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In the previous set of exercises, you worked with the quadratic equation R = x 2 + 100 x 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 = x 2 + 100 x . Find the values of the x -intercepts.

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Writing exercises

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

Answers will vary.

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

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has six rows and four columns. The first row is a header row and it labels each column. The first column is labeled

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.

m 2 40 = 0

m = ± 2 10

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r 2 + 32 = 0

no solution

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4 3 v 2 + 4 = 28

v = ± 3 2

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5 c 2 + 3 = 19

c = ± 4 5 5

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

( p 5 ) 2 + 3 = 19

p = 1 , 9

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( u + 1 ) 2 = 45

u = −1 ± 3 5

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( x 1 4 ) 2 = 3 16

x = 1 4 ± 3 4

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( m 7 ) 2 + 6 = 30

m = 7 ± 2 6

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( n 4 ) 2 50 = 150

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( 5 c + 3 ) 2 = −20

no solution

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m 2 6 m + 9 = 48

m = 3 ± 4 3

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64 a 2 + 48 a + 9 = 81

a = 3 2 , 3 4

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4 b 2 28 b + 49 = 25

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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.

m 2 8 m

( m 4 ) 2

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a 2 3 a

( a 3 2 ) 2

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p 2 + 4 5 p

( p + 2 5 ) 2

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In the following exercises, solve by completing the square.

c 2 + 20 c = 21

c = 1 , −21

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x 2 4 x = 32

x = −4 , 8

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r 2 + 6 r = −100

no solution

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v 2 14 v = −31

v = 7 ± 3 2

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m 2 + 10 m 4 = −13

m = −9 , −1

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a 2 = 3 a + 8

a = 3 2 ± 41 2

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( u + 8 ) ( u + 4 ) = 14

u = −6 ± 2 2

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( z 10 ) ( z + 2 ) = 28

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3 p 2 18 p + 15 = 15

p = 0 , 6

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4 y 2 6 y = 4

y = 1 2 , 2

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3 c 2 + 2 c = 9

c = 1 3 ± 2 7 3

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10.3 Solve Quadratic Equations Using the Quadratic Formula

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

4 x 2 5 x + 1 = 0

x = 1 4 , 1

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r 2 r 42 = 0

r = −6 , 7

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4 v 2 + v 5 = 0

v = 5 4 , 1

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3 m 2 + 8 m + 2 = 0

m = −4 ± 10 3

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6 a 2 5 a + 2 = 0

no real solution

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u ( u 10 ) + 3 = 0

u = 5 ± 22

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1 8 p 2 1 5 p = 1 20

p = 4 ± 6 5

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4 c 2 + 4 c + 1 = 0

c = 1 2

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

  1. 9 x 2 6 x + 1 = 0
  2. 3 y 2 8 y + 1 = 0
  3. 7 m 2 + 12 m + 4 = 0
  4. 5 n 2 n + 1 = 0

1 2 2 none

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  1. 5 x 2 7 x 8 = 0
  2. 7 x 2 10 x + 5 = 0
  3. 25 x 2 90 x + 81 = 0
  4. 15 x 2 8 x + 4 = 0
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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 8 r + 1 = 0
  2. 5 t 2 8 t + 3 = 9
  3. 3 ( c + 2 ) 2 = 15

factor Quadratic Formula square root

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  1. 4 d 2 + 10 d 5 = 21
  2. 25 x 2 60 x + 36 = 0
  3. 6 ( 5 v 7 ) 2 = 150
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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 .

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Questions & Answers

Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
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
Makaila Reply
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
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?
Rafi Reply
integer greater than 2 and less than 12
Emily Reply
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 Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
how
muqtaar
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
Practice Key Terms 6

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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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