# 9.4: Equations with Square Roots

**At Grade**Created by: CK-12

### Let’s Think About It

**License**: CC BY-NC 3.0

Running from one base to the next was a speedy 90 feet for Omar. The job of mowing the field, however, probably wasn’t so fast. Just counting the infield, how many square feet of mowing does the groundskeeper do?

In this concept, you will learn how to solve equations using squares and square roots.

### Guidance

You may already know that **squaring** a number and taking the **square root** of a number are **opposite operations**. If you know one, you can find the other. When working with area and dimensions, the equations most often used are:

\begin{align*}A=s^2\end{align*}

\begin{align*}s= \sqrt{A}\end{align*}

In these equations, both \begin{align*}s\end{align*}**variable** is simply an unknown quantity represented by a letter. Any letter or symbol can be used in a math sentence as a variable. For example, the equation \begin{align*}y=x^2\end{align*}

Here is an example of how to use these equations to solve problems involving squares and square roots.

Solve for \begin{align*}y\end{align*}

\begin{align*}y=5^2\end{align*}

First, you know that \begin{align*}5^2\end{align*}

Next, \begin{align*}5 \times 5=25\end{align*}

Then, \begin{align*}y=25\end{align*}

Your answer is 25.

Here’s another example:

Solve for \begin{align*}x\end{align*}

\begin{align*}x^2 = 36\end{align*}

First, you know that \begin{align*}x\end{align*}

You may also know that in order to find \begin{align*}x\end{align*}

Next, in order to isolate \begin{align*}x\end{align*}

Since a square is attached to the \begin{align*}x\end{align*}

Then, remember, that whatever you do to one side of the equation, you must also do to the other side.

Take the square root of both sides of the equation.

\begin{align*}\sqrt{x^2} = \sqrt{36}\end{align*}

Since you have been given an abstract problem to solve, be sure to include negative roots.

\begin{align*}x = \pm 6\end{align*}

The answer is \begin{align*}\pm 6\end{align*}

### Guided Practice

Solve.

\begin{align*}x^2 + 3 = 12\end{align*}

First, recognize that you are solving for \begin{align*}x\end{align*}

Next, determine which function to remove first from the \begin{align*}x\end{align*}

\begin{align*}x^2 + 3 -3 = 12 -3\end{align*}

After subtracting, you have a new equation:

\begin{align*}x^2 = 9\end{align*}

Then, all that is left is to take the square root of both sides.

\begin{align*}\begin{array}{rcl} \sqrt{x^2} &=& \sqrt{9} \\ x &=& \pm 3 \end{array}\end{align*}

The answer is \begin{align*}\pm 3\end{align*}

### Examples

#### Example 1

Solve for \begin{align*}y\end{align*}

\begin{align*}\sqrt{y-1} = 8\end{align*}

First, recognize the two operations attached to the variable - a 1 by subtraction and a square root.

Next, determine which operation you can perform first to both sides as a step toward isolating \begin{align*}y\end{align*}

Perform the opposite of a square root by squaring both sides of the equation.

\begin{align*}(\sqrt{y-1})^2 = 8^2\end{align*}

This gives you a new equation:

\begin{align*}y-1=64 \end{align*}

Then add to remove the 1 that is attached by subtraction.

\begin{align*}\begin{array}{rcl} y -1 +1 &=& 64 + 1 \\ y &=& 64 +1 \\ y &=& 65 \end{array}\end{align*}

The answer is 65.

#### Example 2

Solve for \begin{align*}x\end{align*}

\begin{align*}x^2=49\end{align*}

First, take the square root of both sides.

\begin{align*}\sqrt{x^2} = \sqrt{49}\end{align*}

Then, \begin{align*}x= \pm 7\end{align*}

The answer is \begin{align*}\pm 7\end{align*}

#### Example 3

Solve for \begin{align*}p\end{align*}

\begin{align*}p^2 + 5=174\end{align*}

First, subtract 5 from both sides of the equation.

\begin{align*}p^2 + 5-5=174-5\end{align*}

Next, rewrite the equation:

\begin{align*}p^2 = 169\end{align*}

Then, take the square root of both sides.

\begin{align*}p=\pm 13\end{align*}

The answer is \begin{align*}p=\pm 13\end{align*}

### Follow Up

**License**: CC BY-NC 3.0

Remember Omar wondering about how many square feet of grass needed to be mowed?

One side of the square baseball diamond infield measured 90 feet.

First, remember the formula for area of a square.

\begin{align*}A=s^2\end{align*}

Next, substitute in what you know.

\begin{align*}A=(90 \ ft)^2\end{align*}

Then solve for \begin{align*}A\end{align*}.

\begin{align*}90 \ ft \times 90 \ ft = 1,800 \ sq \ ft\end{align*}

The answer is 1,800 square feet. Remember that area is measured in squares.

### Video Review

https://www.youtube.com/watch?v=jhX2vY-zU2g

### Explore More

Solve each equation.

- \begin{align*}x^2 = 9\end{align*}
- \begin{align*}x^2 = 49\end{align*}
- \begin{align*}x^2 = 100\end{align*}
- \begin{align*}x^2 = 64\end{align*}
- \begin{align*}x^2 = 225\end{align*}
- \begin{align*}x^2 = 256\end{align*}
- \begin{align*}x^2 + 3 = 12\end{align*}
- \begin{align*}x^2 - 5 = 20\end{align*}
- \begin{align*}x^2 + 3 = 39\end{align*}
- \begin{align*}x^2 - 4 = 60\end{align*}
- \begin{align*}x^2 + 11 = 92\end{align*}
- \begin{align*}\sqrt{x+1} = 10\end{align*}
- \begin{align*}x^2 + 5 = 41\end{align*}
- \begin{align*}x^3 = 8\end{align*}
- \begin{align*}x^3 + 4 = 31\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.4.

### Image Attributions

**[1]****^**License: CC BY-NC 3.0**[2]****^**License: CC BY-NC 3.0

## Description

## Learning Objectives

In this concept, you will learn how to solve equations using squares and square roots.

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Dec 02, 2015## Last Modified:

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