Question:4x^2 - 12x + k = 0In the given equation, k is a constant. The equation has exactly one solution....

1. TRANSLATE the problem information

• Given equation: \(4x^2 - 12x + k = 0\)
• Condition: The equation has exactly one solution
• Need to find: The value of k

2. INFER the mathematical condition

• When a quadratic equation has exactly one solution, its discriminant equals zero
• This is the key insight that transforms the word problem into an algebra problem

3. TRANSLATE the equation into standard form coefficients

• Comparing \(4x^2 - 12x + k = 0\) with \(ax^2 + bx + c = 0\):
  • \(a = 4\)
  • \(b = -12\)
  • \(c = k\)

4. SIMPLIFY using the discriminant formula

• Discriminant = \(b^2 - 4ac = (-12)^2 - 4(4)(k) = 144 - 16k\)
• Set discriminant equal to zero: \(144 - 16k = 0\)
• Solve for k: \(16k = 144\), so \(k = 9\)

Answer: C (9)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "exactly one solution" with discriminant = 0. They might try to solve the quadratic directly or use the quadratic formula without recognizing that the condition gives them information about the discriminant. This leads to confusion and guessing since they can't proceed systematically.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the discriminant correctly but make arithmetic errors. For example, they might calculate \((-12)^2\) as \(-144\) instead of \(+144\), leading to \(-144 - 16k = 0\), which gives \(k = -9\). This may lead them to select Choice A (-9).

The Bottom Line:

This problem tests whether students can translate a verbal condition about solutions into a mathematical property (discriminant). The calculation itself is straightforward once they make this connection.