Square Root Equation Calculator with Work

This square root equation calculator helps you solve equations containing square roots. It shows you the step-by-step work and explains how to verify your results. Whether you're studying algebra or need to solve practical problems, this tool provides clear guidance.

How to Solve Square Root Equations

Solving square root equations involves isolating the square root and then squaring both sides to eliminate the square root. Here's a step-by-step guide:

Isolate the square root term on one side of the equation.
Square both sides of the equation to eliminate the square root.
Solve the resulting equation for the variable.
Check your solutions by substituting them back into the original equation.

Important Notes

When solving square root equations, remember that the square root function yields non-negative results. Therefore, any solution that makes the original equation negative must be discarded.

This process works for equations like √(x + 5) = 3 or √(2x - 1) + 4 = 7. The calculator below demonstrates this method with your specific equation.

Square Root Equation Formula

The general form of a square root equation is:

General Form

√(ax + b) = c

Where a, b, and c are constants, and x is the variable to solve for.

To solve this equation:

Square both sides: (√(ax + b))² = c² → ax + b = c²
Isolate the term with x: ax = c² - b
Solve for x: x = (c² - b)/a

The calculator uses this formula to solve your specific equation. It also checks for extraneous solutions that might result from squaring both sides.

Worked Example

Let's solve the equation √(3x + 2) = 5 step-by-step.

Step 1: Isolate the square root

√(3x + 2) = 5

Step 2: Square both sides

(√(3x + 2))² = 5² → 3x + 2 = 25

Step 3: Solve for x

3x = 25 - 2 → 3x = 23 → x = 23/3 ≈ 7.6667

Step 4: Verify the solution

Substitute x = 23/3 back into the original equation:

√(3*(23/3) + 2) = √(23 + 2) = √25 = 5 ✓

This confirms that x = 23/3 is a valid solution to the equation √(3x + 2) = 5.

FAQ

What is a square root equation?

A square root equation is an equation that contains a square root of an expression with a variable. Examples include √(x + 5) = 3 and √(2x - 1) + 4 = 7.

How do I solve a square root equation?

To solve a square root equation, first isolate the square root term. Then square both sides of the equation to eliminate the square root. Solve the resulting equation for the variable, and check your solutions by substituting them back into the original equation.

What are extraneous solutions?

Extraneous solutions are solutions that emerge from the solving process but do not satisfy the original equation. This can happen when both sides of an equation are squared, as squaring can introduce solutions that don't work in the original equation.

Can I solve equations with nested square roots?

Yes, you can solve equations with nested square roots by working from the outermost square root inward. Isolate each square root term one at a time and square both sides at each step.