How to Solve Radical Equations? (+FREE Worksheet!)

Radical equations are equations that contain a variable inside a radical sign. Solving them requires isolating the radical and then squaring both sides to eliminate it — but this step can sometimes introduce extraneous solutions that do not satisfy the original equation. This guide walks through the complete solving process, including how to check your answers, with four worked examples and two video lessons.

What Is a Radical Equation?

A radical equation is any equation in which the variable appears inside a radical (square root, cube root, etc.). Examples: \(\color{blue}{\sqrt{x} = 5}\), \(\color{blue}{\sqrt{(2x + 1)} = 3}\), \(\color{blue}{\sqrt{(x – 3)} + 4 = 8}\). The main strategy is to isolate the radical, then raise both sides to a power that eliminates it.

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How to Solve Radical Equations

1. Isolate the Radical

Get the radical expression by itself on one side of the equation. Move all other terms to the opposite side first.

• \(\color{blue}{\sqrt{x} + 2 = 7 \rightarrow \sqrt{x} = 5}\)

2. Square Both Sides

Squaring eliminates a square root: \(\color{blue}{(\sqrt{x})^{2} = x}\). Apply this to both sides of the equation.

• \(\color{blue}{(\sqrt{x})^{2} = 5^{2} \rightarrow x = 25}\)

3. Solve the Resulting Equation

After squaring, you will have a linear or quadratic equation. Solve it using standard methods.

4. Check for Extraneous Solutions

Squaring both sides can introduce solutions that do not work in the original equation. Always substitute each solution back into the original equation to verify it is valid. Discard any that do not satisfy the original.

Step-by-Step Summary

1. Isolate the radical on one side.
2. Square both sides to eliminate the radical.
3. Solve the resulting equation.
4. Check every solution in the original equation; discard extraneous ones.

Watch: Solving Radical Equations (Video Lesson)

Khan Academy explains the isolation and squaring process with clear, worked problems:

Radical Equations — Worked Examples

Example 1: Solve \(\color{blue}{\sqrt{x} = 5}\).

Square both sides: \(\color{blue}{x = 25}\).
Check: \(\color{blue}{\sqrt{25} = 5}\) ✓
Answer: \(\color{blue}{x = 25}\)

Example 2: Solve \(\color{blue}{\sqrt{(2x + 1)} = 3}\).

Square both sides: \(\color{blue}{2x + 1 = 9}\) → \(\color{blue}{2x = 8}\) → \(\color{blue}{x = 4}\).
Check: \(\color{blue}{\sqrt{(8 + 1)} = \sqrt{9} = 3}\) ✓
Answer: \(\color{blue}{x = 4}\)

Example 3: Solve \(\color{blue}{\sqrt{(x – 3)} = 4}\).

Square both sides: \(\color{blue}{x – 3 = 16}\) → \(\color{blue}{x = 19}\).
Check: \(\color{blue}{\sqrt{(19 – 3)} = \sqrt{16} = 4}\) ✓
Answer: \(\color{blue}{x = 19}\)

Example 4: Solve \(\color{blue}{\sqrt{(x + 3)} = x – 3}\).

Square both sides: \(\color{blue}{x + 3 = (x – 3)^{2} = x^{2} – 6x + 9}\).
Rearrange: \(\color{blue}{x^{2} – 7x + 6 = 0}\) → \(\color{blue}{(x – 1)(x – 6) = 0}\) → \(\color{blue}{x = 1}\) or \(\color{blue}{x = 6}\).
Check \(\color{blue}{x = 1}\): \(\color{blue}{\sqrt{4} = 2}\) but \(\color{blue}{1 – 3 = -2}\). Since \(\color{blue}{2 \ne -2}\), \(\color{blue}{x = 1}\) is extraneous.
Check \(\color{blue}{x = 6}\): \(\color{blue}{\sqrt{9} = 3}\) and \(\color{blue}{6 – 3 = 3}\) ✓
Answer: \(\color{blue}{x = 6}\)

More Practice: Solving Radical Equations Video

This algebra video covers solving radical equations including cases with extraneous solutions:

Exercises for Radical Equations

Solve each equation. Check for extraneous solutions.

1. \(\color{blue}{\sqrt{x} = 7}\)
2. \(\color{blue}{\sqrt{(x + 4)} = 5}\)
3. \(\color{blue}{\sqrt{(2x – 3)} = 3}\)
4. \(\color{blue}{\sqrt{(x – 1)} = x – 3}\)
5. \(\color{blue}{\sqrt{(3x + 1)} = 4}\)

Answers

1. \(\color{blue}{x = 49}\)
2. \(\color{blue}{x = 21}\)  (\(\color{blue}{x + 4 = 25}\))
3. \(\color{blue}{x = 6}\)  (\(\color{blue}{2x – 3 = 9}\))
4. \(\color{blue}{x = 5}\)  (\(\color{blue}{x = 2}\) is extraneous; check: \(\color{blue}{\sqrt{(5-1)} = 2}\), \(\color{blue}{5-3 = 2}\) ✓)
5. \(\color{blue}{x = 5}\)  (\(\color{blue}{3x + 1 = 16}\))

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Want More Practice?

We haven’t published a worksheet built specifically for Radical Equations just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Solving Quadratics by Square Roots Worksheet
• Download Graphing Square Root Functions Worksheet

Frequently Asked Questions

Why do extraneous solutions appear?

Squaring both sides is not a reversible operation — it can introduce solutions that satisfy the squared equation but not the original. Always check because a square root is defined as non-negative, so \(\color{blue}{\sqrt{x} = -3}\) has no solution even though squaring gives \(\color{blue}{x = 9}\).

What if there are two radicals in the equation?

Isolate one radical on one side, square both sides, then isolate the remaining radical (if any) and square again. This process may require two rounds of squaring.

Does this method work for cube root equations?

Yes. For cube root equations, isolate the cube root and then cube both sides instead of squaring. Cube roots do not produce extraneous solutions because the cube function is one-to-one (both positive and negative values have unique cube roots).

Related Topics

• Simplifying Radical Expressions
• Rationalizing Radical Expressions
• Multiplying Radical Expressions
• Domain and Range of Radical Functions
• Adding and Subtracting Radical Expressions