Question:x^2 - 3x - 10 = 0What is the positive solution to the given equation?

1. TRANSLATE the problem information

• Given: \(\mathrm{x^2 - 3x - 10 = 0}\)
• Find: The positive solution

2. INFER the solution approach

• This is a quadratic equation that can be solved by factoring
• We need to find two numbers that multiply to give the constant term (\(\mathrm{-10}\)) and add to give the coefficient of the middle term (\(\mathrm{-3}\))

3. SIMPLIFY by factoring the quadratic

• Look for factor pairs of \(\mathrm{-10}\): \(\mathrm{(-1, 10), (1, -10), (-2, 5), (2, -5)}\)
• Check which pair adds to \(\mathrm{-3}\):
  • \(\mathrm{(-2) + 5 = 3}\) ✗
  • \(\mathrm{2 + (-5) = -3}\) ✓
• Factor: \(\mathrm{x^2 - 3x - 10 = (x + 2)(x - 5) = 0}\)

4. APPLY the zero product property

• If \(\mathrm{(x + 2)(x - 5) = 0}\), then either:
  • \(\mathrm{x + 2 = 0}\), so \(\mathrm{x = -2}\)
  • \(\mathrm{x - 5 = 0}\), so \(\mathrm{x = 5}\)

5. APPLY CONSTRAINTS to select the final answer

• Since we need the positive solution: \(\mathrm{x = 5}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students struggle to find the correct factor pair for \(\mathrm{-10}\) that adds to \(\mathrm{-3}\). They might try random combinations or get confused about the signs, leading to incorrect factoring like \(\mathrm{(x - 2)(x + 5) = x^2 + 3x - 10}\) instead of the correct \(\mathrm{(x + 2)(x - 5)}\).

This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly factor and find both solutions (\(\mathrm{x = -2}\) and \(\mathrm{x = 5}\)), but then select the negative solution or get confused about which one the question is asking for.

This may lead them to select Choice (E) (-5) if they misread their work, or cause uncertainty between the two solutions they found.

The Bottom Line:

The key challenge is systematically finding the right factor pair and keeping track of positive vs. negative signs throughout the factoring process. Students who rush through factoring or don't double-check their factor pairs often make sign errors that cascade through the rest of the solution.