1.6e: Exercises - Quadratic in Form

Last updated

Jan 14, 2022
Save as PDF
- 1.6: Equations Quadratic in Form
- 1.7: Absolute Value Equations and Inequalities

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

A: Quadratic in Form Polynomial Equations

Exercise $\PageIndex{A}$

$\bigstar$ Solve quadratic in form 4th degree equations.

$x^{4}+x^{2}-72=0 \\[4pt]$
$x^{4}-17 x^{2}-18=0 \\[4pt]$
$x^{4}-13 x^{2}+36=0 \\[4pt]$
$4 x^{4}-17 x^{2}+4=0 \\[4pt]$
$y^{4}-14 y^{2}+46=0 \\[4pt]$
$x^{4}-6 x^{2}+6=0 \\[4pt]$
$x^{4}-6 x^{2}+7=0 \\[4pt]$
$x^{4}-12 x^{2}+31=0$

$x^{4}-8 x^{2}+14=0 \\[4pt]$
$9 x^{4}-30 x^{2}+1=0 \\[4pt]$
$4 x^{4}-16 x^{2}+13=0 \\[4pt]$
$x^{4}-9 x^{2}+18=0 \\[4pt]$
$x^{4}-7 x^{2}+12=0 \\[4pt]$
$x^{4}+5 x^{2}-36=0 \\[4pt]$
$x^{4}-13 x^{2}-30=0 \\[4pt]$
$4 x^{4}-5 x^{2}+1=0$

$2 x^{4}-5 x^{2}+3=0 \\[4pt]$
$3 x^{4}-14 x^{2}+8=0 \\[4pt]$
$2 x^{4}-7 x^{2}+3=0 \\[4pt]$
$x^{4}-2x^{2}-35=0 \\[4pt]$
$x^{4}-11x^{2}+18=0 \\[4pt]$
$6x^{4}-23x^{2}-225=0 \\[4pt]$
$12x^{4}-31x^{2}+9=0 \\[4pt]$
$x^{4}+19x^{2}+48=0$

Answers to odd exercises:

1. $\pm 2 \sqrt{2}, \pm 3 i$
3. $\pm 2, \pm 3$
5. $\pm \sqrt{7-\sqrt{3}}, \pm \sqrt{7+\sqrt{3}}$
7. $\pm 1.26, \pm 2.10$ or $\pm \sqrt {3 \pm \sqrt{2}}$

$\pm 1.61, \pm 2.33$ or

$\pm \sqrt {4 \pm \sqrt{2}}$
11.

$\pm 1.06, \pm 1.69$ or

$\pm \sqrt {2 \pm \frac{1}{2}\sqrt{3}}$
13.

$x=\pm \sqrt{3}, x=\pm 2$
15.

$x=\pm \sqrt{15}, x=\pm \sqrt{2} i$

17.

$x=\pm 1, x=\frac{ \pm \sqrt{6}}{2}$
19.

$x=\pm \sqrt{3}, x=\pm \frac{\sqrt{2}}{2}$
21.

$\{ \pm 3, \pm \sqrt{2} \}$
23.

$\{ \pm \frac {3}{2}. \pm \frac{\sqrt3}{3} \}$

$\bigstar$ Solve quadratic in form polynomial equations.

$x^{6}+7 x^{3}-8=0 \\[4pt]$
$x^{6}+28 x^{3}+27=0 \\[4pt]$
$81 y^{4}-1=0 \\[4pt]$
$x^{6}+16 x^{3}+64=0$

$(x-3)^{2}-5(x-3)-36=0 \\[4pt]$
$(x+2)^{2}-3(x+2)-54=0 \\[4pt]$
$(3 y+2)^{2}+(3 y+2)-6=0 \\[4pt]$
$(5 y-1)^{2}+3(5 y-1)-28=0$

$\left(x^{2}+1\right)^{2} - 5\left(x^{2}+1\right)+4=0 \\[4pt]$
$\left(x^{2}-4\right)^{2}-4\left(x^{2}-4\right)+3=0 \\[4pt]$
$2\left(x^{2}-5\right)^{2}-5\left(x^{2}-5\right)+2=0 \\[4pt]$
$2\left(x^{2}-5\right)^{2}-7\left(x^{2}-5\right)+6=0$

43.

$(x^2-2x+3)^2 -5(x^2-2x+3) + 6 = 0$

44.

$(x^2+x-2)^2 -5(x^2+x-2) - 50 = 0$

Answers to odd exercises

31.

$-2,1,1 \pm i \sqrt{3},-\frac{1}{2} \pm \frac{\sqrt{3}}{2} i$
33.

$\pm \frac{1}{3}, \pm \frac{i}{3}$
35.

$x=12$ ,

$x=-1$

37.

$x=-\frac{5}{3}, x=0$
39.

$x=0, x=\pm \sqrt{3}$

41.

$x=\pm \sqrt{7}, x=\pm \frac{\sqrt{22}}{2}$
43.

$\{ 0, 1, 1, 2 \}$

B. Quadratic in Form Radical and Exponential Equations

Exercise $\PageIndex{B}$

$\bigstar$ Solve quadratic in form radical equations.

$x+2 \sqrt{x}-3=0 \\[4pt]$
$x-\sqrt{x}-2=0 \\[4pt]$
$x-5 \sqrt{x}+6=0 \\[4pt]$
$x-6 \sqrt{x}+5=0 \\[4pt]$
$x-\sqrt{x}-20=0 \\[4pt]$
$x-8 \sqrt{x}+15=0 \\[4pt]$
$x+6 \sqrt{x}-16=0$

$x+4 \sqrt{x}-21=0 \\[4pt]$
$6 x+\sqrt{x}-2=0 \\[4pt]$
$6 x+\sqrt{x}-1=0 \\[4pt]$
$10 x-17 \sqrt{x}+3=0 \\[4pt]$
$12 x+5 \sqrt{x}-3=0 \\[4pt]$
$x+2\sqrt{x}-15=0 \\[4pt]$
$x-10\sqrt{x}+21=0$

$x-8 x^{\frac{1}{2}}+7=0 \\[4pt]$
$2 x-7 x^{\frac{1}{2}}=15 \\[4pt]$
$x^{1 / 2}-3 x^{1 / 4}+2=0 \\[4pt]$
$x+5 \sqrt{x}-50=0 \\[4pt]$
$6x^{1 / 4}- x^{1 / 2}-9=0 \\[4pt]$
$x^{1 / 3}+ x^{1 / 6}-6=0 \\[4pt]$
$2 x^{1 / 3}-3 x^{1 / 6}+1=0 \\[4pt]$
$x^{1 / 3}-x^{1 / 6}-2=0$

Answers to odd exercises:

51.

$1$
53.

$4,9$
55.

$x=25$

57.

$x=4$
59.

$x=\frac{1}{4}$

61.

$x=\frac{1}{25}, x=\frac{9}{4}$
63.

$\{ 9 \}$

65.

$x=1, x=49$
67.

$1, 16$

69.

$\{ 81 \}$
71.

$\frac{1}{64}, 1$

$\bigstar$ Solve quadratic in form radical equations.

$x^{2 / 3}+5 x^{1 / 3}+6=0 \\[4pt]$
$x^{2 / 3}-2 x^{1 / 3}-35=0 \\[4pt]$
$4 x^{2 / 3}-4 x^{1 / 3}+1=0 \\[4pt]$
$3 x^{2 / 3}-2 x^{1 / 3}-1=0 \\[4pt]$
$8 x^{2 / 3}+7 x^{1 / 3}-1=0 \\[4pt]$
$x^{\frac{2}{3}}-3 x^{\frac{1}{3}}=28 \\[4pt]$
$x^{\frac{2}{3}}+4 x^{\frac{1}{3}}=12 \\[4pt]$
$x^{\frac{2}{3}}-11 x^{\frac{1}{3}}+30=0$

$6 x^{\frac{2}{3}}-x^{\frac{1}{3}}=12 \\[4pt]$
$3 x^{\frac{2}{3}}-10 x^{\frac{1}{3}}=8 \\[4pt]$
$8 x^{\frac{2}{3}}-43 x^{\frac{1}{3}}+15=0 \\[4pt]$
$20 x^{\frac{2}{3}}-23 x^{\frac{1}{3}}+6=0 \\[4pt]$
$x^{\frac{2}{3}}+9 x^{\frac{1}{3}}+8=0 \\[4pt]$
$x^{2 / 3}+12 x^{1 / 3}+35=0 \\[4pt]$
$x^{2 / 3}-3 x^{1 / 3}-18=0 \\[4pt]$
$x^{2 / 3}- x^{1 / 3}+4=0$

$x^{2 / 3}-2 x^{1 / 3}-6=0 \\[4pt]$
$9x^{2 / 3}-33 x^{1 / 3}+28=0 \\[4pt]$
$6x^{2 / 3}+11 x^{1 / 3}-10=0 \\[4pt]$
$y^{2 / 3}-9=0 \\[4pt]$
$30 y^{2 / 3}-15 y^{1 / 3}=0 \\[4pt]$
$x^{4 / 3}-2 x^{2 / 3}+1=0 \\[4pt]$
$x^{4 / 3}-13 x^{2 / 3}+36=0 \\[4pt]$
$x^{4 / 3}-5 x^{2 / 3}-6=0$

Answers to odd exercises:

81.

$-27, -8$ ;
83.

$\frac{1}{8}$
85.

$-1, \frac{1}{512}$

87.

$x=8, x=-216$
89.

$x=\frac{27}{8}, x=-\frac{64}{27}$
91.

$x=\frac{27}{512}, x=125$

93.

$x=-1, x=-512$
95.

$\{ 216, -27 \}$
97.

$\{ 22 \pm 10\sqrt {7} \}$

99.

$\{ \frac {8}{27},-\frac{125}{8} \}$
101.

$0, \frac {1}{8}$
103.

$\{ \pm 8, \pm 27 \}$

C: Quadratic in Form Negative Exponential Equations

Exercise $\PageIndex{C}$

$\bigstar$ Solve quadratic in form negative exponential equations.

$5 x^{-2}+9 x^{-1}-2=0 \\[4pt]$
$3 x^{-2}+8 x^{-1}-3=0 \\[4pt]$
$8 x^{-2}+14 x^{-1}-15=0 \\[4pt]$
$9 x^{-2}-24 x^{-1}+16=0 \\[4pt]$
$2 y^{-2}-y^{-1}-1=0$

$10 x^{-2}-19 x^{-1}-2=0 \\[4pt]$
$6 x^{-2}+13 x^{-1}+5=0 \\[4pt]$
$15 x^{-2}-26 x^{-1}+8=0 \\[4pt]$
$8 x^{-2}-2 x^{-1}-3=0 \\[4pt]$
$15 x^{-2}-4 x^{-1}-4=0$

$4 y^{-2}-9=0 \\[4pt]$
$16 y^{-2}+4 y^{-1}=0 \\[4pt]$
$2 x^{-2 / 3}-3 x^{-1 / 3}-2=0 \\[4pt]$
$4 x^{-1}-17 x^{-1 / 2}+4=0 \\[4pt]$
$3 x^{-1}-8 x^{-1 / 2}+4=0$

Answers to odd exercises:

111.

$-\frac{1}{2}, 5$
113.

$-\frac{2}{5}, \frac{4}{3}$

115.

$-2,1$
117.

$x=-10, x=\frac{1}{2}$

119.

$x=\frac{3}{4}, x=\frac{5}{2}$ .
121.

$x=\frac{3}{2}, x=-\frac{5}{2}$

125.

$y = -4$
127.

$\frac{1}{16}, 16$

D: Quadratic in Form Rational Equations

Exercise $\PageIndex{D}$ : Quadratic in Form Rational Equations

$\bigstar$ Solve quadratic in form rational equations.

17. $\left(\dfrac{x-3}{x}\right)^{2}-2\left(\dfrac{x-3}{x}\right)-24=0 \\[4pt]$
$\left(\dfrac{2 x+1}{x}\right)^{2}+9\left(\dfrac{2 x+1}{x}\right)-36=0 \\[4pt]$
$2\left(\dfrac{x}{x+1}\right)^{2}-5\left(\dfrac{x}{x+1}\right)-3=0 \\[4pt]$
$3\left(\dfrac{x}{3 x-1}\right)^{2}+13\left(\dfrac{x}{3 x-1}\right)-10=0 \\[4pt]$
$12\left(\dfrac{x}{2 x-3}\right)^{2}-11\left(\dfrac{x}{2 x-3}\right)+2=0 \\[4pt]$
$5\left(\dfrac{1}{x+2}\right)^{2}-3\left(\dfrac{1}{x+2}\right)-2=0 \\[4pt]$

$\left( \dfrac{2x}{x-5} \right)^{2} +\dfrac{6x}{x-5}-18=0 \\[4pt]$
$\dfrac{x^2+2}{x-6}+\dfrac{x-6}{x^2+2} +\dfrac{13}{6}=0$
$\dfrac{x^2+2}{x^2-2}+\dfrac{x^2-2}{x^2+2} -\dfrac{170}{77}=0 \\[4pt]$
$\sqrt{\dfrac{x+10}{x-6}} + \sqrt{\dfrac{x-6}{x+10} }-\dfrac{34}{15}=0 \\[4pt]$
$\sqrt{\dfrac{x+7}{x-5}} - \sqrt{\dfrac{x-5}{x+7} }-\dfrac{3}{2} =0$

Answers to odd exercises:

131.

$\pm \frac{3}{5}$

133.

$-\frac{3}{2},-\frac{1}{3}$

135.

$-\frac{3}{2}, 6$

137.

$\{ 15, \frac{15}{4} \}$

139.

$\{ \pm 3, \; \pm 3i \}$

141.

$\{ 9 \}$

Search

Text Color

Text Size

Margin Size

Font Type

A: Quadratic in Form Polynomial Equations

B. Quadratic in Form Radical and Exponential Equations

C: Quadratic in Form Negative Exponential Equations

D: Quadratic in Form Rational Equations

Support Center

How can we help?