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ALGEBRA / RADICAL EQUATIONS

Radical Equation Solver with Step-by-Step Working and Extraneous-Root Checks

Updated By Unit Conversion Plus

Solve equations in the form √(ax + b) = cx + d, then substitute every candidate back into the original equation to reject extraneous roots.

Enter the equation

√(ax + b) = cx + d

The square root is the principal, nonnegative root. Coefficients may be integers or decimals.

Why radical equations need verification

Squaring both sides removes a square root but is not a reversible step in every case. It can create a candidate for which the right side is negative or the original equation is false. Such a candidate is called an extraneous root.

How the solver works

Squaring √(ax + b) = cx + d produces ax + b = (cx + d)², usually a quadratic equation. The calculator solves that equation, checks that the radicand is nonnegative and compares both sides of the original equation.

Special cases

If c is zero, the squared equation may be linear. The original equation can have one solution, no solution or—when both sides describe the same constant relationship—infinitely many solutions over an allowed domain.

Frequently asked questions

What is an extraneous solution?

It satisfies an equation produced during the working but fails when substituted into the original equation.

Why must the right side be nonnegative?

The principal square root is always nonnegative, so it cannot equal a negative real number.