61.5 Formula Solution Verification

Formula Solution Verification is the set of checks applied after solving a quadratic equation with the quadratic formula, confirming that both signed branches were computed correctly and that the resulting roots genuinely satisfy the original equation.

Formula-Derived Root Equation Check

Returning to the Original Equation

Verification begins by returning to the original quadratic equation, in the exact form it was originally presented, before any preparation or coefficient extraction took place.

Why the Original Form Is the Reference Point

Using the original equation as the reference ensures that verification catches any error introduced at any stage of the process, including coefficient extraction and discriminant calculation, not only errors within the formula's final arithmetic.

Formula Root Substitution Check

Substituting Each Root Back In

Each root produced by the formula is substituted back into the original equation, replacing the variable everywhere it appears, and both sides are simplified independently.

Confirming Agreement after Substitution

If the simplified result equals zero for a given root, that root is confirmed valid; if it does not, an error occurred somewhere between coefficient extraction and the final formula computation for that particular root.

Positive Branch Verification

Isolating the Positive Branch for Testing

The root produced by the positive branch of the formula is tested on its own, separately from the negative branch, to confirm it independently satisfies the original equation.

Why Separate Testing of This Branch Matters

Because the two branches are calculated through nearly identical but distinct arithmetic, an error confined to just the addition step would affect only this branch, and testing it separately is what allows such a limited error to be detected.

Negative Branch Verification

Isolating the Negative Branch for Testing

The root produced by the negative branch of the formula is tested on its own, separately from the positive branch, to confirm it independently satisfies the original equation.

Why Separate Testing of This Branch Matters

As with the positive branch, an error confined to the subtraction step of this branch alone would otherwise go unnoticed if only the other branch were checked, making independent verification of each branch necessary.

Repeated Formula Root Check

Confirming a Zero Discriminant Produces One Value

When the discriminant was classified as zero, this check confirms that the positive and negative branches indeed collapse into the identical value, consistent with the one-repeated-root classification made earlier.

Why This Special Case Deserves Its Own Check

Since a discrepancy between the two branches in this case would contradict the earlier discriminant classification, confirming their equality serves as a cross-check between the classification step and the final computed result.

Verified Formula Solution Set

The Final Confirmed Answer

The verified solution set is the collection of roots that has passed both branch verification checks, consistent in count and nature with the classification already established by the discriminant.

Why Full Verification Matters

Because the quadratic formula involves several distinct arithmetic operations performed on shared values, confirming both branches independently against the original equation provides a reliable guarantee that neither branch carried forward an unnoticed error from substitution, simplification, or reduction.