In the xy-plane, the parabola y = 9x^2 + 16 and the line y = mx intersect. For which of...

1. INFER the problem setup

• Given: Parabola \(\mathrm{y = 9x^2 + 16}\) and line \(\mathrm{y = mx}\) need to intersect at two distinct points
• Key insight: Two distinct intersection points means the system of equations has two distinct real solutions

2. TRANSLATE intersection condition into algebra

• Set the equations equal: \(\mathrm{9x^2 + 16 = mx}\)
• Rearrange to standard quadratic form: \(\mathrm{9x^2 - mx + 16 = 0}\)

3. INFER the mathematical condition needed

• For two distinct real solutions, we need the discriminant to be positive
• This connects to the quadratic formula: when \(\mathrm{\Delta \gt 0}\), we get two distinct real roots

4. SIMPLIFY the discriminant calculation

• For \(\mathrm{9x^2 - mx + 16 = 0}\): \(\mathrm{a = 9, b = -m, c = 16}\)
• Discriminant: \(\mathrm{\Delta = b^2 - 4ac = (-m)^2 - 4(9)(16) = m^2 - 576}\)
• Condition: \(\mathrm{m^2 - 576 \gt 0}\)

5. SIMPLIFY the inequality

• \(\mathrm{m^2 - 576 \gt 0}\)
• \(\mathrm{m^2 \gt 576}\)
• \(\mathrm{|m| \gt 24}\) (taking square root of both sides)

6. APPLY CONSTRAINTS to test answer choices

• \(\mathrm{|m| \gt 24}\) means \(\mathrm{m \gt 24}\) or \(\mathrm{m \lt -24}\)
• Test each option:
  • (A) -24: \(\mathrm{|-24| = 24}\), not greater than 24 ❌
  • (B) 0: \(\mathrm{|0| = 0}\), not greater than 24 ❌
  • (C) 20: \(\mathrm{|20| = 20}\), not greater than 24 ❌
  • (D) 30: \(\mathrm{|30| = 30 \gt 24}\) ✅

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not connect "two distinct intersection points" with "discriminant \(\mathrm{\gt 0}\)." Instead, they might try to solve the system directly or assume any non-zero value of m works. Without this key insight, they get stuck trying different approaches and end up guessing. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify the need for \(\mathrm{\Delta \gt 0}\) but make algebraic errors in the discriminant calculation or solving \(\mathrm{m^2 \gt 576}\). For example, they might incorrectly calculate the discriminant as \(\mathrm{m^2 - 144}\) instead of \(\mathrm{m^2 - 576}\), leading to \(\mathrm{|m| \gt 12}\). This may lead them to select Choice C (20) since \(\mathrm{20 \gt 12}\).

The Bottom Line:

This problem requires connecting the geometric concept of "two distinct intersection points" with the algebraic condition that a quadratic has two distinct real roots. Students who miss this connection or execute the discriminant algebra incorrectly will struggle to find the right answer systematically.