Quadratic Equations and Functions
Solve Applications of Quadratic Equations
Learning Objectives
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.
Solve Applications Modeled by Quadratic Equations
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.
Let’s first summarize the methods we now have to solve quadratic equations.
- Factoring
- Square Root Property
- Completing the Square
- Quadratic Formula
As you solve each equation, choose the method that is most convenient for you to work the problem. As a reminder, we will copy our usual Problem-Solving Strategy here so we can follow the steps.
- Read the problem. Make sure all the words and ideas are understood.
- Identify what we are looking for.
- Name what we are looking for. Choose a variable to represent that quantity.
- 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 algebraic equation.
- Solve the equation using algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence
We have solved number applications that involved consecutive even and 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.
Some applications of odd or even consecutive 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.
The product of two consecutive odd integers is 99. Find the integers.
The two consecutive odd integers whose product is 99 are 9, 11, and −9, −11
The product of two consecutive even integers is 168. Find the integers.
The two consecutive even integers whose product is 128 are 12, 14 and −12, −14.
We will use the formula for the area of a triangle to solve the next example.
For a triangle with base, b, and height, h, the area, A, is given by the formula
Recall that when we solve geometric applications, it is helpful to draw the figure.
An architect is designing the entryway of a restaurant. She wants to put a triangular window above the doorway. Due to energy restrictions, the window can only have an area of 120 square feet and the architect wants the base to be 4 feet more than twice the height. Find the base and height of the window.
| Step 1. Read the problem.
Draw a picture. |
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| Step 2. Identify what we are looking for.