# 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 $-19.$

Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost$5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make? Cherry Reply Nga and Lauren bought a chest at a flea market for$50. They re-finished it and then added a 350 % mark - up
the sum of two Numbers is 19 and their difference is 15
2, 17
Jose
interesting
saw
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
integer greater than 2 and less than 12
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
He charges $125 per job. His monthly expenses are$1,600. How many jobs must he work in order to make a profit of at least $2,400? Alicia Reply at least 20 Ayla what are the steps? Alicia 6.4 jobs Grahame 32 Grahame 1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32 Orlando He must work 32 jobs to make a profit POP what is algebra Azhar Reply repeated addition and subtraction of the order of operations. i love algebra I'm obsessed. Shemiah hi Krekar Eric here. I'm a parent. 53 years old. I have never taken algebra. I want to learn. Eric One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag? Leanna Reply they are 92 candies in the bag POP rectangular field solutions Navin Reply What is this? Donna t muqtaar the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is anas Reply ? Choli a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190 Usman Stella bought a dinette set on sale for$725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount? Manhwa Reply 44.19% Scott 40.22% Terence 44.2% Orlando I don't know Donna if you want the discounted price subtract$725 from $1299. then divide the answer by$1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif
Tausif
how
muqtaar
Can you explain further
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was$1,585. What is the amount of each loan?