Radical Equations Formula

The Formula

If \sqrt{f(x)} = g(x), then f(x) = [g(x)]^2 (check for extraneous solutions)

When to use: 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.

Quick Example

\sqrt{x + 3} = 5 \to x + 3 = 25 \to x = 22
Check: \sqrt{22 + 3} = \sqrt{25} = 5. Valid.

Notation

Isolate the radical: \sqrt{\ldots} = \ldots. Then square both sides: (\sqrt{\ldots})^2 = (\ldots)^2. Extraneous solutions must be rejected.

What This Formula Means

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.

Formal View

If \sqrt{f(x)} = g(x), then f(x) = [g(x)]^2 and g(x) \geq 0 (domain constraint). Squaring may introduce extraneous roots: solutions of f(x) = [g(x)]^2 must be verified against g(x) \geq 0 and the original equation.

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.

Common Mistakes

  • Forgetting to check for extraneous solutions—some answers from squaring do not satisfy the original equation
  • Squaring before isolating the radical: \sqrt{x} + 2 = 5 should become \sqrt{x} = 3 first, THEN square
  • When there are two radicals, you may need to isolate and square twice

Why This Formula Matters

Radical equations appear in geometry (distance formulas), physics (kinematic equations), and any context where square roots arise naturally.

Frequently Asked Questions

What is the Radical Equations formula?

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.

How do you use the Radical Equations formula?

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.

What do the symbols mean in the Radical Equations formula?

Isolate the radical: \sqrt{\ldots} = \ldots. Then square both sides: (\sqrt{\ldots})^2 = (\ldots)^2. Extraneous solutions must be rejected.

Why is the Radical Equations formula important in Math?

Radical equations appear in geometry (distance formulas), physics (kinematic equations), and any context where square roots arise naturally.

What do students get wrong about Radical Equations?

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

What should I learn before the Radical Equations formula?

Before studying the Radical Equations formula, you should understand: radical operations, solving linear equations.

← Back to Radical Equations See All Examples Practice Problems