x^2 + kx + 36 = 0 In the given equation, k is a positive constant. If the equation has...

1. TRANSLATE the problem information

• Given information:
  • Quadratic equation: \(\mathrm{x^2 + kx + 36 = 0}\)
  • k is a positive constant
  • Equation has exactly one distinct real solution
• Need to find: value of k

2. INFER the key mathematical condition

• "Exactly one distinct real solution" is the critical phrase here
• This condition occurs when a quadratic's discriminant equals zero
• For \(\mathrm{ax^2 + bx + c = 0}\), discriminant = \(\mathrm{b^2 - 4ac = 0}\)

3. TRANSLATE the coefficients from our equation

• Comparing \(\mathrm{x^2 + kx + 36 = 0}\) to \(\mathrm{ax^2 + bx + c = 0}\):
• \(\mathrm{a = 1, b = k, c = 36}\)

4. SIMPLIFY by setting up and solving the discriminant equation

• Discriminant = \(\mathrm{b^2 - 4ac = 0}\)
• \(\mathrm{k^2 - 4(1)(36) = 0}\)
• \(\mathrm{k^2 - 144 = 0}\)
• \(\mathrm{k^2 = 144}\)
• \(\mathrm{k = ±12}\)

5. APPLY CONSTRAINTS to select the final answer

• The problem states k is positive
• Therefore: \(\mathrm{k = 12}\)

Answer: B (12)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "exactly one distinct real solution" to the discriminant condition. Instead, they might try to factor the quadratic directly or use the quadratic formula without realizing they need the discriminant to equal zero. This leads to confusion about how to proceed systematically, causing them to abandon the problem and guess randomly.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS reasoning: Students correctly find \(\mathrm{k = ±12}\) but forget that k must be positive, selecting the negative solution. This may lead them to look for Choice containing -12 (though none exists) or become confused about which value to choose, potentially leading to guessing.

The Bottom Line:

This problem tests whether students can translate a verbal condition ("exactly one distinct real solution") into a precise mathematical constraint (discriminant = 0). The algebraic work is straightforward once this connection is made, but missing this key insight leaves students without a clear solution path.