\((x-2)(x^2-36)=0\)What is the sum of the positive solutions to the given equation?26814

1. INFER the solving strategy

• Given: \(\mathrm{(x-2)(x^2-36)=0}\)
• Key insight: When a product equals zero, at least one factor must equal zero
• Strategy: Apply Zero Product Property by setting each factor equal to zero

2. SIMPLIFY each factor equation

• From \(\mathrm{x - 2 = 0}\):
  • Add 2 to both sides: \(\mathrm{x = 2}\)
• From \(\mathrm{x^2 - 36 = 0}\):
  • Add 36 to both sides: \(\mathrm{x^2 = 36}\)
  • Take square root of both sides: \(\mathrm{x = ±\sqrt{36} = ±6}\)
  • So \(\mathrm{x = 6}\) or \(\mathrm{x = -6}\)

3. CONSIDER ALL CASES to find complete solution set

• All solutions to the original equation: \(\mathrm{x = -6, x = 2, x = 6}\)
• The question asks specifically for positive solutions
• Positive solutions: \(\mathrm{x = 2}\) and \(\mathrm{x = 6}\)

4. APPLY CONSTRAINTS to answer the question

• Sum only the positive solutions: \(\mathrm{2 + 6 = 8}\)

Answer: C (8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students solve \(\mathrm{x^2 - 36 = 0}\) but only consider \(\mathrm{x = 6}\), forgetting that \(\mathrm{x = -6}\) is also a solution. Even though this doesn't affect the final answer (since -6 is negative anyway), this incomplete thinking pattern can cause confusion about whether they found all solutions correctly.

This may lead them to doubt their work and guess randomly.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly find all three solutions (-6, 2, 6) but then add all of them together instead of only the positive ones. They calculate \(\mathrm{-6 + 2 + 6 = 2}\).

This may lead them to select Choice A (2).

The Bottom Line:

This problem tests whether students can systematically apply the Zero Product Property while carefully tracking which solutions meet the specified criteria. Success requires both complete algebraic execution and careful reading comprehension.