Skip to ContentGo to accessibility page
OpenStax Logo
Algebra 1

8.1.3 Formulating a Quadratic Equation to Represent the Model

Algebra 18.1.3 Formulating a Quadratic Equation to Represent the Model

8.1.3 Formulating a Quadratic Equation to Represent the Model

8.1.3 • Formulating a Quadratic Equation to Represent the Model

Activity

Here is a diagram that shows the picture with a frame that is the same width all the way around.

The picture is 7 inches by 4 inches. The frame is created from 10 square inches of framing material (in the form of a rectangle measuring 4 inches by 2.5 inches).

1.

Write an expression to represent the length of the framed picture on its shorter side. Use xx to represent the width of the frame.

2.

Write an expression to represent the length of the framed picture on its longer side. Use xx to represent the width of the frame.

3.

Use the expressions from the previous exercises to write an equation to represent the relationship between the measurements of the framed picture and its total area. Be prepared to explain what each part of your equation represents.

4.

What would a solution to this equation mean in this situation?

Video: Formulating a Quadratic Equation to Represent the Model

Watch the following video to learn more about formulating a quadratic equation to represent the model.

Formulating a Quadratic Equation to Represent the Model

Self Check

Suppose a different picture is 5 inches by 5 inches. The frame is created from 11 square inches of framing material. The frame will be uniform in width, and all of the framing material is used. Which equation would you need to solve to determine the value of x , the width of the frame?

  1. 4 x 2 + 20 x = 36
  2. x 2 + 25 = 11
  3. x 2 + 10 x + 25 = 36
  4. 4 x 2 + 20 x + 25 = 36

Additional Resources

Formulating a Quadratic Equation to Represent the Model

Let’s look at a similar situation to the previous activities but using a picture with different measurements.

Imagine a picture 8 inches by 6 inches. The frame is created from 32 square inches of framing material. The frame will be uniform in width, and all of the framing material is used.

How do we write an equation representing the total area of the framed picture?

Let’s break down the measurements that we know.

Let xx represent the thickness of the frame in inches.

Since the picture is 8 inches by 6 inches, it has an area of 48 square inches.

The frame is created from 32 square inches of framing material, and we must use all of it. So, the total area of the framed picture is 48+3248+32, or 80 square inches.

The length of one side of the framed picture is x+6+x=2x+6x+6+x=2x+6.

The length of the other side of the framed picture is x+8+x=2x+8x+8+x=2x+8.

So, the area of the framed picture is (2x+8)(2x+6)(2x+8)(2x+6). We can set this equal to 80 square inches.

(2x+8)(2x+6)=80(2x+8)(2x+6)=80

Let’s multiply.

4x2+12x+16x+48=804x2+12x+16x+48=80

4x2+28x+48=804x2+28x+48=80

So, the equation representing the total area of the framed picture is 4x2+28x+48=804x2+28x+48=80.

This equation is a quadratic equation. A quadratic equation is defined as one that can be written in the form ax2+bx+c=0ax2+bx+c=0, where aa is not 0.

Try it

Formulating a Quadratic Equation to Represent the Model

A picture measures 9 inches by 7 inches. Its frame is created from 36 square inches of framing material. The frame will be uniform in width, and all of the framing material is used.

Write an equation representing the total area of the framed picture.

PreviousNext
Order a print copy
Citation/Attribution
Reuse and redistribution of this content in digital or print format:
  • This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's prior written permission.
  • This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License, which means that you can reuse and modify the material only for noncommercial purposes, must attribute OpenStax, and must distribute any derivative works under the same license.
  • Any commercial printing of this textbook, including using a local or custom printer, must be approved by OpenStax, and proper citation provided.
  • OpenStax-copyrighted images, activities, assessments, and similar components of this book are subject to the same licensing – CC-BY-NC-SA. They can be used for noncommercial purposes with attribution. Commercial use requires permission.
  • Permission requests: Anyone who intends to incorporate this content (including text, images, and other components) into large language models, use it in AI offerings, use it commercially (including in print), and/or has questions about another use case is welcome to complete our reuse request form.
Attribution information
  • If you are redistributing all or part of this book in a noncommercial print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

  • If you are redistributing all or part of this book in a noncommercial digital format, then for every page that includes OpenStax content, you must license the derivative work under the same CC-BY-NC-SA license as the original, and include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

The information below includes the information needed to generate citations in most major styles (APA, MLA, etc.); you must reformat and organize the information as needed to fit the requirements of the style. Use the information below to generate a citation. We recommend using a citation tool such as this one.

© Apr 23, 2026 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo, and Rice University name, and Rice University logo trademarks, or wordmarks are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.