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Variable Calculator with Square Root

Reviewed by Calculator Editorial Team

This variable calculator with square root helps you solve equations where variables are under square roots. Whether you're solving for x in √(x + 5) = 7 or working with more complex expressions, this tool provides accurate results and explains the steps involved.

What is a Variable Calculator with Square Root?

A variable calculator with square root is a specialized tool designed to solve equations where variables appear under square root signs. These calculators are particularly useful in algebra, physics, engineering, and other scientific fields where square root functions are common.

The calculator handles equations of the form √(ax + b) = c, where a, b, and c are known constants, and x is the variable to solve for. It eliminates the square root by squaring both sides of the equation, then isolates the variable term.

This calculator assumes the square root is defined (i.e., the expression inside the square root is non-negative) and that the equation has a real solution.

How to Use This Calculator

  1. Enter the coefficients for the terms inside the square root in the designated fields.
  2. Enter the value on the right side of the equation.
  3. Click "Calculate" to solve for the variable.
  4. Review the result and the step-by-step solution provided.

The calculator will display the solution to the equation and show the steps taken to arrive at that solution, including any necessary algebraic manipulations.

Formulas and Assumptions

The calculator uses the following formula to solve equations of the form √(ax + b) = c:

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

Assumptions:

  • The equation must have a real solution, which requires that c² - b ≥ 0.
  • The coefficient a must not be zero.
  • The calculator assumes the square root function is defined (i.e., ax + b ≥ 0).

Worked Examples

Example 1

Solve √(3x + 4) = 5

Solution:

  1. Square both sides: 3x + 4 = 25
  2. Subtract 4: 3x = 21
  3. Divide by 3: x = 7

Result: x = 7

Example 2

Solve √(2x - 1) = 3

Solution:

  1. Square both sides: 2x - 1 = 9
  2. Add 1: 2x = 10
  3. Divide by 2: x = 5

Result: x = 5

Frequently Asked Questions

What types of equations can this calculator solve?

This calculator can solve equations of the form √(ax + b) = c, where a, b, and c are constants, and x is the variable to solve for.

What if the equation has no real solution?

The calculator will display an error message if the equation has no real solution, which occurs when c² - b is negative.

Can I solve equations with variables under multiple square roots?

This calculator is designed for equations with a single square root. For more complex equations, consider using a more advanced algebraic solver.

Is the solution always exact?

Yes, the solution provided by this calculator is exact, assuming the input values are exact and the equation has a real solution.