The parabola with equation y = 3x^2 - 12x + k touches the x-axis at exactly one point. What is...

1. TRANSLATE the problem information

• Given information:
  • Parabola equation: \(\mathrm{y = 3x^2 - 12x + k}\)
  • Condition: touches x-axis at exactly one point

• What this tells us: The quadratic equation \(\mathrm{3x^2 - 12x + k = 0}\) has exactly one solution

2. INFER the mathematical requirement

• Key insight: When a parabola touches the x-axis at exactly one point, it means the quadratic has a repeated root
• This happens when the discriminant equals zero
• Strategy: Use discriminant formula and set it equal to zero

3. TRANSLATE to discriminant formula

• For quadratic \(\mathrm{ax^2 + bx + c = 0}\), discriminant = \(\mathrm{b^2 - 4ac}\)
• Here: \(\mathrm{a = 3, b = -12, c = k}\)
• Discriminant = \(\mathrm{(-12)^2 - 4(3)(k) = 144 - 12k}\)

4. SIMPLIFY by setting discriminant to zero

• Set up equation: \(\mathrm{144 - 12k = 0}\)
• Solve for k: \(\mathrm{144 = 12k}\)
• Therefore: \(\mathrm{k = 12}\)

Answer: D) 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "touches x-axis at exactly one point" with discriminant = 0. They might try to find where \(\mathrm{y = 0}\) directly or get confused about what "exactly one point" means mathematically.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Calculation error during SIMPLIFY: Students set up the discriminant correctly but make arithmetic mistakes when solving \(\mathrm{144 - 12k = 0}\). Common errors include forgetting the negative sign or incorrectly dividing 144 by 12.

This may lead them to select Choice A (-12) if they get \(\mathrm{k = -12}\), or other incorrect values.

The Bottom Line:

This problem requires connecting geometric language ("touches at one point") with algebraic concepts (discriminant). Students who memorize the discriminant formula but don't understand its geometric meaning will struggle with the setup.