5.4E: Exercises
- Page ID
- 58472
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Practice Makes Perfect
In the following exercises, solve.
- \(x^{4}-7 x^{2}+12=0\)
- \(x^{4}-9 x^{2}+18=0\)
- \(x^{4}-13 x^{2}-30=0\)
- \(x^{4}+5 x^{2}-36=0\)
- \(2 x^{4}-5 x^{2}+3=0\)
- \(4 x^{4}-5 x^{2}+1=0\)
- \(2 x^{4}-7 x^{2}+3=0\)
- \(3 x^{4}-14 x^{2}+8=0\)
- \((x-3)^{2}-5(x-3)-36=0\)
- \((x+2)^{2}-3(x+2)-54=0\)
- \((3 y+2)^{2}+(3 y+2)-6=0\)
- \((5 y-1)^{2}+3(5 y-1)-28=0\)
- \(\left(x^{2}+1\right)^{2}-5\left(x^{2}+1\right)+4=0\)
- \(\left(x^{2}-4\right)^{2}-4\left(x^{2}-4\right)+3=0\)
- \(2\left(x^{2}-5\right)^{2}-5\left(x^{2}-5\right)+2=0\)
- \(2\left(x^{2}-5\right)^{2}-7\left(x^{2}-5\right)+6=0\)
- \(x-\sqrt{x}-20=0\)
- \(x-8 \sqrt{x}+15=0\)
- \(x+6 \sqrt{x}-16=0\)
- \(x+4 \sqrt{x}-21=0\)
- \(6 x+\sqrt{x}-2=0\)
- \(6 x+\sqrt{x}-1=0\)
- \(10 x-17 \sqrt{x}+3=0\)
- \(12 x+5 \sqrt{x}-3=0\)
- \(x^{\frac{2}{3}}+9 x^{\frac{1}{3}}+8=0\)
- \(x^{\frac{2}{3}}-3 x^{\frac{1}{3}}=28\)
- \(x^{\frac{2}{3}}+4 x^{\frac{1}{3}}=12\)
- \(x^{\frac{2}{3}}-11 x^{\frac{1}{3}}+30=0\)
- \(6 x^{\frac{2}{3}}-x^{\frac{1}{3}}=12\)
- \(3 x^{\frac{2}{3}}-10 x^{\frac{1}{3}}=8\)
- \(8 x^{\frac{2}{3}}-43 x^{\frac{1}{3}}+15=0\)
- \(20 x^{\frac{2}{3}}-23 x^{\frac{1}{3}}+6=0\)
- \(x-8 x^{\frac{1}{2}}+7=0\)
- \(2 x-7 x^{\frac{1}{2}}=15\)
- \(6 x^{-2}+13 x^{-1}+5=0\)
- \(15 x^{-2}-26 x^{-1}+8=0\)
- \(8 x^{-2}-2 x^{-1}-3=0\)
- \(15 x^{-2}-4 x^{-1}-4=0\)
- 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. \(x=-2, x=\frac{4}{3}\)
- Explain how to recognize an equation in quadratic form.
- 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.
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?