Radical Equation Calculator

Solve equations containing square roots, cube roots, and nth roots. Get step-by-step solutions and check for extraneous solutions.

Use standard operators: +, -, *, /, ^. Example: x + 2, 2x - 5, x^2 + 1

Solution

-

Verification

-

Extraneous Solutions

-

Step-by-Step Solution

Enter a radical equation and click "Solve" to see the step-by-step solution.

How to Solve Radical Equations

  • Step 1: Isolate the radical term on one side
  • Step 2: Raise both sides to the power of the root index
  • Step 3: Solve the resulting equation
  • Step 4: Check for extraneous solutions

Important Rules

  • √(a) = b → a = b² (b ≥ 0)
  • ∛(a) = b → a = b³ (no restriction)
  • Always check extraneous solutions!
  • Even roots (√, ∜) have domain restrictions

Common Examples

  • √(x + 2) = 3 → x = 7
  • √(2x - 1) = 5 → x = 13
  • √(x + 3) = √(2x - 1) → x = 4
  • ∛(x - 5) = 2 → x = 13

Frequently Asked Questions

What is a radical equation?

A radical equation is an equation in which the variable appears inside a radical (square root, cube root, etc.). For example: √(x + 2) = 5 or ∛(2x - 1) = 3.

How do you solve radical equations?

Step 1: Isolate the radical on one side of the equation

Step 2: Raise both sides to the power of the root index

Step 3: Solve the resulting equation

Step 4: Check for extraneous solutions (always required!)

What are extraneous solutions?

Extraneous solutions are solutions that satisfy the squared equation but not the original equation. They often occur when both sides of an equation are squared. Always check your solutions in the original equation!

How to solve √(x + 2) = 3?

Step 1: Square both sides: (√(x + 2))² = 3²

Step 2: Simplify: x + 2 = 9

Step 3: Solve: x = 7

Step 4: Check: √(7 + 2) = √9 = 3 ✓

What is the domain of a square root equation?

The expression inside a square root must be ≥ 0. So for √(x + 2) = 3, we need x + 2 ≥ 0 → x ≥ -2. This is an additional constraint that helps identify extraneous solutions.

Can cube roots have negative values?

Yes! Cube roots (and odd roots in general) are defined for all real numbers. ∛(-8) = -2. There's no domain restriction for odd roots.