Which of the following is a solution to the given equation? 3x^2 - x - 2 = 0...

1. INFER the solution approach

• Given: \(\mathrm{3x^2 - x - 2 = 0}\)
• This is a quadratic equation in standard form
• Since we need to find solutions and have answer choices, factoring is the most efficient approach
• For factoring by grouping, I need two numbers that multiply to \(\mathrm{ac = (3)(-2) = -6}\) and add to \(\mathrm{b = -1}\)

2. INFER the key factoring numbers

• What two numbers multiply to \(\mathrm{-6}\) and add to \(\mathrm{-1}\)?
• Try different factor pairs of \(\mathrm{-6}\):
  • 6 and -1: \(\mathrm{(6) + (-1) = 5}\) ✗
  • -6 and 1: \(\mathrm{(-6) + (1) = -5}\) ✗
  • 3 and -2: \(\mathrm{(3) + (-2) = 1}\) ✗
  • -3 and 2: \(\mathrm{(-3) + (2) = -1}\) ✓
• The numbers are -3 and 2

3. SIMPLIFY by rewriting and factoring

• Rewrite the middle term using -3 and 2:
  \(\mathrm{3x^2 - x - 2 = 3x^2 - 3x + 2x - 2}\)
• Factor by grouping:
  \(\mathrm{3x(x - 1) + 2(x - 1) = 0}\)
  \(\mathrm{(3x + 2)(x - 1) = 0}\)

4. SIMPLIFY using zero product property

• If \(\mathrm{(3x + 2)(x - 1) = 0}\), then:
  \(\mathrm{3x + 2 = 0}\) or \(\mathrm{x - 1 = 0}\)
• Solving each equation:
  \(\mathrm{3x = -2}\) → \(\mathrm{x = -\frac{2}{3}}\)
  \(\mathrm{x = 1}\)

5. INFER which solution matches the choices

• The solutions are \(\mathrm{x = -\frac{2}{3}}\) and \(\mathrm{x = 1}\)
• Looking at the answer choices: only \(\mathrm{-\frac{2}{3}}\) appears as option (B)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students struggle to find the correct pair of numbers that multiply to \(\mathrm{ac}\) and add to \(\mathrm{b}\). They might try random factor pairs without systematic thinking, or incorrectly calculate that they need numbers multiplying to \(\mathrm{-2}\) (just the constant term) instead of \(\mathrm{ac = -6}\).

This leads to incorrect factoring attempts and confusion, causing them to abandon the systematic approach and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the correct numbers (-3 and 2) but make algebraic errors during the factoring by grouping process. They might incorrectly distribute when factoring out common terms, or make sign errors when solving \(\mathrm{3x + 2 = 0}\).

This may lead them to select Choice A (\(\mathrm{-\frac{3}{2}}\)) if they incorrectly solve \(\mathrm{3x + 2 = 0}\) as \(\mathrm{x = -\frac{3}{2}}\) instead of \(\mathrm{x = -\frac{2}{3}}\).

The Bottom Line:

This problem tests whether students can systematically factor quadratics using the ac method. Success requires both strategic thinking (finding the right number pair) and careful algebraic execution.