53.5 Factorable Quadratic Outcomes

Factorable Quadratic Outcomes are rational equations that, once every denominator has been cleared using the equation-wide LCD, reduce to a quadratic polynomial equation whose standard form can be factored into two binomial factors, allowing the zero-product property to generate candidate solutions. This is the more elaborate of the two outcome types a rational equation can produce after clearing, since it requires the trinomial or special-pattern factoring techniques already established, applied to the cleared equation rather than directly to the original rational expressions.

Because factoring can produce two distinct candidates rather than the single candidate a linear outcome provides, this outcome type requires the domain-screening step to be applied to each candidate independently before either one is accepted.

Recognizing the Outcome

Cleared Quadratic Outcome

After the LCD has been distributed across every term of the original rational equation, the resulting equation is examined and found to contain a squared variable term, classifying it as a quadratic rational equation rather than a linear one.

Preparing to Factor

Rational Quadratic Standard Form

The cleared equation is rearranged so that every term is moved to one side, leaving zero on the other, and arranged in the standard descending-degree order expected before factoring begins.

Cleared-Equation GCF Extraction

Before attempting trinomial factoring, the standard-form equation is checked for an overall greatest common factor across all its terms, extracting any such factor first exactly as in ordinary trinomial factoring preparation.

Factoring the Quadratic

Factorable Quadratic Reduction

The standard-form quadratic is factored into two binomial factors using the monic or nonmonic trinomial factoring technique, or a special factoring pattern if the structure matches one, exactly as covered in trinomial and special-pattern factoring.

Generating the Candidates

Zero-Product Candidate Set

With the equation factored and set equal to zero, the zero-product property is applied, setting each binomial factor equal to zero individually and solving, producing a set of up to two candidate solutions.

Screening Every Candidate

Quadratic Candidate Domain Screening

Each of the candidate solutions generated by the zero-product property is checked individually against the domain restriction determined during equation restriction preparation, since either, both, or neither candidate may match an originally excluded value.

Quadratic Candidate Substitution Check

Each candidate that survives the domain screening is substituted directly into the original, uncleared rational equation, confirming it produces a true statement before being finally accepted.

Reaching the Final Answer

Accepted Quadratic Solutions

Every candidate that passes both the domain screening and the substitution check is included in the final solution set; since factoring can produce up to two distinct candidates, the final solution set may contain zero, one, or two valid solutions, depending on how many candidates survive both checks.