Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
Decide quadratic equivalence by coefficient comparison
Problem
Determine whether \(x^{2}-6x+8\) and \((x-2)(x-4)\) define the same quadratic for every \(x\). Expand the product, write \(\text{both}~\text{coefficient}~\text{triples}\) \((a,b,c)\), and base the equivalence decision on \(\text{all}~\text{three}~\text{coefficients}\) rather than \(\text{one}~\text{test}~\text{input}\).
Big Picture
What this problem is really about
Polynomial equivalence is an all-input claim, so one matching test value is not enough. We’ll rewrite both quadratics in the same standard form, compare the coefficients of every power of x, and use a complete match to establish identity.
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