56.6 Radical Solution Verification

Radical Solution Verification is the collection of checks used to confirm that a candidate solution to a radical equation is genuinely valid, centering on direct substitution into the original, unaltered equation as the definitive test, since this substitution alone reliably distinguishes a true solution from an extraneous one introduced by the power-raising step. Because raising both sides of an equation to a power can create false solutions, particularly with even roots, this verification treats the original equation, not the derived polynomial equation, as the ultimate authority on which candidates are genuinely correct.

These checks apply uniformly across single-radical, repeated-radical, odd-root, and rational-power equations, since every one of these cases relies on the same underlying isolation-and-power technique that can introduce the same category of extraneous solution.

The Central Verification Step

Original Radical Equation Reuse

The original equation, exactly as it was first written before any isolation or power-raising was applied, is retrieved and used as the basis for verification, rather than any intermediate cleared or squared version of the equation.

Candidate-by-Candidate Substitution

Each candidate solution produced by solving the cleared polynomial equation is substituted individually into this original equation, evaluated using ordinary arithmetic on both sides.

Evaluating the Substituted Sides

Principal Root Value Check

When evaluating the radical side of the substituted equation, the principal root convention is applied, taking the nonnegative root for an even index, ensuring the evaluated value matches the same convention the original equation was built on.

Radical Equation Side Agreement

The two sides of the substituted equation, one now a numerical radical value and the other a numerical constant or expression, are compared for exact agreement; only a candidate producing a true numerical statement here is accepted.

Confirming the Domain

Radical Domain Condition Recheck

Independently of the substitution check, each candidate is also checked against the original domain condition determined for the equation's radicand, confirming the candidate does not violate the nonnegativity requirement that an even-indexed radical imposes.

Identifying and Excluding Bad Candidates

Power-Generated Extraneous Candidate

A candidate that satisfies the cleared polynomial equation but fails the original-equation substitution check is classified as extraneous, a false solution generated specifically by the loss of sign information during the power-raising step, and it is excluded from the final answer regardless of how cleanly it emerged from the polynomial solving step.

Reporting the Outcome

Radical No-Real-Solution Outcome

If every candidate produced by solving the cleared equation turns out to be extraneous, the radical equation is reported as having no real solution, even though the cleared polynomial equation itself may have been solvable.

Radical Solution Completeness Check

The final accepted solution set is checked to confirm it includes every candidate that passed the original-equation substitution, and excludes every candidate that did not, ensuring the reported answer is neither missing a valid solution nor retaining an extraneous one.