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In the previous set of exercises, you worked with the quadratic equation
$R=\text{\u2212}{x}^{2}+100x$ that modeled the revenue received from selling backpacks at a price of
$x$ dollars. You found the selling price that would give the maximum revenue and calculated the maximum revenue. Now you will look at more characteristics of this model.
ⓐ Graph the equation
$R=\text{\u2212}{x}^{2}+100x$ .
ⓑ Find the values of the
x -intercepts.
For the revenue model in [link] and [link] , explain what the x -intercepts mean to the computer store owner.
Answers will vary.
For the revenue model in [link] and [link] , explain what the x -intercepts mean to the backpack retailer.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
In the following exercises, solve using the Square Root Property.
${y}^{2}=144$
${m}^{2}-40=0$
$m=\pm \phantom{\rule{0.2em}{0ex}}2\sqrt{10}$
${n}^{2}-80=0$
$2{b}^{2}=72$
${t}^{2}+18=0$
$\frac{4}{3}{v}^{2}+4=28$
$v=\pm \phantom{\rule{0.2em}{0ex}}3\sqrt{2}$
$\frac{2}{3}{w}^{2}-20=30$
$5{c}^{2}+3=19$
$c=\pm \phantom{\rule{0.2em}{0ex}}\frac{4\sqrt{5}}{5}$
$3{d}^{2}-6=43$
In the following exercises, solve using the Square Root Property.
${\left(q+4\right)}^{2}=9$
${\left(z-5\right)}^{2}=50$
${\left(x-\frac{1}{4}\right)}^{2}=\frac{3}{16}$
$x=\frac{1}{4}\pm \frac{\sqrt{3}}{4}$
${\left(y-\frac{2}{3}\right)}^{2}=\frac{2}{9}$
${\left(n-4\right)}^{2}-50=150$
${\left(4c-1\right)}^{2}=\mathrm{-18}$
${n}^{2}+10n+25=12$
$4{b}^{2}-28b+49=25$
In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
${y}^{2}+6y$
${n}^{2}-10n$
${b}^{2}+13b$
${q}^{2}-\frac{1}{3}q$
In the following exercises, solve by completing the square.
${d}^{2}+14d=\mathrm{-13}$
${y}^{2}-16y=36$
${t}^{2}-12t=\mathrm{-40}$
${w}^{2}-20w=100$
${n}^{2}-6n+11=34$
${b}^{2}=11b-5$
$\left(u+8\right)\left(u+4\right)=14$
$u=\mathrm{-6}\pm 2\sqrt{2}$
$\left(z-10\right)\left(z+2\right)=28$
$5{q}^{2}+70q+20=0$
$2{x}^{2}+2x=4$
$4{d}^{2}-2d=8$
In the following exercises, solve by using the Quadratic Formula.
$7{y}^{2}+4y-3=0$
${t}^{2}+13t+22=0$
$2{w}^{2}+9w+2=0$
$5{n}^{2}+2n-1=0$
$4{b}^{2}-b+8=0$
$5z(z-2)=3$
$\frac{1}{8}{p}^{2}-\frac{1}{5}p=-\frac{1}{20}$
$p=\frac{4\pm \sqrt{6}}{5}$
$\frac{2}{5}{q}^{2}+\frac{3}{10}q=\frac{1}{10}$
$9{d}^{2}-12d=\mathrm{-4}$
In the following exercises, determine the number of solutions to each quadratic equation.
ⓐ 1 ⓑ 2 ⓒ 2 ⓓ none
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation.
ⓐ factor ⓑ Quadratic Formula ⓒ square root
In the following exercises, solve by using methods of factoring, the square root principle, or the quadratic formula.
Find two consecutive odd numbers whose product is 323.
Two consecutive odd numbers whose product is 323 are 17 and 19, and $\mathrm{-17}$ and $\mathrm{-19}.$
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