Square Root Calculator with Variables on Both Sides Worksheet

This worksheet helps you solve equations where square roots appear on both sides. Whether you're studying algebra or preparing for exams, this tool provides step-by-step guidance and an interactive calculator to help you master solving equations with square roots.

How to Use This Calculator

To solve an equation with square roots on both sides:

Enter the equation in the format √(a + bx) = √(c + dx)
Input the coefficients a, b, c, and d
Click "Calculate" to solve the equation
Review the step-by-step solution and the final answer

Note: This calculator assumes the equation is solvable and that the square roots are defined (non-negative radicands).

The Formula Explained

The general form of the equation we're solving is:

√(a + bx) = √(c + dx)

To solve this equation:

Square both sides to eliminate the square roots: a + bx = c + dx
Rearrange terms to isolate the variable: bx - dx = c - a
Factor out the variable: x(b - d) = c - a
Solve for x: x = (c - a)/(b - d)

This gives us the solution to the original equation.

Worked Examples

Example 1

Solve √(5 + 3x) = √(2 + x)

Square both sides: 5 + 3x = 2 + x
Rearrange: 3x - x = 2 - 5
Simplify: 2x = -3
Solution: x = -1.5

Example 2

Solve √(8 + 2x) = √(3 + x)

Square both sides: 8 + 2x = 3 + x
Rearrange: 2x - x = 3 - 8
Simplify: x = -5

Frequently Asked Questions

What if the equation has negative radicands?: The square root function is only defined for non-negative numbers. If either radicand is negative, the equation has no real solution.
Can I solve equations with more complex expressions under the square roots?: This calculator handles linear expressions under square roots. For more complex cases, you may need to use algebraic techniques beyond this tool's scope.
What if the coefficients make the denominator zero?: If b - d = 0, the equation either has infinitely many solutions (if c - a = 0) or no solution (if c - a ≠ 0).