x^2 + mx + 9 = 0 In the equation above, m is a constant. If the equation has two...

1. TRANSLATE the problem information

• Given: \(\mathrm{x^2 + mx + 9 = 0}\) where m is a constant
• Need: Value of m that gives two distinct real solutions
• Answer choices: -8, -6, 4, 6

2. INFER the mathematical condition needed

• For any quadratic \(\mathrm{ax^2 + bx + c = 0}\) to have two distinct real solutions, the discriminant \(\mathrm{b^2 - 4ac}\) must be positive
• In our equation: \(\mathrm{a = 1, b = m, c = 9}\)

3. SIMPLIFY the discriminant inequality

• Set up: \(\mathrm{m^2 - 4(1)(9) \gt 0}\)
• Simplify: \(\mathrm{m^2 - 36 \gt 0}\)
• Rearrange: \(\mathrm{m^2 \gt 36}\)

4. APPLY CONSTRAINTS to interpret the solution

• Taking square root: \(\mathrm{|m| \gt 6}\)
• This means: \(\mathrm{m \gt 6}\) OR \(\mathrm{m \lt -6}\)
• Check each choice:
  • (A) -8: Since \(\mathrm{-8 \lt -6}\) ✓
  • (B) -6: Gives \(\mathrm{discriminant = 0}\) (one solution only)
  • (C) 4: Falls between \(\mathrm{-6}\) and \(\mathrm{6}\)
  • (D) 6: Gives \(\mathrm{discriminant = 0}\) (one solution only)

Answer: A (-8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may confuse the discriminant conditions, thinking that any positive discriminant value automatically works without considering the "distinct" requirement. They might check if the discriminant is simply non-negative instead of strictly positive.

This leads them to incorrectly accept choices B or D (where \(\mathrm{discriminant = 0}\)), potentially selecting Choice B (-6) or Choice D (6).

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly find \(\mathrm{m^2 \gt 36}\) but misinterpret the absolute value inequality. They might think \(\mathrm{|m| \gt 6}\) means only \(\mathrm{m \gt 6}\), forgetting about the negative case \(\mathrm{m \lt -6}\).

This causes them to reject choice A (-8) and incorrectly focus only on values greater than 6, leading to confusion and guessing.

The Bottom Line:

This problem tests whether students truly understand the discriminant's role in determining solution types, not just its calculation. The key insight is recognizing that "two distinct" means strictly greater than zero, and that absolute value inequalities have two parts.