# 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 $-19.$

4x+7y=29,x+3y=11 substitute method of linear equation
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
divide 3x⁴-4x³-3x-1 by x-3
how to multiply the monomial
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Brandon has a cup of quarters and dimes with a total of 5.55\$. The number of quarters is five less than three times the number of dimes
app is wrong how can 350 be divisible by 3.
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
I'm getting "math processing error" on math problems. Anyone know why?
Can you all help me I don't get any of this
4^×=9
Did anyone else have trouble getting in quiz link for linear inequalities?
operation of trinomial