9x^2 + bx + 16 = 0 In the given equation, b is a constant. For which of the following...

1. INFER the key relationship

• Since we need to determine when the equation has "no real solutions," this is a discriminant problem
• For quadratic \(\mathrm{ax^2 + bx + c = 0}\): discriminant \(\Delta = \mathrm{b^2 - 4ac}\) determines solution count
• No real solutions means \(\Delta \lt 0\)

2. TRANSLATE the equation components

• Given: \(\mathrm{9x^2 + bx + 16 = 0}\)
• Identify coefficients: \(\mathrm{a = 9, c = 16, b}\) = variable we're testing

3. SIMPLIFY the discriminant inequality

• Set up: \(\mathrm{b^2 - 4(9)(16) \lt 0}\)
• Calculate 4ac: \(\mathrm{4 \times 9 \times 16 = 576}\)
• The inequality becomes: \(\mathrm{b^2 - 576 \lt 0}\)
• Rearrange: \(\mathrm{b^2 \lt 576}\)
• Take square root: \(\mathrm{|b| \lt 24}\)
• This gives us the interval: \(\mathrm{-24 \lt b \lt 24}\)

4. APPLY CONSTRAINTS to boundary values

• Important: This is a strict inequality, so -24 and 24 are excluded
• At \(\mathrm{b = \pm 24}\), discriminant equals zero → exactly one solution (not no solutions)

5. INFER which answer choice fits

• Test each option against interval \(\mathrm{-24 \lt b \lt 24}\):
  • (A) -48: Outside interval → two solutions
  • (B) -24: Boundary value → one solution
  • (C) 12: Inside interval → no solutions ✓
  • (D) 25: Outside interval → two solutions

Answer: C (12)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the discriminant conditions, thinking that \(\Delta \gt 0\) means no real solutions instead of \(\Delta \lt 0\). They set up \(\mathrm{b^2 - 576 \gt 0}\), leading to \(\mathrm{|b| \gt 24}\), and incorrectly conclude that values outside the interval \(\mathrm{(-24, 24)}\) give no solutions.

This may lead them to select Choice A (-48) or Choice D (25).

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly find \(\mathrm{-24 \lt b \lt 24}\) but don't understand that this is a strict inequality. They think the boundary values \(\mathrm{b = \pm 24}\) also work because they're "at the edge" of having no solutions.

This may lead them to select Choice B (-24) since it appears to be a limiting case.

The Bottom Line:

This problem tests both discriminant knowledge AND careful inequality reasoning. The key insight is recognizing that "no real solutions" requires a strictly negative discriminant, making boundary cases (where \(\Delta = 0\)) incorrect choices.