3x^2 - 18x + k = 0In the given quadratic equation, k is a constant. The equation has two distinct...

1. TRANSLATE the problem condition

• Given information:
  • Quadratic equation: \(\mathrm{3x^2 - 18x + k = 0}\)
  • Condition: equation has two distinct real solutions when \(\mathrm{k \lt m}\)
  • Need to find: value of m

• What this tells us: We need to find when the discriminant is positive for two distinct real solutions.

2. TRANSLATE the mathematical condition

• For any quadratic \(\mathrm{ax^2 + bx + c = 0}\) to have two distinct real solutions: \(\mathrm{b^2 - 4ac \gt 0}\)
• In our equation: \(\mathrm{a = 3, b = -18, c = k}\)

3. SIMPLIFY the discriminant inequality

• Set up: \(\mathrm{(-18)^2 - 4(3)(k) \gt 0}\)
• Calculate: \(\mathrm{324 - 12k \gt 0}\)
• Isolate k: \(\mathrm{324 \gt 12k}\)
• Divide by 12: \(\mathrm{27 \gt k}\), or \(\mathrm{k \lt 27}\)

4. INFER the final answer

• The problem states: equation has two distinct real solutions when \(\mathrm{k \lt m}\)
• We found: equation has two distinct real solutions when \(\mathrm{k \lt 27}\)
• Therefore: \(\mathrm{m = 27}\)

Answer: C) 27

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not connect "two distinct real solutions" with the discriminant being positive. They might try to solve the quadratic directly or use the quadratic formula incorrectly, missing that this is about when solutions exist, not what the solutions are.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{b^2 - 4ac \gt 0}\) but make algebraic errors when solving \(\mathrm{324 - 12k \gt 0}\). Common mistakes include sign errors (getting \(\mathrm{k \gt 27}\) instead of \(\mathrm{k \lt 27}\)) or arithmetic errors when dividing 324 by 12.

This may lead them to select Choice E (39) if they incorrectly get \(\mathrm{k \lt 39}\), or other wrong choices based on calculation mistakes.

The Bottom Line:

This problem tests whether students understand the discriminant concept and can connect abstract conditions about solution types to concrete mathematical inequalities. The key insight is recognizing that "two distinct real solutions" translates directly to "positive discriminant."