If the quadratic function \(\mathrm{f(x) = 3x^2 - 18x + m}\) has no x-intercepts, which of the following could be...

1. TRANSLATE the problem information

• Given information:
  • Quadratic function: \(\mathrm{f(x) = 3x^2 - 18x + m}\)
  • The function has no x-intercepts
  • Need to find possible values of m from given choices

2. INFER what "no x-intercepts" means mathematically

• X-intercepts occur where the function equals zero: \(\mathrm{f(x) = 0}\)
• "No x-intercepts" means the equation \(\mathrm{3x^2 - 18x + m = 0}\) has no real solutions
• For a quadratic to have no real solutions, its discriminant must be negative

3. SIMPLIFY by calculating the discriminant

• For quadratic \(\mathrm{ax^2 + bx + c = 0}\), discriminant = \(\mathrm{b^2 - 4ac}\)
• Here: \(\mathrm{a = 3, b = -18, c = m}\)
• Discriminant = \(\mathrm{(-18)^2 - 4(3)(m) = 324 - 12m}\)

4. SIMPLIFY the inequality condition

• For no real solutions: \(\mathrm{324 - 12m \lt 0}\)
• Rearrange: \(\mathrm{324 \lt 12m}\)
• Divide by 12: \(\mathrm{m \gt 27}\)

5. APPLY CONSTRAINTS to select the correct answer

• Check each choice against \(\mathrm{m \gt 27}\):
  • Choice A (25): \(\mathrm{25 \gt 27}\)? No
  • Choice B (26): \(\mathrm{26 \gt 27}\)? No
  • Choice C (27): \(\mathrm{27 \gt 27}\)? No (equal, not greater)
  • Choice D (29): \(\mathrm{29 \gt 27}\)? Yes

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "no x-intercepts" to the discriminant concept. They might try to solve \(\mathrm{3x^2 - 18x + m = 0}\) directly for specific values of m, or attempt to complete the square without understanding the underlying condition.

This leads to confusion about what the question is actually asking, causing them to get stuck and guess randomly.

Second Most Common Error:

Incomplete APPLY CONSTRAINTS reasoning: Students correctly find \(\mathrm{m \gt 27}\) but then select Choice C (27) because 27 seems like the "boundary" value. They miss that the inequality is strict (greater than, not greater than or equal to).

The Bottom Line:

This problem requires students to translate a graphical concept (x-intercepts) into an algebraic condition (discriminant), then carefully apply that condition. The key insight is recognizing that "no x-intercepts" is equivalent to "negative discriminant."