Applications of Quadratic Equations

This section describes using quadratic equations to solve word problems involving numbers, geometrical figures, and motion. Read this section and work through the examples.

Solve Applications Modeled by Quadratic Equations

Learning Objectives

By the end of this section, you will be able to:

Solve applications modeled by Quadratic Equations

BE PREPARED 10.10

The sum of two consecutive odd numbers is -100. Find the numbers.

BE PREPARED 10.11

The area of triangular mural is 64 square feet. The base is 16 feet. Find the height.

BE PREPARED 10.12

Find the length of the hypotenuse of a right triangle with legs 5 inches and 12 inches.

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.

HOW TO

Use the problem solving strategy.

Step 1. Read the problem. Make sure all the words and ideas are understood.

Step 2. Identify what we are looking for.

Step 3. Name what we are looking for. Choose a variable to represent that quantity.

Step 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.

Step 5. Solve the equation using good algebra techniques.

Step 6. Check the answer in the problem and make sure it makes sense.

Step 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 $n +2 + 2$ 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.

	Consecutive even integers		Consecutive odd integers
	$64, 66, 68$		$77, 79, 81$
$n$	1st even integer	$n$	1st odd integer
$n+2$	2nd consecutive even integer	$n+2$	2nd consecutive odd integer
$n+4$	3rd consecutive even integer	$n+4$	3nd consecutive odd integer

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.

EXAMPLE 10.38

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

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 $n+2$ $13+2$ $15$
	First odd integer $n=-15$ next odd integer $n+2$ $-15+2$ $-13$
Step 6. Check the answer. Do these pairs work? Are they consecutive odd integers? Is their product 195?	$\begin{array}{cl} 13,15, \text { yes } & -13,-15, \text { yes } \\ 13 \cdot 15=195, \text { yes } & -13(-15)=195, \text { yes } \end{array}$
Step 7. Answer the question.	The two consecutive odd integers whose product is 195 are 13, 15, and −13, −15.