Question:x^2 - 40x + k = 0In the given equation, k is a constant. The equation has real solutions if...

1. TRANSLATE the problem information

• Given information:
  • Quadratic equation: \(\mathrm{x^2 - 40x + k = 0}\)
  • Condition: "equation has real solutions if \(\mathrm{k \lt m}\)"
  • Find: greatest possible value of m

• What this tells us: We need to find the largest value m such that whenever \(\mathrm{k \lt m}\), the equation definitely has real solutions.

2. INFER the approach needed

• Key insight: Real solutions for quadratics depend on the discriminant
• Strategy: Use discriminant analysis to find when real solutions exist, then determine the boundary for m

3. SIMPLIFY using discriminant analysis

• For quadratic \(\mathrm{ax^2 + bx + c = 0}\): discriminant \(\mathrm{D = b^2 - 4ac}\)
• Our equation: \(\mathrm{a = 1, b = -40, c = k}\)
• Calculate: \(\mathrm{D = (-40)^2 - 4(1)(k) = 1600 - 4k}\)

4. INFER the condition for real solutions

• Real solutions exist when \(\mathrm{D \geq 0}\)
• Therefore: \(\mathrm{1600 - 4k \geq 0}\)
• Solving: \(\mathrm{1600 \geq 4k}\), so \(\mathrm{k \leq 400}\)

5. INFER the meaning of "greatest possible m"

• The condition "has real solutions if \(\mathrm{k \lt m}\)" means: for ALL \(\mathrm{k \lt m}\), real solutions must exist
• Since real solutions exist when \(\mathrm{k \leq 400}\), the boundary is at \(\mathrm{k = 400}\)
• Therefore: \(\mathrm{m = 400}\) is the greatest value that works
• If \(\mathrm{m \gt 400}\), we could pick k between 400 and m, giving no real solutions

Answer: C) 400

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may interpret "\(\mathrm{k \lt m}\)" as simply needing \(\mathrm{k \leq 400}\), leading them to think any \(\mathrm{m \gt 400}\) works.

They might reason: "Since \(\mathrm{k \leq 400}\) gives real solutions, I can make m as large as I want, like \(\mathrm{m = 1600}\)." This misses the critical logical constraint that m must be the boundary value.

This may lead them to select Choice E (1600).

Second Most Common Error Path:

Inadequate SIMPLIFY execution: Students correctly identify discriminant analysis but make calculation errors in \(\mathrm{D = 1600 - 4k}\).

Common mistakes include using \(\mathrm{b = 40}\) instead of \(\mathrm{b = -40}\), leading to \(\mathrm{D = 40^2 - 4k = 1600 - 4k}\) (accidentally correct) or computing \(\mathrm{(-40)^2}\) incorrectly as 800 instead of 1600.

This may lead them to select Choice D (800) or Choice B (200).

The Bottom Line:

This problem requires both technical discriminant skills and logical reasoning about boundary conditions. Students often handle one aspect well but struggle with connecting the mathematical constraint \(\mathrm{k \leq 400}\) to the logical requirement for the greatest possible m.