# 10.5 Graphing quadratic equations  (Page 8/15)

 Page 8 / 15

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.

Answers will vary.

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

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

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

$v=7±3\sqrt{2}$

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

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

$m=-9,-1$

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

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

$a=\frac{3}{2}±\frac{\sqrt{41}}{2}$

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

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

$u=-6±2\sqrt{2}$

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

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

$p=0,6$

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

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

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

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

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

$c=-\frac{1}{3}±\frac{2\sqrt{7}}{3}$

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

## 10.3 Solve Quadratic Equations Using the Quadratic Formula

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. $3{\left(c+2\right)}^{2}=15$

factor Quadratic Formula square root

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 $-19.$

#### Questions & Answers

where's the answers?
Ed Reply
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
What does 3 times your weight right now
Cherokee Reply
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Kendra Reply
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
Malik Reply
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
Adda Reply
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Adda
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
Adda
(×+1)(×-4) = x^2-3×-4
Gee
Quadratic equations involving factorization
Winner Reply
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs$4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon? Mohamed Reply (a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r) muhammad Reply 4x-7y=8 2x-7y=1 what is the answer? Ramil Reply x=7/2 & y=6/7 Pbp x=7/2 & y=6/7 use Elimination Debra true bismark factoriz e usman 4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2 Peggie Ok cool answer peggie Frank thanks Ramil copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x Sandra Reply What is c+4=8 Penny Reply 2 Letha 4 Lolita 4 Rich 4 thinking C+4=8 -4 -4 C =4 thinking I need to study Letha 4+4=8 William During two years in college, a student earned$9,500. The second year, she earned \$500 more than twice the amount she earned the first year.
Nicole Reply
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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