60.5 Square Completion Verification

Square Completion Verification is a method to confirm the correctness of completing the square in quadratic equations, ensuring algebraic accuracy and proper form.

Square Completion Verification is the set of checks applied after solving a quadratic equation by completing the square, confirming that the completion term was calculated correctly, that the rewritten binomial genuinely matches the trinomial it replaced, and that the final roots satisfy the original equation.

Completion Term Recalculation

Recomputing the Added Constant

Verification begins by independently recalculating the completion term, halving the linear coefficient and squaring the result a second time, and confirming this recalculated value matches the one originally added to both sides.

{(\frac{b}{2})}^{2} recalculated independently

Why This Recalculation Matters

Because the entire completion process depends on this single value being correct, an error here would propagate through every subsequent step; recalculating it independently is a direct way to catch such an error at its source.

Perfect-Square Expansion Check

Expanding the Rewritten Binomial

The binomial that replaced the trinomial is expanded back out, multiplying it by itself, to confirm that this expansion reproduces the exact trinomial from before the rewrite.

{(x + \frac{b}{2})}^{2} = x^{2} + b x + {(\frac{b}{2})}^{2}

Why This Check Is Distinct from Recalculation

While recalculating the completion term checks the value added to the equation, expanding the binomial checks the separate claim that this trinomial can indeed be written as that binomial squared; both checks address different points where an error could occur.

Equation Balance Check

Confirming Both Sides Received the Same Addition

This check confirms that the completion term added during preparation was applied identically to both sides of the equation, preserving the original equation's balance.

Why an Imbalance Would Be a Serious Error

If the completion term had been added to only one side, the resulting equation would no longer be equivalent to the original, meaning any roots found afterward would not actually solve the equation that was originally given.

Original Equation Substitution Check

Testing the Final Roots

Each root found through the completion method is substituted back into the original, unprepared quadratic equation, confirming that both sides of that original equation match once the substitution is simplified.

a x^{2} + b x + c = 0 tested with the found root

Why the Original Form Is Used for This Check

Testing against the original equation, rather than any of the intermediate prepared forms, ensures that the entire chain of preparation and solving steps produced a root that is valid for the equation as it was initially stated.

Root-Branch Completeness Check

Confirming Both Branches Were Carried Through

This check confirms that both the positive and negative square-root branches were followed all the way through to a final isolated value, unless the right-side value was exactly zero, in which case only a single repeated root is expected.

Guarding Against a Dropped Branch

Because the two branches are solved through nearly identical steps, it is possible to complete one branch and mistakenly treat it as the entire solution; this check specifically guards against that oversight.

Verified Completion Solution Set

The Final Confirmed Answer

The verified solution set is the collection of roots that has passed the substitution check, drawn from a completion process whose completion term, binomial expansion, and equation balance have all been independently confirmed.

Why Full Verification Matters

Because completing the square involves several distinct calculated values built one upon another, confirming each of them separately, rather than only checking the final roots, provides a more reliable guarantee that no single miscalculation was carried silently through the entire process.