Which of the following is a solution to the given equation?w^2 + 12w - 40 = 0

1. TRANSLATE the problem information

• Given equation: \(\mathrm{w^2 + 12w - 40 = 0}\)
• Need to find: which answer choice is a solution

2. INFER the solution approach

• This is a quadratic equation that can be solved by completing the square
• The coefficient of w is 12, so we'll need to add \(\mathrm{(12/2)^2 = 36}\) to complete the square

3. SIMPLIFY by rearranging and completing the square

• Add 40 to both sides: \(\mathrm{w^2 + 12w = 40}\)
• Add 36 to both sides to complete the square: \(\mathrm{w^2 + 12w + 36 = 76}\)
• Factor the left side: \(\mathrm{(w + 6)^2 = 76}\)

4. SIMPLIFY by taking square roots

• Take square root of both sides: \(\mathrm{w + 6 = ±\sqrt{76}}\)
• Simplify \(\mathrm{\sqrt{76} = \sqrt{4 × 19} = 2\sqrt{19}}\)
• So: \(\mathrm{w + 6 = ±2\sqrt{19}}\)

5. SIMPLIFY to solve for w

• Subtract 6 from both sides: \(\mathrm{w = -6 ± 2\sqrt{19}}\)
• This gives two solutions: \(\mathrm{w = -6 + 2\sqrt{19}}\) and \(\mathrm{w = -6 - 2\sqrt{19}}\)

6. INFER which solution matches the choices

• Compare with answer choices - only \(\mathrm{-6 + 2\sqrt{19}}\) appears as Choice D

Answer: D. \(\mathrm{-6 + 2\sqrt{19}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make calculation errors when completing the square, particularly in determining what constant to add.

Common mistake: Adding \(\mathrm{(12)^2 = 144}\) instead of \(\mathrm{(12/2)^2 = 36}\), or making sign errors when rearranging terms. They might also incorrectly simplify \(\mathrm{\sqrt{76}}\), perhaps writing \(\mathrm{\sqrt{76} = 4\sqrt{19}}\) instead of \(\mathrm{2\sqrt{19}}\).

This leads to incorrect solutions that don't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Inadequate INFER reasoning: Students attempt to factor the quadratic directly without recognizing it doesn't factor nicely with integers.

They might try to find factors of -40 that add to 12, get frustrated when simple factoring fails, and either guess or abandon the systematic approach. Some may remember the quadratic formula exists but not feel confident using it.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

Completing the square requires precise algebraic manipulation and careful attention to signs. Students who haven't mastered this technique often struggle with the multi-step process and make computational errors that derail their solution.