56.3 Single Even-Root Equations

Single Even-Root Equations are radical equations containing exactly one radical with an even index, most commonly a square root, solved by isolating that radical, squaring both sides to eliminate it, solving the resulting polynomial equation, and then checking every candidate solution against the original equation to reject any extraneous root the squaring step may have introduced. This is the foundational case of radical equation solving, and the mandatory final check exists specifically because squaring both sides of an equation is not a fully reversible operation in the way addition or multiplication by a nonzero constant are.

Because squaring eliminates any sign information a quantity carried before being squared, a candidate that fails when substituted back into the original unsquared equation must be rejected even though it correctly satisfies the squared polynomial equation.

Preparing the Equation

Isolated Square-Root Form

Following radical equation preparation, the equation is arranged so that the single square-root expression stands completely alone on one side, with every other term on the opposite side.

Nonnegative Opposite-Side Requirement

The side of the equation opposite the isolated radical is checked for its own sign; since a square root always represents a nonnegative principal value, this opposite side must also be nonnegative for any real solution to exist, and a negative value there signals immediately that the equation has no solution.

Eliminating the Radical

Isolated Radical Squaring

Both sides of the prepared equation are squared, matching the radical's index, which eliminates the square root on the isolated side entirely and leaves a polynomial expression on the opposite side.

Post-Squaring Polynomial Equation

The result of squaring is a polynomial equation, free of any radical, ready to be solved using ordinary linear or quadratic equation-solving techniques.

Solving and Screening

Post-Squaring Candidate Set

The polynomial equation is solved, producing one or more candidate solutions, exactly as any linear or quadratic equation would be solved on its own.

Original Radical Candidate Check

Every candidate obtained from the polynomial equation is substituted back into the original, unsquared radical equation, confirming it produces a true statement; this is the mandatory step that catches any extraneous root introduced by the squaring process, since squaring can make a false statement, involving a mismatched sign, appear true.

Presenting the Final Answer

Accepted Radical Solution Set

Only the candidates that pass the original-equation substitution check are included in the final solution set; any candidate that fails, despite satisfying the squared polynomial equation, is rejected as extraneous, and if no candidate survives the check, the equation is reported as having no solution.