If w^2 + 12w + 36 = 64, what is one value of w?

1. TRANSLATE the problem information

• Given: \(\mathrm{w^2 + 12w + 36 = 64}\)
• Find: One value of w (as an integer)

2. INFER the most efficient approach

• Look at the left side: \(\mathrm{w^2 + 12w + 36}\)
• This matches the perfect square pattern \(\mathrm{a^2 + 2ab + b^2}\) where \(\mathrm{a = w}\) and \(\mathrm{b = 6}\)
• Key insight: \(\mathrm{w^2 + 12w + 36 = (w + 6)^2}\)
• This transforms our equation into \(\mathrm{(w + 6)^2 = 64}\)

3. SIMPLIFY using square root properties

• Take the square root of both sides: \(\mathrm{\sqrt{(w + 6)^2} = \sqrt{64}}\)
• This gives us: \(\mathrm{w + 6 = \pm 8}\)

4. CONSIDER ALL CASES from the ± solution

• Case 1: \(\mathrm{w + 6 = 8}\) → \(\mathrm{w = 2}\)
• Case 2: \(\mathrm{w + 6 = -8}\) → \(\mathrm{w = -14}\)
• Both solutions are valid integers

Answer: 2 or -14 (either answer is acceptable)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the perfect square trinomial pattern

Students often try to solve \(\mathrm{w^2 + 12w + 36 = 64}\) by moving everything to one side and getting \(\mathrm{w^2 + 12w - 28 = 0}\), then attempting to factor or use the quadratic formula. While this approach works, it's much more complex and creates more opportunities for arithmetic errors. Missing the perfect square pattern makes a 30-second problem into a 3-minute problem.

This leads to confusion with complex factoring or messy quadratic formula calculations.

Second Most Common Error:

Poor CONSIDER ALL CASES reasoning: Forgetting the ± when taking square roots

Students correctly identify \(\mathrm{(w + 6)^2 = 64}\), take the square root to get \(\mathrm{w + 6 = 8}\), solve for \(\mathrm{w = 2}\), but completely forget about the negative case \(\mathrm{w + 6 = -8}\). They submit only one answer when the problem asks for "one value" but both are equally valid.

This may lead them to miss that \(\mathrm{w = -14}\) is also a correct answer.

The Bottom Line:

Pattern recognition is crucial in algebra. When you see a quadratic expression, always check if it's a perfect square trinomial before diving into more complex solution methods. The perfect square shortcut can turn a difficult problem into a simple one.