\((\mathrm{w}^2 - 4\mathrm{w} + 4) \cdot 3 = 12\) If w is a solution to the given equation, which of...

1. INFER the structure of the quadratic expression

• Given: \((\mathrm{w}^2 - 4\mathrm{w} + 4) \cdot 3 = 12\)
• Key insight: Look at \(\mathrm{w}^2 - 4\mathrm{w} + 4\) carefully. This follows the pattern \(\mathrm{a}^2 - 2\mathrm{ab} + \mathrm{b}^2\) where \(\mathrm{a} = \mathrm{w}\) and \(\mathrm{b} = 2\)
• This means \(\mathrm{w}^2 - 4\mathrm{w} + 4 = (\mathrm{w} - 2)^2\)

2. SIMPLIFY by substituting the factored form

• Replace the trinomial: \((\mathrm{w} - 2)^2 \cdot 3 = 12\)
• Divide both sides by 3: \((\mathrm{w} - 2)^2 = 4\)

3. SIMPLIFY further by taking square roots

• Take the square root of both sides: \(\mathrm{w} - 2 = \pm 2\)
• This gives us two possibilities: \(\mathrm{w} - 2 = 2\) or \(\mathrm{w} - 2 = -2\)

4. CONSIDER ALL CASES and match to answer choices

• The question asks for "a possible value of \(\mathrm{w} - 2\)"
• From our work: \(\mathrm{w} - 2\) could be 2 or -2
• Looking at choices: 2 appears as choice (C)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the perfect square trinomial pattern in \(\mathrm{w}^2 - 4\mathrm{w} + 4\)

Students might try to use the quadratic formula on the original equation or attempt to expand everything out, leading to unnecessarily complex calculations. Without seeing the factorization shortcut, they may get bogged down in algebra or make computational errors.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate CONSIDER ALL CASES execution: Forgetting that square roots produce both positive and negative solutions

Students correctly factor to get \((\mathrm{w} - 2)^2 = 4\), but when taking the square root, they only consider \(\mathrm{w} - 2 = 2\) and miss \(\mathrm{w} - 2 = -2\). While this doesn't affect the final answer since 2 is among the choices, it reflects incomplete mathematical reasoning.

This might cause them to doubt their answer or second-guess their work.

The Bottom Line:

The key to this problem is pattern recognition. Students who immediately spot the perfect square trinomial can solve this in just a few steps, while those who don't recognize the pattern face much more complex algebra.