10.10: Exercise Supplement
- Page ID
- 60075
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise Supplement
Solving Quadratic Equations - Solving Quadratic Equations by Factoring
For the following problems, solve the equations.
\((x−2)(x−5)=0\)
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\(x = 2,5\)
\((b+1)(b−6)=0\)
\((a+10)(a−5)=0\)
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\(a=−10,5\)
\((y−3)(y−4)=0\)
\((m−8)(m+1)=0\)
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\(m=8,−1\)
\((4y+1)(2y+3)=0\)
\((x+2)(3x−1)=0\)
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\(x = -2, \dfrac{1}{3}\)
\((5a−2)(3a−10)=0\)
\(x(2x+3)=0\)
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\(x = 0, -\dfrac{3}{2}\)
\((a-5)^2 = 0\)
\((y + 3)^2 = 0\)
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\(y=−3\)
\(c^2= 36\)
\(16y^2 - 49 = 0\)
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\(y = \pm \dfrac{7}{4}\)
\(6r^2 - 36 = 0\)
\(a^2 + 6a 8 = 0\)
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\(a=−4,−2\)
\(r^2 + 7r + 10 = 0\)
\(s^2 - 9s + 8 = 0\)
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\(s=1,8\)
\(y^2 = -10y - 9\)
\(11y - 2 = -6y^2\)
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\(y = \dfrac{1}{6}, -2\)
\(16x^2 - 3 = -2x\)
\(m^2 = 4m - 4\)
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\(m=2\)
\(3(y^2 - 8) = -7y\)
\(a(4b + 7) = 0\)
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\(a = 0; b = -\dfrac{7}{4}\)
\(x^2 - 64 = 0\)
\(m^2 - 81 = 0\)
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\(m= \pm 9\)
\(9x^2 - 25 = 0\)
\(5a^2 - 125 = 0\)
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\(a = \pm 5\)
\(8r^3 - 6r = 0\)
\(m^2 - 6m + 5 = 0\)
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\(m=5,1\)
\(x^2 + 2x - 24 = 0\)
\(x^2 + 3x = 28\)
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\(x=−7,4\)
\(20a^2 - 3 = 7a\)
\(2y^2 - 6y = 8\)
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\(y=4,−1\)
\(a^2 + 2a = -1\)
\(2r^2 = 5 - 3r\)
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\(r = -\dfrac{5}{2}, 1\)
Solving Quadratic Equations Using the Method of Extraction of Roots
For the following problems, solve the equations using the extraction of roots.
\(y^2 = 81\)
\(a^2 = 121\)
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\(a = \pm 11\)
\(x^2 = 35\)
\(m^2 = 2\)
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\(m = \pm \sqrt{2}\)
\(r^2 = 1\)
\(s^2 - 10 = 0\)
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\(s = \pm \sqrt{10}\)
\(4x^2 - 64 = 0\)
\(-3y^2 = -75\)
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\(y = \pm 5\)
Solve \(y^2 = 4a^2\) for \(y\)
Solve \(m^2 = 16n^2p^4\) for \(m\)
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\(m = \pm 4np^2\)
Solve \(x^2 = 25y^4z^{10}w^8\) for \(x\).
Solve \(x^2 - y^2 = 0\) for \(y\)
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\(y = \pm x\)
Solve \(a^4b^8 - x^6y^{12}z^2 = 0\) for \(a^2\)
\((x-2)^2 = 9\)
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\(x=5,−1\)
\((y + 3)^2 = 25\)
\((a + 10)^2 = 1\)
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\(a=−11,−9\)
\((m + 12)^2 = 6\)
\((r - 8)^2 = 10\)
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\(r = 8 \pm \sqrt{10}\)
\((x - 1)^2 = 5\)
\((a - 2)^2 = -2\)
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No real number solution.
Solve \((x - 2b)^2 = b^2\) for \(x\)
Solve \((y + 6)^2 = a\) for \(y\)
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\(y = -6 \pm \sqrt{a}\)
Solve \((2a - 5)^2 = c\) for \(a\)
Solve \((3m - 11)^2 = 2a^2\) for \(m\)
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\(m = \dfrac{11 \pm a\sqrt{2}}{3}\)
Solving Quadratic Equations Using the Method of Completing the Square - Solving Quadratic Equations Using the Quadratic Formula
For the following problems, solve the equations by completing the square or by using the quadratic formula.
\(y^2 - 8y - 12 = 0\)
\(s^2 + 2s - 24 = 0\)
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\(s=4,−6\)
\(a^2 + 3a - 9 = 0\)
\(b^2 + b - 8 = 0\)
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\(b = \dfrac{-1 \pm \sqrt{33}}{2}\)
\(3x^2 - 2x - 1 = 0\)
\(5a^2 + 2a - 6 = 0\)
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\(a = \dfrac{-1 \pm \sqrt{31}}{5}\)
\(a^2 = a + 4\)
\(y^2 = 2y + 1\)
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\(y = 1 \pm \sqrt{2}\)
\(m^2 - 6 = 0\)
\(r^2 + 2r = 9\)
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\(r = -1 \pm \sqrt{10}\)
\(3p^2 + 2p = 7\)
\(10x^3 + 2x^2 - 22x = 0\)
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\(x = 0, \dfrac{-1 \pm \sqrt{221}}{10}\)
\(6r^3 + 6r^2 - 3r = 0\)
\(15x^2 + 2x^3 = 12x^4\)
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\(x = 0, \dfrac{1 \pm \sqrt{181}}{12}\)
\(6x^3 - 6x = -6x^2\)
\((x+3)(x-4) = 3\)
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\(x = \dfrac{1 \pm \sqrt{61}}{2}\)
\((y−1)(y−2)=6\)
\((a+3)(a+4)=−10\)
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No real number solution.
\((2m+1)(3m−1)=−2\)
\((5r+6)(r−1)=2\)
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\(r = \dfrac{-1 \pm \sqrt{161}}{10}\)
\(4x^2 + 2x - 3 = 3x^2 + x + 1\)
\(5a^2 + 5a + 4 = 3a^2 + 2a + 5\)
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\(a = \dfrac{-3 \pm \sqrt{17}}{4}\)
\((m + 3)^2 = 11\)
\((r - 8)^2 = 70\)
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\(r = 8 \pm \sqrt{70}\)
\((2x + 7)^2 = 51\)
Applications
For the following problems, find the solution.
The revenue \(R\), in dollars, collected by a certain manufacturer of inner tubes is related to the number \(x\) of inner tubes sold by \(R = 1400 - 16x + 3x^2\). How many inner tubes must be sold to produce a profit of $1361?
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No solution.
A study of the air quality in a particular city by an environmental group suggests that \(t\) years from now the level of carbon monoxide, in parts per million, in the air will be \(A = 0.8t^2 + 0.5t + 3.3\).
a) What is the level, in parts per million, of carbon monoxide in the air now?
b) How many years from now will the carbon monoxide level be at 6 parts per million?
A contractor is to pour a concrete walkway around a community garden that is 15 feet wide and 50 feet long. The area of the walkway and garden is to be 924 square feet and of uniform width. How wide should the contractor make it?
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\(x \approx 1.29\) feet
A ball is thrown vertically into the air has the equation of motion \(h = 144 + 48t - 16t^2\)
a) How high is the ball at \(t = 0\)?
b) How high is the ball at \(t = 1\)?
c) When does the ball hit the ground?
The length of a rectangle is 5 feet longer than three times its width. Find the dimensions if the area is to be 138 square feet.
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\(w=6\)
The area of a triangle is 28 square centimeters. The base is 3 cm longer than the height. Find both the length of the base and the height.
The product of two consecutive integers is 210. Find them.
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\(x=−15,−14\), or \(14,15\)
The product of two consecutive negative integers is 272. Find them.
A box with no top and a square base is to be made by cutting out 3-inch squares from each corner and folding up the sides of a piece of cardboard. The volume of the box is to be 25 cubic inches. What size should the piece of cardboard be?
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\(x = \dfrac{18 + 5 \sqrt{3}}{3}\)