    15. \(\frac{4}{3} x^{2}+2=110\)

    16. \(\frac{2}{3} y^{2}-8=-2\)

    17. \(\frac{2}{5} a^{2}+3=11\)

    18. \(\frac{3}{2} b^{2}-7=41\)

    19. \(7 p^{2}+10=26\)

    20. \(2 q^{2}+5=30\)

    21. \(5 y^{2}-7=25\)

    22. \(3 x^{2}-8=46\)

    Answer

    1. \(a=\pm 7\)

    3. \(r=\pm 2 \sqrt{6}\)

    5. \(u=\pm 10 \sqrt{3}\)

    7. \(m=\pm 3\)

    9. \(x=\pm 6\)

    11. \(x=\pm 5 i\)

    13. \(x=\pm 3 \sqrt{7} i\)

    15. \(x=\pm 9\)

    17. \(a=\pm 2 \sqrt{5}\)

    19. \(p=\pm \frac{4 \sqrt{7}}{7}\)

    21. \(y=\pm \frac{4 \sqrt{10}}{5}\)

    ExerciseS 23 - 46: Solve Quadratic Equations of the Form \(a(x-h)^{2}=k\) Using the Square Root Property

    In the following exercises, solve each equation.

    23. \((u-6)^{2}=64\)

    24. \((v+10)^{2}=121\)

    25. \((m-6)^{2}=20\)

    26. \((n+5)^{2}=32\)

    27. \(\left(r-\frac{1}{2}\right)^{2}=\frac{3}{4}\)

    28. \(\left(x+\frac{1}{5}\right)^{2}=\frac{7}{25}\)

    29. \(\left(y+\frac{2}{3}\right)^{2}=\frac{8}{81}\)

    30. \(\left(t-\frac{5}{6}\right)^{2}=\frac{11}{25}\)

    31. \((a-7)^{2}+5=55\)

    32. \((b-1)^{2}-9=39\)

    33. \(4(x+3)^{2}-5=27\)

    34. \(5(x+3)^{2}-7=68\)

    35. \((5 c+1)^{2}=-27\)

    36. \((8 d-6)^{2}=-24\)

    37. \((4 x-3)^{2}+11=-17\)

    38. \((2 y+1)^{2}-5=-23\)

    39. \(m^{2}-4 m+4=8\)

    40. \(n^{2}+8 n+16=27\)

    41. \(x^{2}-6 x+9=12\)

    42. \(y^{2}+12 y+36=32\)

    43. \(25 x^{2}-30 x+9=36\)

    44. \(9 y^{2}+12 y+4=9\)

    45. \(36 x^{2}-24 x+4=81\)

    46. \(64 x^{2}+144 x+81=25\)

    Answer

    23. \(u=14, u=-2\)

    25. \(m=6 \pm 2 \sqrt{5}\)

    27. \(r=\frac{1}{2} \pm \frac{\sqrt{3}}{2}\)

    29. \(y=-\frac{2}{3} \pm \frac{2 \sqrt{2}}{9}\)

    31. \(a=7 \pm 5 \sqrt{2}\)

    33. \(x=-3 \pm 2 \sqrt{2}\)

    35. \(c=-\frac{1}{5} \pm \frac{3 \sqrt{3}}{5} i\)

    37. \(x=\frac{3}{4} \pm \frac{\sqrt{7}}{2} i\)

    39. \(m=2 \pm 2 \sqrt{2}\)

    41. \(x=3+2 \sqrt{3}, x=3-2 \sqrt{3}\)

    43. \(x=-\frac{3}{5}, x=\frac{9}{5}\)

    45. \(x=-\frac{7}{6}, x=\frac{11}{6}\)

    ExerciseS 47 - 68: Mixed Practice

    In the following exercises, solve using the Square Root Property.

    47. \(2 r^{2}=32\)

    48. \(4 t^{2}=16\)

    49. \((a-4)^{2}=28\)

    50. \((b+7)^{2}=8\)

    51. \(9 w^{2}-24 w+16=1\)

    52. \(4 z^{2}+4 z+1=49\)

    53. \(a^{2}-18=0\)

    54. \(b^{2}-108=0\)

    55. \(\left(p-\frac{1}{3}\right)^{2}=\frac{7}{9}\)

    56. \(\left(q-\frac{3}{5}\right)^{2}=\frac{3}{4}\)

    57. \(m^{2}+12=0\)

    58. \(n^{2}+48=0\)

    59. \(u^{2}-14 u+49=72\)

    60. \(v^{2}+18 v+81=50\)

    61. \((m-4)^{2}+3=15\)

    62. \((n-7)^{2}-8=64\)

    63. \((x+5)^{2}=4\)

    64. \((y-4)^{2}=64\)

    65. \(6 c^{2}+4=29\)

    66. \(2 d^{2}-4=77\)

    67. \((x-6)^{2}+7=3\)

    68. \((y-4)^{2}+10=9\)

    Answer

    47. \(r=\pm 4\)

    49. \(a=4 \pm 2 \sqrt{7}\)

    51. \(w=1, w=\frac{5}{3}\)

    53. \(a=\pm 3 \sqrt{2}\)

    55. \(p=\frac{1}{3} \pm \frac{\sqrt{7}}{3}\)

    57. \(m=\pm 2 \sqrt{2 i}\)

    59. \(u=7 \pm 6 \sqrt{2}\)

    61. \(m=4 \pm 2 \sqrt{3}\)

    63. \(x=-3, x=-7\)

    65. \(c=\pm \frac{5 \sqrt{6}}{6}\)

    67. \(x=6 \pm 2 i\)

    ExerciseS 69 - 70: Writing exercises

    69. In your own words, explain the Square Root Property.

    70. In your own words, explain how to use the Square Root Property to solve the quadratic equation \((x+2)^{2}=16\).

    Answer

    69. Answers will vary.

    Self Check

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

    This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement “I can solve quadratic equations of the form a times x squared equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”
    Figure 9.1.23

    Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of \(x\) minus \(h\) equals \(k\) using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”

    b. If most of your checks were:

    …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

    …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

    …no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.