59.5 Quadratic Factoring Verification

Quadratic Factoring Verification ensures correct factorization by checking if the product of factors matches the original quadratic expression.

Quadratic Factoring Verification is the set of checks applied after solving a quadratic equation by factoring, confirming that both the factorization itself and the resulting roots are correct. It closes the factoring process by tracing the solution back to the original equation, ensuring no error was introduced during preparation, factoring, or root resolution.

Original Quadratic Equation Reuse

Returning to the Starting Point

Verification begins by returning to the original quadratic equation, before any preparation, consolidation, or factoring was applied, since this is the statement the final roots must actually satisfy.

Original Equation \equiv Statement the roots must satisfy

Why the Original, Not the Prepared Form, Is Used

Using the original equation rather than the prepared or factored form ensures the verification catches errors that might have occurred anywhere in the process, including during the earlier preparation steps, not only during factoring itself.

Root-by-Root Substitution

Testing Each Root Individually

Each root found through the zero-product solution is substituted back into the original equation, one at a time, replacing the variable everywhere it appears.

a p^{2} + b p + c tested for root p

Why Each Root Is Tested Separately

Because each root arose from a separate factor equation, an error affecting only one of the factors would produce one correct root and one incorrect root; testing each root individually is what allows such a partial error to be detected.

Quadratic Side Agreement

Confirming Both Sides Match

After substitution, both sides of the original equation are simplified independently and compared, confirming that they produce the same numerical value.

Left Side = Right Side

What Agreement Confirms

Agreement between both sides confirms that the tested value genuinely satisfies the original equation as written, independent of whatever factoring steps were used to find that value in the first place.

Factored Product Reconstruction

Redistributing the Factors

Separately from root substitution, the factored expression itself is verified by multiplying its factors back together and confirming the result matches the prepared standard-form expression.

(x - p) (x - q) = a x^{2} + b x + c

Distinguishing This Check from Root Substitution

Reconstructing the factored product checks the factoring step directly, at the level of the expression itself, while root substitution checks the solving step indirectly, at the level of the final numerical answers; the two checks catch different kinds of errors.

Missing Quadratic Root Check

Confirming No Root Was Overlooked

This check counts the number of distinct factors produced during factoring and confirms that a root was resolved from every one of them, guarding against a factor being skipped during the zero-product step.

Special Attention for Repeated Factors

This check is applied carefully for equations with a repeated factor, confirming that the single resulting root is intentional and not the result of accidentally overlooking a second, distinct factor that should have been present.

Verified Quadratic Solution Set

The Final Confirmed Answer

The verified solution set is the collection of roots that has passed both the root-by-root substitution check and the missing-root check, together with a factorization that has passed the product reconstruction check.

Why Full Verification Matters

Because factoring involves several sequential steps, each depending on the correctness of the one before it, a single unnoticed error early in the process can silently produce an incorrect final answer; full verification is what confirms the entire chain of steps was carried out correctly.