Question:\(2(\mathrm{x} - 3)^2 - 11 = 0\)What are the solutions to the given equation?

1. INFER the approach from answer choices

• The answer choices all have the form \(\mathrm{(12 ± \sqrt{...}) / 4}\), which matches quadratic formula output
• This tells us we need to convert our equation \(\mathrm{2(x - 3)^2 - 11 = 0}\) to standard form \(\mathrm{ax^2 + bx + c = 0}\)

2. SIMPLIFY by expanding to standard form

• Expand \(\mathrm{(x - 3)^2}\): \(\mathrm{(x - 3)^2 = x^2 - 6x + 9}\)
• Substitute back: \(\mathrm{2(x^2 - 6x + 9) - 11 = 0}\)
• Distribute the 2: \(\mathrm{2x^2 - 12x + 18 - 11 = 0}\)
• SIMPLIFY by combining like terms: \(\mathrm{2x^2 - 12x + 7 = 0}\)

3. INFER the quadratic formula parameters

• From \(\mathrm{2x^2 - 12x + 7 = 0}\), we identify: \(\mathrm{a = 2, b = -12, c = 7}\)
• Note that \(\mathrm{b = -12}\) (negative twelve)

4. SIMPLIFY using the quadratic formula

• \(\mathrm{x = [-b ± \sqrt{b^2 - 4ac}] / (2a)}\)
• \(\mathrm{x = [-(-12) ± \sqrt{(-12)^2 - 4(2)(7)}] / (2(2))}\)
• \(\mathrm{x = [12 ± \sqrt{144 - 56}] / 4}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when applying the quadratic formula, particularly with \(\mathrm{b = -12}\).

Students often write \(\mathrm{x = [-12 ± \sqrt{...}]}\) instead of \(\mathrm{x = [12 ± \sqrt{...}]}\), forgetting that \(\mathrm{-b = -(-12) = +12}\). This leads them to select Choice A \(\mathrm{((-12 ± \sqrt{144 - 56}) / 4)}\).

Second Most Common Error:

Poor SIMPLIFY execution: Calculation errors when computing the discriminant \(\mathrm{b^2 - 4ac}\).

Students might calculate \(\mathrm{(-12)^2 - 4(2)(7)}\) incorrectly, getting \(\mathrm{144 + 56 = 200}\) instead of \(\mathrm{144 - 56 = 88}\), or \(\mathrm{144 + 16 = 160}\). This leads them to select Choice B \(\mathrm{((12 ± \sqrt{144 + 16}) / 4)}\) or Choice D \(\mathrm{((12 ± \sqrt{144 + 56}) / 4)}\).

The Bottom Line:

This problem tests careful algebraic manipulation and precise application of the quadratic formula. The vertex form starting point requires students to recognize the need for standard form conversion, then execute multiple algebraic steps without arithmetic errors.