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Mathematics LibreTexts

9.5E: Exercises

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    Practice Makes Perfect

    Exercise \(\PageIndex{11}\) Solve equations in quadratic form

    In the following exercises, solve.

    1. \(x^{4}-7 x^{2}+12=0\)
    2. \(x^{4}-9 x^{2}+18=0\)
    3. \(x^{4}-13 x^{2}-30=0\)
    4. \(x^{4}+5 x^{2}-36=0\)
    5. \(2 x^{4}-5 x^{2}+3=0\)
    6. \(4 x^{4}-5 x^{2}+1=0\)
    7. \(2 x^{4}-7 x^{2}+3=0\)
    8. \(3 x^{4}-14 x^{2}+8=0\)
    9. \((x-3)^{2}-5(x-3)-36=0\)
    10. \((x+2)^{2}-3(x+2)-54=0\)
    11. \((3 y+2)^{2}+(3 y+2)-6=0\)
    12. \((5 y-1)^{2}+3(5 y-1)-28=0\)
    13. \(\left(x^{2}+1\right)^{2}-5\left(x^{2}+1\right)+4=0\)
    14. \(\left(x^{2}-4\right)^{2}-4\left(x^{2}-4\right)+3=0\)
    15. \(2\left(x^{2}-5\right)^{2}-5\left(x^{2}-5\right)+2=0\)
    16. \(2\left(x^{2}-5\right)^{2}-7\left(x^{2}-5\right)+6=0\)
    17. \(x-\sqrt{x}-20=0\)
    18. \(x-8 \sqrt{x}+15=0\)
    19. \(x+6 \sqrt{x}-16=0\)
    20. \(x+4 \sqrt{x}-21=0\)
    21. \(6 x+\sqrt{x}-2=0\)
    22. \(6 x+\sqrt{x}-1=0\)
    23. \(10 x-17 \sqrt{x}+3=0\)
    24. \(12 x+5 \sqrt{x}-3=0\)
    25. \(x^{\frac{2}{3}}+9 x^{\frac{1}{3}}+8=0\)
    26. \(x^{\frac{2}{3}}-3 x^{\frac{1}{3}}=28\)
    27. \(x^{\frac{2}{3}}+4 x^{\frac{1}{3}}=12\)
    28. \(x^{\frac{2}{3}}-11 x^{\frac{1}{3}}+30=0\)
    29. \(6 x^{\frac{2}{3}}-x^{\frac{1}{3}}=12\)
    30. \(3 x^{\frac{2}{3}}-10 x^{\frac{1}{3}}=8\)
    31. \(8 x^{\frac{2}{3}}-43 x^{\frac{1}{3}}+15=0\)
    32. \(20 x^{\frac{2}{3}}-23 x^{\frac{1}{3}}+6=0\)
    33. \(x-8 x^{\frac{1}{2}}+7=0\)
    34. \(2 x-7 x^{\frac{1}{2}}=15\)
    35. \(6 x^{-2}+13 x^{-1}+5=0\)
    36. \(15 x^{-2}-26 x^{-1}+8=0\)
    37. \(8 x^{-2}-2 x^{-1}-3=0\)
    38. \(15 x^{-2}-4 x^{-1}-4=0\)
    Answer

    1. \(x=\pm \sqrt{3}, x=\pm 2\)

    3. \(x=\pm \sqrt{15}, x=\pm \sqrt{2} i\)

    5. \(x=\pm 1, x=\frac{ \pm \sqrt{6}}{2}\)

    7. \(x=\pm \sqrt{3}, x=\pm \frac{\sqrt{2}}{2}\)

    9. \(x=-1, x=12\)

    11. \(x=-\frac{5}{3}, x=0\)

    13. \(x=0, x=\pm \sqrt{3}\)

    15. \(x=\pm \frac{11}{2}, x=\pm \frac{\sqrt{22}}{2}\)

    17. \(x=25\)

    19. \(x=4\)

    21. \(x=\frac{1}{4}\)

    23. \(x=\frac{1}{25}, x=\frac{9}{4}\)

    25. \(x=-1, x=-512\)

    27. \(x=8, x=-216\)

    29. \(x=\frac{27}{8}, x=-\frac{64}{27}\)

    31. \(x=27, x=64,000\)

    33. \(x=1, x=49\)

    35. \(x=-2, x=-\frac{3}{5}\)

    37. \(x=-2, x=\frac{4}{3}\)

    Exercise \(\PageIndex{12}\) writing exercises
    1. Explain how to recognize an equation in quadratic form.
    2. Explain the procedure for solving an equation in quadratic form.
    Answer

    1. 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 equations in quadratic form.” “Confidently,” “with some help,” or “No, I don’t get it.”
    Figure 9.4.43

    b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

    9.5E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.