Radical Equations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Radical Equations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Solving equations that contain variable expressions under a radical by isolating the radical, raising both sides to the appropriate power to eliminate it, solving the resulting equation, and checking for extraneous solutions.

A radical 'traps' the variable inside a square root. To free it, isolate the radical on one side, then square both sides to undo the square root. But squaring can introduce fake solutions (extraneous solutions) that do not actually satisfy the original equation, so you MUST check every answer.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Isolate the radical, then square both sides. Always check answers because squaring is not a reversible operation.

Common stuck point: Extraneous solutions: squaring both sides can create solutions that do not work in the original equation. Always substitute back to verify.

Sense of Study hint: Isolate the radical on one side first, then square both sides and solve. Always check your answer in the original.

Worked Examples

Example 1

easy
Solve \sqrt{x + 3} = 5.

Solution

  1. 1
    Step 1: Square both sides: x + 3 = 25.
  2. 2
    Step 2: Subtract 3: x = 22.
  3. 3
    Step 3: Check: \sqrt{22 + 3} = \sqrt{25} = 5 ✓

Answer

x = 22
To solve a radical equation, isolate the radical then square both sides. Always check the solution because squaring can introduce extraneous solutions.

Example 2

hard
Solve \sqrt{2x + 1} = x - 1.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Solve \sqrt{x} = 7.

Example 2

medium
Solve \sqrt{3x - 2} = 4.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

radical operationssolving linear equations