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Solve by Taking Square Root of Both Sides Calculator

Reviewed by Calculator Editorial Team

Solving equations by taking the square root of both sides is a fundamental algebraic technique used to isolate variables in quadratic equations. This method is particularly useful when dealing with equations where the variable is squared. Our calculator simplifies this process, providing accurate solutions and explanations for each step.

How to Use This Calculator

To solve an equation by taking the square root of both sides:

  1. Enter the equation in the format x² = a where a is a positive number.
  2. Click the "Calculate" button to see the solution.
  3. Review the step-by-step solution and the graphical representation of the equation.

The calculator will display the solution in the form x = ±√a, which represents both the positive and negative roots of the equation.

When to Use This Method

This method is appropriate when:

  • The equation is quadratic (contains x²).
  • The variable is isolated on one side of the equation.
  • The other side of the equation is a positive number.

Important Note

This method only works when the equation is in the form x² = a. If the equation is more complex, additional algebraic steps may be required before applying this method.

Step-by-Step Solution

To solve the equation x² = a by taking the square root of both sides:

  1. Start with the original equation: x² = a
  2. Take the square root of both sides: √(x²) = √a
  3. Simplify the left side: |x| = √a
  4. Consider both positive and negative roots: x = ±√a

Formula Used

If x² = a, then x = ±√a where a ≥ 0.

Worked Examples

Example 1: Simple Equation

Solve x² = 16.

  1. Take the square root of both sides: √(x²) = √16
  2. Simplify: |x| = 4
  3. Find both roots: x = ±4

The solutions are x = 4 and x = -4.

Example 2: Equation with Variables

Solve x² = 9y².

  1. Take the square root of both sides: √(x²) = √(9y²)
  2. Simplify: |x| = 3|y|
  3. Find both roots: x = ±3y

The solutions are x = 3y and x = -3y.

Frequently Asked Questions

Can I use this method for any quadratic equation?

No, this method only works for equations in the form x² = a. If the equation is more complex, you may need to perform additional algebraic steps before applying this method.

Why do I get two solutions when taking the square root?

Because squaring both a positive and negative number yields the same result. For example, both 4² = 16 and (-4)² = 16 are true, so both solutions are valid.

What if the right side of the equation is negative?

If the right side is negative, the equation has no real solutions because the square of any real number is non-negative. For example, x² = -4 has no real solutions.