10.5 Graphing quadratic equations  (Page 8/15)

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In the previous set of exercises, you worked with the quadratic equation $R=\text{−}{x}^{2}+100x$ 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 $R=\text{−}{x}^{2}+100x$ . Find the values of the x -intercepts.

Writing exercises

For the revenue model in [link] and [link] , explain what the x -intercepts mean to the computer store owner.

For the revenue model in [link] and [link] , explain what the x -intercepts mean to the backpack retailer.

Self check

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?

10.1 Solve Quadratic Equations Using the Square Root Property

In the following exercises, solve using the Square Root Property.

${x}^{2}=100$

$x=±\phantom{\rule{0.2em}{0ex}}10$

${y}^{2}=144$

${m}^{2}-40=0$

$m=±\phantom{\rule{0.2em}{0ex}}2\sqrt{10}$

${n}^{2}-80=0$

$4{a}^{2}=100$

$a=±\phantom{\rule{0.2em}{0ex}}5$

$2{b}^{2}=72$

${r}^{2}+32=0$

no solution

${t}^{2}+18=0$

$\frac{4}{3}{v}^{2}+4=28$

$v=±\phantom{\rule{0.2em}{0ex}}3\sqrt{2}$

$\frac{2}{3}{w}^{2}-20=30$

$5{c}^{2}+3=19$

$c=±\phantom{\rule{0.2em}{0ex}}\frac{4\sqrt{5}}{5}$

$3{d}^{2}-6=43$

In the following exercises, solve using the Square Root Property.

${\left(p-5\right)}^{2}+3=19$

$p=1,9$

${\left(q+4\right)}^{2}=9$

${\left(u+1\right)}^{2}=45$

$u=-1±3\sqrt{5}$

${\left(z-5\right)}^{2}=50$

${\left(x-\frac{1}{4}\right)}^{2}=\frac{3}{16}$

$x=\frac{1}{4}±\frac{\sqrt{3}}{4}$

${\left(y-\frac{2}{3}\right)}^{2}=\frac{2}{9}$

${\left(m-7\right)}^{2}+6=30$

$m=7±2\sqrt{6}$

${\left(n-4\right)}^{2}-50=150$

${\left(5c+3\right)}^{2}=-20$

no solution

${\left(4c-1\right)}^{2}=-18$

${m}^{2}-6m+9=48$

$m=3±4\sqrt{3}$

${n}^{2}+10n+25=12$

$64{a}^{2}+48a+9=81$

$a=-\frac{3}{2},\frac{3}{4}$

$4{b}^{2}-28b+49=25$

10.2 Solve Quadratic Equations Using Completing the Square

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

${x}^{2}+22x$

${\left(x+11\right)}^{2}$

${y}^{2}+6y$

${m}^{2}-8m$

${\left(m-4\right)}^{2}$

${n}^{2}-10n$

${a}^{2}-3a$

${\left(a-\frac{3}{2}\right)}^{2}$

${b}^{2}+13b$

${p}^{2}+\frac{4}{5}p$

${\left(p+\frac{2}{5}\right)}^{2}$

${q}^{2}-\frac{1}{3}q$

In the following exercises, solve by completing the square.

${c}^{2}+20c=21$

$c=1,-21$

${d}^{2}+14d=-13$

${x}^{2}-4x=32$

$x=-4,8$

${y}^{2}-16y=36$

${r}^{2}+6r=-100$

no solution

${t}^{2}-12t=-40$

${v}^{2}-14v=-31$

$v=7±3\sqrt{2}$

${w}^{2}-20w=100$

${m}^{2}+10m-4=-13$

$m=-9,-1$

${n}^{2}-6n+11=34$

${a}^{2}=3a+8$

$a=\frac{3}{2}±\frac{\sqrt{41}}{2}$

${b}^{2}=11b-5$

$\left(u+8\right)\left(u+4\right)=14$

$u=-6±2\sqrt{2}$

$\left(z-10\right)\left(z+2\right)=28$

$3{p}^{2}-18p+15=15$

$p=0,6$

$5{q}^{2}+70q+20=0$

$4{y}^{2}-6y=4$

$y=-\frac{1}{2},2$

$2{x}^{2}+2x=4$

$3{c}^{2}+2c=9$

$c=-\frac{1}{3}±\frac{2\sqrt{7}}{3}$

$4{d}^{2}-2d=8$

In the following exercises, solve by using the Quadratic Formula.

$4{x}^{2}-5x+1=0$

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

$7{y}^{2}+4y-3=0$

${r}^{2}-r-42=0$

$r=-6,7$

${t}^{2}+13t+22=0$

$4{v}^{2}+v-5=0$

$v=-\frac{5}{4},1$

$2{w}^{2}+9w+2=0$

$3{m}^{2}+8m+2=0$

$m=\frac{-4±\sqrt{10}}{3}$

$5{n}^{2}+2n-1=0$

$6{a}^{2}-5a+2=0$

no real solution

$4{b}^{2}-b+8=0$

$u\left(u-10\right)+3=0$

$u=5±\sqrt{22}$

$5z\left(z-2\right)=3$

$\frac{1}{8}{p}^{2}-\frac{1}{5}p=-\frac{1}{20}$

$p=\frac{4±\sqrt{6}}{5}$

$\frac{2}{5}{q}^{2}+\frac{3}{10}q=\frac{1}{10}$

$4{c}^{2}+4c+1=0$

$c=-\frac{1}{2}$

$9{d}^{2}-12d=-4$

In the following exercises, determine the number of solutions to each quadratic equation.

1. $9{x}^{2}-6x+1=0$
2. $3{y}^{2}-8y+1=0$
3. $7{m}^{2}+12m+4=0$
4. $5{n}^{2}-n+1=0$

1 2 2 none

1. $5{x}^{2}-7x-8=0$
2. $7{x}^{2}-10x+5=0$
3. $25{x}^{2}-90x+81=0$
4. $15{x}^{2}-8x+4=0$

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation.

1. $16{r}^{2}-8r+1=0$
2. $5{t}^{2}-8t+3=9$
3. $3{\left(c+2\right)}^{2}=15$

1. $4{d}^{2}+10d-5=21$
2. $25{x}^{2}-60x+36=0$
3. $6{\left(5v-7\right)}^{2}=150$

10.4 Solve Applications Modeled by Quadratic Equations

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 $-17$ and $-19.$