11.7: Quadratic Equations and Applications- Answers to the Homework Exercises
- Page ID
- 45116
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Square Root Property
- \(7,-1\)
- \(9, 5\)
- \(4, -14\)
- \(-9\pm\sqrt{37}\)
- \(2\pm\sqrt{70}\)
- \(-4\pm 3\sqrt{7}\)
- \(\dfrac{2}{9},-\dfrac{4}{9}\)
- \(\dfrac{11}{3},\dfrac{1}{3}\)
- \(-6\pm 5\sqrt{3}\)
- \(-\dfrac{1}{2}\)
- \(4,-10\)
- \(\pm 2\sqrt{2}\)
- \(-9\pm 2\sqrt{7}\)
- \(-8\pm 3\sqrt{5}\)
- \(0,-3\)
- \(-\dfrac{2}{7},-\dfrac{10}{7}\)
- \(\dfrac{7}{4},\dfrac{3}{4}\)
Completing the Square
- \(225\); \((x-15)^2\)
- \(\dfrac{225}{4}\); \(\left(x-\dfrac{15}{2}\right)^2\)
- \(144\); \((a-12)^2\)
- \(\dfrac{1}{324}\); \(\left(r-\dfrac{1}{18}\right)^2\)
- \(11,5\)
- \(-1\pm i\sqrt{\dfrac{19}{2}}\)
- \(-5+\sqrt{86}, -5-\sqrt{86}\)
- \(-1\pm i\sqrt{21}\)
- \(-5\pm 2i\)
- \(1\pm i\sqrt{2}\)
- \(4\pm i\sqrt{39}\)
- \(-1\pm i\sqrt{\dfrac{43}{6}}\)
- \(-\dfrac{5}{2}\pm\dfrac{i\sqrt{87}}{2}\)
- \(\dfrac{7}{2}\pm\dfrac{i\sqrt{139}}{2}\)
- \(\dfrac{9+\sqrt{21}}{2},\dfrac{9-\sqrt{21}}{2}\)
- \(1,-\dfrac{5}{2}\)
- \(4+2\sqrt{7},4-2\sqrt{7}\)
- \(1\pm 2i\sqrt{\dfrac{11}{3}}\)
- \(8+2\sqrt{29}, 8-2\sqrt{29}\)
- \(1,-3\)
- \(7+\sqrt{85},7-\sqrt{85}\)
- \(x=1\pm 1\sqrt{\dfrac{21}{5}}\)
- \(-1,-7\)
- \(-2-i\sqrt{\dfrac{37}{3}}\)
- \(\dfrac{-7+\sqrt{181}}{2},\dfrac{-7-\sqrt{181}}{2}\)
- \(\dfrac{5}{2}\pm 3i\sqrt{7}\)
- \(\dfrac{1}{2}\pm i\sqrt{163}\)
- \(8,7\)
Quadratic Formula
- \(\dfrac{\sqrt{6}}{2},-\dfrac{\sqrt{6}}{2}\)
- \(\dfrac{\sqrt{6}}{2},-\dfrac{\sqrt{6}}{2}\)
- \(3,-3\)
- \(\dfrac{-3\pm i\sqrt{55}}{4}\)
- \(\dfrac{-3+\sqrt{401}}{14},\dfrac{-3-\sqrt{401}}{14}\)
- \(\dfrac{-1\pm i\sqrt{3}}{2}\)
- \(\dfrac{7+3\sqrt{21}}{10},\dfrac{7-3\sqrt{21}}{10}\)
- \(-1,-\dfrac{3}{2}\)
- \(4,-9\)
- \(\dfrac{-3+\sqrt{345}}{14},\dfrac{-3-\sqrt{345}}{14}\)
- \(\pm\dfrac{i\sqrt{6}}{3}\)
- \(\dfrac{-1\pm i\sqrt{29}}{5}\)
- \(\pm\dfrac{i\sqrt{2}}{2}\)
- \(\dfrac{3\pm i\sqrt{159}}{12}\)
- \(\dfrac{-5+\sqrt{137}}{8},\dfrac{-5-\sqrt{137}}{8}\)
- \(3,-\dfrac{1}{3}\)
- \(\dfrac{-5+\sqrt{337}}{12},\dfrac{-5-\sqrt{337}}{12}\)
- \(2\sqrt{2},-2\sqrt{2}\)
- \(\dfrac{3\pm 3i\sqrt{5}}{7}\)
- \(\dfrac{\sqrt{6}}{2},-\dfrac{\sqrt{6}}{2}\)
Graph Quadratic Functions
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Figure 11.7.1
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Figure 11.7.2
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Figure 11.7.3
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Figure 11.7.4
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Figure 11.7.5
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Figure 11.7.6
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Figure 11.7.7
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Figure 11.7.8
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Figure 11.7.9
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Figure 11.7.10
- \(f(x)=3(x-1)^2\)
- \(f(x)=\dfrac{1}{3}x^2\)
- \(f(x)=4(x+4)^2\)
- \(f(x)=3x^2-4\)
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Figure 11.7.11
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Figure 11.7.12
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Figure 11.7.13
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Figure 11.7.14
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Figure 11.7.15
Quadratic Inequalities
- \((-\infty, 3)\cup (6,\infty )\)
- \([-4,6]\)
- \((-1,3)\)
- \((-\infty, -6]\cup [-4,\infty )\)
- \((-\infty ,2)\cup (2,\infty )\)
- \((-\infty, -4]\cup [2,\infty )\)
- \([1,3]\)
- \(\left[-4,\dfrac{5}{3}\right]\)
- \((-\infty ,-4]\cup\left(\dfrac{5}{4},\infty\right)\)
- \(\left(-1,\dfrac{1}{3}\right)\)
- \(\left[0,\dfrac{7}{4}\right]\)
- \((-\infty, -6]\cup [0,\infty )\)
- No Solution
- All real numbers
Applications with Quadratic Functions
- \(\left(\dfrac{5}{4},-\dfrac{57}{8}\right)\); \(-\dfrac{57}{8}\)
- \(\left(\dfrac{1}{2},-\dfrac{11}{2}\right)\); \(-\dfrac{11}{2}\)
- \((-1,1)\); \(1\)
- \(27.65\); \(3,896.99\)
- \(19.39\); \(2,236.84\)
- \(22.76\); \(2,666.19\)
- \(25\); \(8.88\)
- \(45\); \(14.88\)
- \(11\); \(551.25\) or \(550\)
- \(16\); \(480.5\) or \(480\)
- \(10\); \(361\) or \(360\)