# 9.4: Equations with Square Roots

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

**Practice**Equations with Square Roots

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.

### Solving Equations with Square Roots

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:

In these equations, both **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

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

Solve for

First, you know that

Next,

Then,

Your answer is 25.

Here’s another example:

Solve for

First, you know that

You may also know that in order to find

Next, in order to isolate

Since a square is attached to the

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.

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

The answer is

### Examples

#### Example 1

Earlier, you were given a problem about Omar, who was 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.

Next, substitute in what you know.

Then solve for

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

#### Example 2

Solve.

First, recognize that you are solving for

Next, determine which function to remove first from the

After subtracting, you have a new equation:

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

The answer is

#### Example 3

Solve for

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

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

This gives you a new equation:

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

The answer is 65.

#### Example 4

Solve for

First, take the square root of both sides.

Then,

The answer is

#### Example 5

Solve for

\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*}

### Review

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*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.4.

### Resources

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Perfect Square

A perfect square is a number whose square root is an integer.Radical

The , or square root, sign.Radical Expression

A radical expression is an expression with numbers, operations and radicals in it.Square Root

The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.### Image Attributions

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

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