x^2 + x - 12 = 0 If a is a solution of the equation above and a gt 0,...

1. INFER the solution strategy

• Given: \(\mathrm{x^2 + x - 12 = 0}\) and we need the positive solution
• Since this is a quadratic equation in standard form, factoring will reveal the solutions most efficiently
• We need to find two numbers that multiply to \(\mathrm{-12}\) and add to \(\mathrm{+1}\)

2. SIMPLIFY through factoring

• Test factor pairs of \(\mathrm{-12}\):
  • Try \(\mathrm{-3}\) and \(\mathrm{4}\): \(\mathrm{(-3) \times 4 = -12}\) ✓ and \(\mathrm{(-3) + 4 = 1}\) ✓
• Rewrite: \(\mathrm{x^2 + x - 12 = (x + 4)(x - 3) = 0}\)

3. INFER solutions from factored form

• Using zero product property: if \(\mathrm{(x + 4)(x - 3) = 0}\), then:
  • \(\mathrm{x + 4 = 0}\) → \(\mathrm{x = -4}\)
  • \(\mathrm{x - 3 = 0}\) → \(\mathrm{x = 3}\)

4. APPLY CONSTRAINTS to select final answer

• We have two solutions: \(\mathrm{x = -4}\) and \(\mathrm{x = 3}\)
• Since the problem states \(\mathrm{a \gt 0}\), we need the positive solution
• Therefore: \(\mathrm{a = 3}\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students struggle to find the correct factor pair for \(\mathrm{-12}\) that also sums to \(\mathrm{1}\). They might try obvious pairs like \(\mathrm{1}\) and \(\mathrm{-12}\) or \(\mathrm{2}\) and \(\mathrm{-6}\), but miss that \(\mathrm{-3}\) and \(\mathrm{4}\) work. This leads to incorrect factoring attempts like \(\mathrm{(x + 1)(x - 12)}\) or \(\mathrm{(x + 2)(x - 6)}\), producing wrong solutions entirely.

Second Most Common Error:

Forgetting to APPLY CONSTRAINTS: Students correctly factor to get \(\mathrm{(x + 4)(x - 3) = 0}\) and find both solutions \(\mathrm{x = -4}\) and \(\mathrm{x = 3}\), but then forget about the condition \(\mathrm{a \gt 0}\). They might report both solutions or randomly pick one, leading to the wrong final answer of \(\mathrm{-4}\).

The Bottom Line:

This problem tests whether students can systematically work through factoring (finding the right number pair) and remember to apply given constraints to their final answer selection. The factoring requires patience and organization, while the constraint application requires careful reading.