x^2 - ax + 12 = 0 In the equation above, a is a constant and a gt 0. If...

1. INFER the key relationship

• Given: \(\mathrm{x^2 - ax + 12 = 0}\) has two integer solutions, and \(\mathrm{a \gt 0}\)
• Key insight: If a quadratic has integer solutions, it can be factored using integers
• If the solutions are integers r and s, then: \(\mathrm{(x - r)(x - s) = 0}\)

2. TRANSLATE the factored form

• Expanding \(\mathrm{(x - r)(x - s)}\): \(\mathrm{x^2 - (r + s)x + rs}\)
• Comparing with \(\mathrm{x^2 - ax + 12 = 0}\):
  • Coefficient of x: \(\mathrm{-(r + s) = -a}\) → \(\mathrm{r + s = a}\)
  • Constant term: \(\mathrm{rs = 12}\)

3. CONSIDER ALL CASES for integer factor pairs

• We need all integer pairs (r, s) where \(\mathrm{rs = 12}\):

Positive factor pairs:

• \(\mathrm{r = 1, s = 12}\) → sum = 13
• \(\mathrm{r = 2, s = 6}\) → sum = 8
• \(\mathrm{r = 3, s = 4}\) → sum = 7

Negative factor pairs:

• \(\mathrm{r = -1, s = -12}\) → sum = -13
• \(\mathrm{r = -2, s = -6}\) → sum = -8
• \(\mathrm{r = -3, s = -4}\) → sum = -7

4. APPLY CONSTRAINTS to select valid solutions

• Since \(\mathrm{a \gt 0}\) and \(\mathrm{r + s = a}\), we need \(\mathrm{r + s \gt 0}\)
• Valid sums: 7, 8, 13
• Therefore: \(\mathrm{a = 7, 8, or 13}\)

Answer: 7, 8, or 13

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "integer solutions" to "integer factorization." They might try to use the quadratic formula or other methods without recognizing that integer solutions mean the quadratic must factor nicely with integer coefficients.

This leads to confusion about how to approach the problem systematically, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students find some factor pairs of 12 but miss others, particularly the negative pairs. They might only consider positive factors like (3,4) and conclude \(\mathrm{a = 7}\) is the only answer.

This causes them to provide an incomplete answer, missing valid options like \(\mathrm{a = 8}\) or \(\mathrm{a = 13}\).

The Bottom Line:

This problem requires recognizing that "integer solutions" is the key that unlocks a systematic factorization approach, rather than trying to solve the quadratic directly. The constraint \(\mathrm{a \gt 0}\) then filters which factor pairs are actually valid.