Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Algebra 1

9.5.3 Finding Irrational Solutions by Completing the Square

Algebra 19.5.3 Finding Irrational Solutions by Completing the Square

Search for key terms or text.

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Here is an example of an equation being solved by graphing and by completing the square.

Graph of a parabola on a coordinate plane. Two points (negative 4.414, 0) and (negative 1.586, 0) have been labeled on the parabola. The x-axis has a scale of 1 extending from negative 8 to 4. The y-axis extends from negative 4 to 4 with a scale of 1.

x 2 + 6 x + 7 = 0 x 2 + 6 x + 9 = 2 ( x + 3 ) 2 = 2 x + 3 = ± 2 x = 3 ± 2 x 2 + 6 x + 7 = 0 x 2 + 6 x + 9 = 2 ( x + 3 ) 2 = 2 x + 3 = ± 2 x = 3 ± 2

Verify: 2 2 is approximately 1.414. So 3 + 2 1.586 3 + 2 1.586 and 3 2 4.414 3 2 4.414 .

For each equation, find the exact solutions by completing the square and the approximate solutions by graphing. Then, verify that the solutions found using the two methods are close. If you get stuck, refer back to the example.

1.

1. x 2 + 4 x + 1 = 0 x 2 + 4 x + 1 = 0

a. Find the exact solutions by completing the square.

b. Graph the equation to find approximate solutions.

Use the Desmos graphing tool or technology outside the course.

2.

3.

c. Are the solutions you found using the two methods close?

4.

2. x 2 10 x + 18 = 0 x 2 10 x + 18 = 0

a. Find the exact solutions by completing the square.

b. Graph the equation to find approximate solutions.

Use the Desmos graphing tool or technology outside the course.

5.

6.

c. Are the solutions you found using the two methods close?

7.

3. x 2 + 5 x + 1 4 = 0 x 2 + 5 x + 1 4 = 0

a. Find the exact solutions by completing the square.

b. Graph the equation to find approximate solutions.

Use the Desmos graphing tool or technology outside the course.

8.

9.

c. Are the solutions you found using the two methods close?

10.

4. x 2 + 8 3 x + 14 9 = 0 x 2 + 8 3 x + 14 9 = 0

a. Find the exact solutions by completing the square.

b. Graph the equation to find approximate solutions.

Use the Desmos graphing tool or technology outside the course.

11.

12.

c. Are the solutions you found using the two methods close?

Are you ready for more?

Extending Your Thinking

1.

What quadratic equation of the form a x 2 + b x + c = 0 a x 2 + b x + c = 0 would have the solutions x = 5 2 x = 5 2 and x = 5 + 2 x = 5 + 2 ?

Self Check

Which of the following are the solutions to ( x 2 ) 2 = 12 , rounded to the nearest thousandth?
  1. x 3.162
  2. x = 14
  3. x 1.464 , x 5.464
  4. x 3.162 , x 3.162

Additional Resources

Using the Calculator for Approximate Solutions

Let's look at the solutions to the equations from 9.5.2: Additional Resources:

A more compact way to write the two solutions to the equation ( x + 4 ) 2 = 11 ( x + 4 ) 2 = 11 is: x = 4 ± 11 x = 4 ± 11 .

How large or small are those numbers? Are they positive or negative? We can use a calculator to compute the approximate values of both expressions:

4 + 11 0.683 4 + 11 0.683 or 4 11 7.317 4 11 7.317

We can also approximate the solutions by graphing. The equation ( x + 4 ) 2 = 11 ( x + 4 ) 2 = 11 is equivalent to ( x + 4 ) 2 11 = 0 ( x + 4 ) 2 11 = 0 , so we can graph the function y = ( x + 4 ) 2 11 y = ( x + 4 ) 2 11 and find its zeros by locating the x x -intercepts of the graph.

Graph of a parabola on a coordinate grid. The points (negative 7.317, 0) and (negative 0.683, 0) are labeled on the parabola. The x-axis extends from negative 14 to 6 with a scale of 2. The y-axis extends from negative 14 to 4 with a scale of 2.

Try it

Try It: Using the Calculator for Approximate Solutions

Approximate the solutions to x = 3 ± 19 x = 3 ± 19 .

Citation/Attribution

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 permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a 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 digital format, then you must 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

© May 21, 2025 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 are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.