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

9.5.2 Solutions Written as Square Roots

Algebra 19.5.2 Solutions Written as Square Roots
Search for key terms or text.

Activity

Solve each equation. Use the notation when appropriate and express radicals in simplest form.

1.

x 2 − 13 = − 12 x 2 − 13 = − 12

2.

Enter the solution to the equation.

( x − 6 ) 2 = 0 ( x − 6 ) 2 = 0

3.

x 2 + 9 = 0 x 2 + 9 = 0

So far in this activity you encountered radical solutions that were easy to simplify. Sometimes you will get radical solutions like 1212. Let’s review how to simplify more difficult radicals using perfect squares.

Step 1 - Look at the ways to factor the number under the radical.

For example, factor 12 from 1212.

12=6*212=6*2 and 12=4*312=4*3

Step 2 - Choose the factorization of the number that includes a perfect square. Rewrite the number under the radical using that factorization.

12=4*312=4*3

Step 3 - Recall that a*b=a*ba*b=a*b.

4*3=4*34*3=4*3

Step 4 - Simplify the perfect squares.

4*3=234*3=23

So 12=2312=23

4.

x 2 = 18 x 2 = 18

5.

x 2 + 1 = 18 x 2 + 1 = 18

6.

( x + 1 ) 2 = 18 ( x + 1 ) 2 = 18

Video: Solving Quadratics with Square Roots

Watch the following video to learn more about equations with solutions that are square roots.

Access multimedia content

Self Check

Solve x 2 − 3 = 19 .
  1. x = ± 22
  2. x = ± 22
  3. x = ± 19 + 3
  4. x = − 4 , x = 4

Additional Resources

Solving Quadratics with Radical Solutions

When solving quadratic equations, it is important to remember that:

  • Any positive number has two square roots, one positive and one negative, because there are two numbers that can be squared to make that number. (For example, 6262and (−6)2(−6)2 both equal 36, so 6 and -6 are both square roots of 36.)
  • The square root symbol ()() can be used to express the positive square root of a number. For example, the square root of 36 is 6, but it can also be written as 3636 because 36·36=3636·36=36.
  • To express the negative square root of a number, say 36, we can write -6 or −36−36.
  • When a number is not a perfect square (for example, 40), we can express its square roots by writing 4040 and −40−40.

How could we write the solutions to an equation like (x+4)2=11(x+4)2=11? This equation is saying, “something squared is 11.” To make the equation true, that something must be 1111 or −11−11.

We can write:

x+4=11x+4=11      or      x+4=−11x+4=−11

x=−4+11x=−4+11      or      x=−4+−11x=−4+−11

A more compact way to write the two solutions to the equation is: x=−4±11x=−4±11.

Try it

Try It: Solving Quadratics with Radical Solutions

Solve (x−2)2=17(x−2)2=17.

PreviousNext
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.