# 10.4 Solve applications modeled by quadratic equations

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By the end of this section, you will be able to:
• Solve applications modeled by Quadratic Equations

Before you get started, take this readiness quiz.

1. The sum of two consecutive odd numbers is $-100$ . Find the numbers.
If you missed this problem, review [link] .
2. The area of triangular mural is 64 square feet. The base is 16 feet. Find the height.
If you missed this problem, review [link] .
3. Find the length of the hypotenuse of a right triangle with legs 5 inches and 12 inches.
If you missed this problem, review [link] .

## Solve applications of the quadratic formula

We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications. To get us started, we will copy our usual Problem Solving Strategy here so we can follow the steps.

## Use the problem solving strategy.

1. Read the problem. Make sure all the words and ideas are understood.
2. Identify what we are looking for.
3. Name what we are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

We have solved number applications that involved consecutive even integers    and consecutive odd integers    by modeling the situation with linear equations. Remember, we noticed each even integer is 2 more than the number preceding it. If we call the first one n, then the next one is $n+2$ . The next one would be $n+2+2$ or $n+4$ . This is also true when we use odd integers. One set of even integers and one set of odd integers are shown below.

$\begin{array}{cccc}\hfill \mathbf{\text{Consecutive even integers}}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\mathbf{\text{Consecutive odd integers}}\hfill \\ \hfill 64,66,68\hfill & & & \hfill \phantom{\rule{4em}{0ex}}77,79,81\hfill \\ \hfill \begin{array}{cccc}n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{even integer}\hfill \\ n+2\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive even integer}\hfill \\ n+4\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive even integer}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\begin{array}{cccc}n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{odd integer}\hfill \\ n+2\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive odd integer}\hfill \\ n+4\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive odd integer}\hfill \end{array}\hfill \end{array}$

Some applications of consecutive odd integers    or consecutive even integers    are modeled by quadratic equations. The notation above will be helpful as you name the variables.

The product of two consecutive odd integers is 195. Find the integers.

## Solution

 Step 1. Read the problem. Step 2. Identify what we are looking for. We are looking for two consecutive odd integers. Step 3. Name what we are looking for. Let $n=$ the first odd integer. $n+2=$ the next odd integer Step 4. Translate into an equation. State the problem in one sentence. "The product of two consecutive odd integers is 195." The product of the first odd integer and the second odd integer is 195. Translate into an equation Step 5. Solve the equation. Distribute. Subtract 195 to get the equation in standard form. Identify the a, b, c values. Write the quadratic equation. Then substitute in the values of a, b, c. . Simplify. Simplify the radical. Rewrite to show two solutions. Solve each equation. There are two values of n that are solutions. This will give us two pairs of consecutive odd integers for our solution. First odd integer $n=13$ next odd integer $n+2$ $\phantom{\rule{6.8em}{0ex}}13+2$ $\phantom{\rule{8em}{0ex}}15$ First odd integer $n=-15$ next odd integer $n+2$ $\phantom{\rule{6.1em}{0ex}}-15+2$ $\phantom{\rule{7em}{0ex}}-13$ Step 6. Check the answer. Do these pairs work? Are they consecutive odd integers? Is their product 195? $\begin{array}{cccc}\hfill 13,15,\phantom{\rule{0.2em}{0ex}}\text{yes}& & & -13,-15,\phantom{\rule{0.2em}{0ex}}\text{yes}\hfill \\ \hfill 13\cdot 15=195,\phantom{\rule{0.2em}{0ex}}\text{yes}& & & -13\left(-15\right)=195,\phantom{\rule{0.2em}{0ex}}\text{yes}\hfill \end{array}$ Step 7. Answer the question. The two consecutive odd integers whose product is 195 are 13, 15, and −13, −15.

a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
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Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost$3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use? Jake Reply enrique borrowed$23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total$23,500
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost$20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be \$10 per square foot?
The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water.
Bridget
help me understand graphs
what kind of graphs?
bruce
function f(x) to find each value
Marlene
I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much.
Marlene
Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6
Melissa
if they say find f(5) then the answer would be f(5)=12
Melissa
I need you to help me Melissa. Wish I can show you my homework
Marlene
How is f(1) =0 I am really confused
Marlene
what's the formula given? f(x)=?
Melissa
It shows a graph that I wish I could send photo of to you on here
Marlene
Which problem specifically?
Melissa
which problem?
Melissa
I don't know any to be honest. But whatever you can help me with for I can practice will help
Marlene
I got it. sorry, was out and about. I'll look at it now.
Melissa
Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things.
Marlene
I just responded.
Melissa
Thank you
Marlene
-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
Rich
write in this form a/b answer should be in the simplest form 5%
convert to decimal 9/11
August
0.81818
Rich
5/100 = .05 but Rich is right that 9/11 = .81818
Melissa
Equation in the form of a pending point y+2=1/6(×-4)
write in simplest form 3 4/2
August