# 3.3 Quadratic equations: applications

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This module is from Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr. Methods of solving quadratic equations as well as the logic underlying each method are discussed. Factoring, extraction of roots, completing the square, and the quadratic formula are carefully developed. The zero-factor property of real numbers is reintroduced. The chapter also includes graphs of quadratic equations based on the standard parabola, y = x^2, and applied problems from the areas of manufacturing, population, physics, geometry, mathematics (numbers and volumes), and astronomy, which are solved using the five-step method.Objectives of this module: become more proficient at using the five-step method for solving applied problems.

## Overview

• The Five-Step Method
• Examples

## The five-step method

We are now in a position to study some applications of quadratic equations. Quadratic equations can arise from a variety of physical (applied) and mathematical (logical) problems.

We will, again, apply the five-step method for solving word problems.

## Five-step method of solving word problems

• Step 1:   Let $x$ (or some other letter) represent the unknown quantity.
• Step 2:   Translate the verbal expression to mathematical symbols and form an equation.
• Step 3:   Solve this equation.
• Step 4:   Check the solution by substituting the result into the equation found in step 2.
• Step 5:   Write a conclusion.

Remember, step 1 is very important.

ALWAYS START BY INTRODUCING A VARIABLE.

Once the quadratic equation is developed (step 2), try to solve it by factoring. If factoring doesn’t work, use the quadratic formula. A calculator may help to make some of the calculations a little less tedious.

## Sample set a

A producer of personal computer mouse covers determines that the number $N$ of covers sold is related to the price $x$ of a cover by $N=35x-{x}^{2}.$ At what price should the producer price a mouse cover in order to sell 216 of them?

$\begin{array}{l}\begin{array}{ccc}\text{Step\hspace{0.17em}}1:& & \text{Let}\text{\hspace{0.17em}}x=\text{the}\text{\hspace{0.17em}}\text{price}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{mouse}\text{\hspace{0.17em}}\text{cover}\text{.}\end{array}\hfill \\ \begin{array}{lll}\text{Step\hspace{0.17em}}2:\hfill & \hfill & \text{Since}\text{\hspace{0.17em}}N\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}\text{216,}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{equation}\text{\hspace{0.17em}}\text{is}\hfill \\ \hfill & \hfill & 216=35x-{x}^{2}\hfill \end{array}\hfill \\ \begin{array}{lllllll}\text{Step\hspace{0.17em}}3:\hfill & \hfill & 216\hfill & =\hfill & 35x-{x}^{2}\hfill & \hfill & \text{Rewrite}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{standard}\text{\hspace{0.17em}}\text{form}\text{.}\hfill \\ \hfill & \hfill & {x}^{2}-35x+216\hfill & =\hfill & 0\hfill & \hfill & \text{Try}\text{\hspace{0.17em}}\text{factoring}\text{.}\hfill \\ \hfill & \hfill & \left(x-8\right)\left(x-27\right)\hfill & =\hfill & 0\hfill & \hfill & \hfill \\ \hfill & \hfill & x-8=0\hfill & or\hfill & \hfill & x-27=0\hfill & \hfill \\ \hfill & \hfill & x=8\hfill & or\hfill & \hfill & x=27\hfill & \hfill \end{array}\hfill \\ \begin{array}{ccc}& & \text{Check}\text{\hspace{0.17em}}\text{these}\text{\hspace{0.17em}}\text{potential}\text{\hspace{0.17em}}\text{solutions}\text{.}\end{array}\hfill \\ \begin{array}{llllllllll}\text{Step\hspace{0.17em}}4:\hfill & \hfill & \text{If}\text{\hspace{0.17em}}x=8,\hfill & \hfill & \hfill & \hfill & \text{If}\text{\hspace{0.17em}}x=27,\hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 35\text{\hspace{0.17em}}·\text{\hspace{0.17em}}8-{8}^{2}& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill 35\text{\hspace{0.17em}}·\text{\hspace{0.17em}}27-{27}^{2}& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 280-64& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill 945-729& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 216& =\hfill & 216\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill & \hfill 216& =\hfill & 216\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \end{array}\hfill \\ \begin{array}{ccc}& & \text{These}\text{\hspace{0.17em}}\text{solutions}\text{\hspace{0.17em}}\text{check}\text{.}\end{array}\hfill \\ \hfill \begin{array}{ccc}\text{Step}\text{\hspace{0.17em}}5:& & \text{The}\text{\hspace{0.17em}}\text{computer}\text{\hspace{0.17em}}\text{mouse}\text{\hspace{0.17em}}\text{covers}\text{\hspace{0.17em}}\text{can}\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}\text{priced}\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}\text{either}\text{\hspace{0.17em}}\text{8}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{27}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{order}\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{sell}\text{\hspace{0.17em}}\text{216}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{them}\text{.}\text{\hspace{0.17em}}\end{array}\end{array}$

## Practice set a

A manufacturer of cloth personal computer dust covers notices that the number $N$ of covers sold is related to the price of covers by $N=30x-{x}^{2}.$ At what price should the manufacturer price the covers in order to sell 216 of them?

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:   In order to sell 216 covers, the manufacturer should price them at either or .

12 or 18

It is estimated that $t$ years from now the population of a particular city will be

$P={t}^{2}-24t+96,000.$

How many years from now will the population be 95,865?

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

In 9 and 15 years, the population of the city will be 95,865.

#### Questions & Answers

can someone help me with some logarithmic and exponential equations.
sure. what is your question?
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Cesar
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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