Problem

Compare \(x^2~-~5x~+~6\) with \((x~-~2)(x~-~3)\) for identifying the zeros. State the preferred form, the zeros it reveals, and why the other form requires more work.

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What this problem is really about

Equivalent forms can emphasize different features of the same quadratic. When the goal is to read the zeros, factored form is the efficient representation because each zero comes directly from one factor. Expanded form is still useful, but it requires another algebraic step before those intercepts become visible.

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Neither form is preferred because neither directly reveals a zero; both require completing the square before any zero can be identified.

The expanded form x²−5x+6 is preferred because its constant 6 directly reveals zeros x=2 and x=3; the factored form hides both zeros.

The factored form (x−2)(x−3) is preferred because setting each factor to zero directly reveals x=2 and x=3; the expanded form requires factoring or another solving method.

The factored form (x−2)(x−3) is preferred because it directly reveals zeros x=−2 and x=−3; the expanded form reveals x=2 and x=3.

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Compare \(x^2~-~5x~+~6\) with \((x~-~2)(x~-~3)\) for identifying the zeros. State the preferred form, the zeros it reveals, and why the other form requires more work.

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