# 11.3: Radical Equations

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

## Learning Objectives

At the end of this lesson, students will be able to:

- Solve a radical equation.
- Solve radical equations with radicals on both sides.
- Identify extraneous solutions.
- Solve real-world problems using square root functions.

## Vocabulary

Terms introduced in this lesson:

- radical equation
- radical expression
- extraneous solutions

## Teaching Strategies and Tips

Up to this point, students have been solving linear and quadratic equations. In this lesson, they now look at solving radical equations.

The following steps are used to solve radical equations:

- Isolate a radical.
- Square both sides of the equation (or use another appropriate power).
- Solve the new polynomial equation now free of radicals.
- Check answers in the original equation.

Use Example 2 to show that radical equations containing radicals of *any* index – not just square roots – can be solved.

- The steps are identical except for a change in the power that each side of the equation is raised to.

Use Example 4 to show that radical equations containing *more than one* radical expression can be solved.

- Isolate the most complicated radical expression and raise the equation to the appropriate power.
- Repeat the process until all radical signs are eliminated. In Example 4 and
*Review Questions*11-16, students must square both sides, twice.

Some radical equations can be made easier by reducing all terms by a common factor.

- This should occur at the beginning of the problem or after the step when both sides have been raised to a power.

Example:

*Find the real solutions of:*

\begin{align*}4\sqrt{2x + 1} = 12 - 8 \sqrt{x}\end{align*}

Hint: Divide by \begin{align*}4\end{align*} first.

\begin{align*}\sqrt{2x + 1} = 3 - 2\sqrt{3}\end{align*}

After squaring both sides and simplifying, the equation is:

\begin{align*}12\sqrt{x} = 2x + 8\end{align*}

Divide again, this time by \begin{align*}2\end{align*} before squaring both sides.

Point out that an equation such as \begin{align*}\sqrt{x + 5} = -3\end{align*} can be readily answered as not having any real solutions.

## Error Troubleshooting

In *Review Questions* 13-16, remind students to isolate a radical first. By not isolating, some radical will always remain in the equation and can even make the equation more complicated.

General Tip: Remind students to raise each *side* of the equation to the appropriate power, rather than term by term. Encourage students to use parentheses for each side.

General Tip: Have students always check their answers for extraneous solutions.