7.8: Applications of Quadratic Equations
Two cars leave an intersection at the same time. One car travels north and the other car travels west. When the car traveling north had gone 24 miles, the distance between the cars was four miles more than three times the distance traveled by the car heading west. Find the distance between the cars at that time.
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Khan Academy Applying Quadratic Equations
Guidance
Quadratic functions can be used to help solve many different real world problems. Here are two hints for solving quadratic word problems:
 It is often helpful to start by drawing a picture in order to visualize what you are asked to solve.
 Once you have solved the problem, it is important to make sure that your answers are realistic given the context of the problem. For example, if you are solving for the age of a person and one of your answers is a negative number, that answer does not make sense in the context of the problem and is not actually a solution.
Example A
The number of softball games that must be scheduled in a league with
Solution: You are given the function
Start by setting the equation equal to zero:
Now solve for
There are 7 teams in the softball league.
Example B
When a homemade rocket is launched from the ground, it goes up and falls in the pattern of a parabola. The height, in feet, of a homemade rocket is given by the equation
Solution: The formula for the path of the rocket is
This means
The rocket will hit the ground after 10 seconds.
Example C
Using the information in Example B, what is the height of the rocket after 2 seconds?
Solution: To solve this problem, you need to replace
Therefore, after 2 seconds, the height of the rocket is 256 feet.
Concept Problem Revisited
Two cars leave an intersection at the same time. One car travels north and the other car travels west. When the car traveling north had gone 24 miles the distance between the cars was four miles more than three times the distance traveled by the car heading west. Find the distance between the cars at that time.
First draw a diagram. Since the cars are traveling north and west from the same starting position, the triangle made to connect the distance between them is a right triangle. Since you have a right triangle, you can use the Pythagorean Theorem to set up an equation relating the lengths of the sides of the triangle.
The Pythagorean Theorem is a geometry theorem that says that for all right triangles,
Now set the equation equal to zero and factor the quadratic expression so that you can use the zero product property.
So you now know that
Vocabulary
 Factor
 To factor means to rewrite an expression as a product.
 Pythagorean Theorem

The Pythagorean Theorem is a right triangle theorem that relates all three sides of a right triangle according to the equation
a2+b2=c2 wherea andb are legs of the triangle andc is the hypotenuse.
 Quadratic Expression

A quadratic expression is a polynomial of degree 2. The general form of a quadratic expression is
ax2+bx+c .
Guided Practice
1. A rectangle is known to have an area of 520 square inches. The lengths of the sides are shown in the diagram below. Solve for both the length and the width.
2. The height of a ball in feet can be found by the quadratic function
3. A manufacturer measures the number of cell phones sold using the binomial
Answers:
1. The rectangle has an area of 520 square inches and you know that the area of a rectangle has the formula:

520520000=(x+7)(2x)=2x2+14x=2x2+14x−520=2(x2+7x−260)=2(x−13)(x+20) 
Therefore the value of
x is 13. This means that the width is2x or2(13)=26 inches . The length isx+7=13+7=20 inches .
2. The equation for the ball being thrown is \begin{align*}h(t)=16t^2+80t+5\end{align*}. If you drew the path of the thrown ball, you would see something like that shown below.
 You are asked to find the time(s) when the ball hits a height of 69 feet. In other words, solve for:

\begin{align*}69=16t^2+80t+5\end{align*}
 To solve for \begin{align*}t\end{align*}, you have to factor the quadratic and then solve for the value(s) of \begin{align*}t\end{align*}.

\begin{align*}&\quad \qquad 0=16t^2+80t64\\
&\quad \qquad 0=16(t^25t+4)\\
&\quad \qquad 0=16(t1)(t4)\\
&\qquad \quad \qquad \swarrow \qquad \qquad \searrow\\
& \quad t  1 = 0 \qquad \qquad t4 = 0\\
&\qquad \ \ t = 1 \quad \qquad \qquad \ \ t = 4\end{align*}
 Since both values are positive, you can conclude that there are two times when the ball hits a height of 69 feet. These times are at 1 second and at 4 seconds.
3. The number of cell phones sold is the binomial \begin{align*}0.015c + 2.81\end{align*}. The wholesale price on these phones is the binomial \begin{align*}0.011c +3.52.\end{align*} The revenue she takes in is the wholesale price times the number that she sells. Therefore:
 \begin{align*}R(c)=(0.015c+2.81)(0.011c+3.52)\end{align*}
 First, let’s expand the expression for \begin{align*}R\end{align*} to get the quadratic expression. Therefore:

\begin{align*}R(c)&=(0.015c+2.81)(0.011c+3.52)\\
R(c)&=0.000165c^2+0.08371c+9.8912\end{align*}
 The question then asks if she sold 100,000 cell phones, what would her revenue be. Therefore what is \begin{align*}R(c)\end{align*} when \begin{align*}c = 100,000\end{align*}.

\begin{align*}R(c)&=0.000165c^2+0.08371c+9.8912\\
R(c)&=0.000165(100,000)^2+0.08371(100,000)+9.8912\\
R(c)&=1,658,380.89\end{align*}
 Therefore she would make $1,658,380.89 in revenue.
Practice
 A rectangle is known to have an area of 234 square feet. The length of the rectangle is given by \begin{align*}x+3\end{align*} and the width of the rectangle is given by \begin{align*}x+8\end{align*}. What is the value of \begin{align*}x\end{align*}?
 Solve for \begin{align*}x\end{align*} in the rectangle below given that the area is 9 units.
 Solve for \begin{align*}x\end{align*} in the triangle below given that the area is 10 units.
A pool is treated with a chemical to reduce the amount of algae. The amount of algae in the pool \begin{align*}t\end{align*} days after the treatment can be approximated by the function \begin{align*}A(t)=40t^2300t+500\end{align*}.
 How many days after treatment will the pool have the no algae?
 How much algae is in the pool before treatments are started?
 How much less algae is in the pool after 1 day?
A football is kicked into the air. The height of the football in meters can be found by the quadratic function \begin{align*}h(t)=5t^2+25t\end{align*} where \begin{align*}t\end{align*} is the time in seconds since the ball has been kicked.
 How high is the ball after 3 seconds? At what other time is the ball the same height?
 When will the ball be 20 meters above the ground?
 After how many seconds will the ball hit the ground?
A ball is thrown into the air. The height of the ball in meters can be found by the quadratic function \begin{align*}h(t)=5t^2+30t\end{align*} where \begin{align*}t\end{align*} is the time in seconds since the ball has been thrown.
 How high is the ball after 3 seconds?
 When will the ball be 25 meters above the ground?
 After how many seconds will the ball hit the ground?
Kim is drafting the windows for a new building. Their shape can be modeled by the function \begin{align*}h(w)=w^2+4\end{align*}, where \begin{align*}h\end{align*} is the height and \begin{align*}w\end{align*} is the width of points on the window frame, measured in meters.
 Find the width of each window at its base.
 Find the width of each window when the height is 3 meters.
 What is the height of the window when the width is 1 meter?
Parabola
A parabola is the characteristic shape of a quadratic function graph, resembling a "U".Pythagorean Theorem
The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by , where and are legs of the triangle and is the hypotenuse of the triangle.quadratic function
A quadratic function is a function that can be written in the form , where , , and are real constants and .Zero Product Rule
The zero product rule states that if the product of two expressions is equal to zero, then at least one of the original expressions much be equal to zero.Image Attributions
Description
Learning Objectives
Here you'll learn how to apply your knowledge of factoring quadratic expressions to solve real world application problems.