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Square Roots Equation Calculator

Reviewed by Calculator Editorial Team

This square roots equation calculator helps you solve equations containing square roots. Learn how to find solutions to equations like √(x + 5) = 3 or √(2x - 1) + √(x + 3) = 5 using our step-by-step guide and examples.

How to Solve Square Roots Equations

Solving equations with square roots requires careful handling of the radical expressions. Here's a general approach:

  1. Isolate the square root term on one side of the equation.
  2. Square both sides to eliminate the square root.
  3. Solve the resulting equation for the variable.
  4. Check all potential solutions in the original equation to ensure they're valid.

Remember that squaring both sides of an equation can introduce extraneous solutions, so always verify your answers.

Methods for Solving Square Roots Equations

Method 1: Isolating the Square Root

For equations like √(x + 5) = 3:

  1. Square both sides: (√(x + 5))² = 3² → x + 5 = 9
  2. Solve for x: x = 9 - 5 → x = 4
  3. Check: √(4 + 5) = √9 = 3 (valid)

Method 2: Two Square Roots

For equations like √(2x - 1) + √(x + 3) = 5:

  1. Let u = √(2x - 1) and v = √(x + 3)
  2. Equation becomes u + v = 5
  3. Square both sides: u² + 2uv + v² = 25
  4. Substitute u² = 2x - 1 and v² = x + 3
  5. Solve the resulting equation for x

Formula: For √(ax + b) = c, the solution is x = (c² - b)/a

Worked Examples

Equation Solution Verification
√(3x + 2) = 4 x = 4 √(12 + 2) = √14 ≈ 3.74 (≠4)
√(2x - 1) + √(x + 3) = 5 x = 1 or x = 9 Check both solutions in original equation

FAQ

What if the equation has a negative square root?
Square roots are defined as non-negative in real numbers, so equations like √(x) = -2 have no real solutions.
How do I solve equations with nested square roots?
Isolate the innermost square root first, then work your way out. For example, in √(√x + 3) = 2, first solve √x + 3 = 4, then √x = 1, then x = 1.
Why do I get extraneous solutions when solving square root equations?
Squaring both sides of an equation can introduce solutions that don't satisfy the original equation. Always verify your solutions by plugging them back into the original equation.