64x^2 + bx + 25 = 0In the given equation, b is a constant. For which of the following values...

1. TRANSLATE the problem information

• Given information:
  • Quadratic equation: \(64\mathrm{x}^2 + \mathrm{bx} + 25 = 0\)
  • Need to find which value of b gives "more than one real solution"

• What this tells us: We're looking for values of b that make the equation have exactly two distinct real solutions

2. INFER the mathematical approach

• Key insight: "More than one real solution" for a quadratic means exactly two distinct real solutions
• This happens when the discriminant is positive: \(\mathrm{b}^2 - 4\mathrm{ac} \gt 0\)
• We need to set up and solve this inequality

3. SIMPLIFY the discriminant condition

• For our equation \(64\mathrm{x}^2 + \mathrm{bx} + 25 = 0\):
  • \(\mathrm{a} = 64, \mathrm{b} = \mathrm{b}, \mathrm{c} = 25\)
  • Discriminant = \(\mathrm{b}^2 - 4(64)(25) = \mathrm{b}^2 - 6400\)

• Set up inequality: \(\mathrm{b}^2 - 6400 \gt 0\)
• Solve: \(\mathrm{b}^2 \gt 6400\)
• Take square root: \(|\mathrm{b}| \gt 80\) (use calculator: \(\sqrt{6400} = 80\))
• This means: \(\mathrm{b} \lt -80\) or \(\mathrm{b} \gt 80\)

4. APPLY CONSTRAINTS to select the correct answer

• Check each option against \(\mathrm{b} \lt -80\) or \(\mathrm{b} \gt 80\):
  • A. -91: Since \(-91 \lt -80\), this works ✓
  • B. -80: This equals -80, but we need strict inequality ✗
  • C. 5: This is between -80 and 80 ✗
  • D. 40: This is between -80 and 80 ✗

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about discriminant conditions: Students often confuse when a quadratic has "more than one" versus "at least one" real solution, or they mix up the discriminant conditions (thinking discriminant ≥ 0 instead of > 0 for two distinct solutions).

They might set up \(\mathrm{b}^2 - 6400 \geq 0\) instead of \(\mathrm{b}^2 - 6400 \gt 0\), leading them to accept \(\mathrm{b} = \pm 80\) as valid solutions. This could make them select Choice B (-80) since it satisfies the incorrect condition.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly identify the need for discriminant > 0 but make algebraic errors when solving \(\mathrm{b}^2 \gt 6400\). They might forget to consider both positive and negative cases when taking the square root, only finding \(\mathrm{b} \gt 80\) and missing \(\mathrm{b} \lt -80\).

This incomplete solution might lead them to eliminate choice A (-91) incorrectly and select Choice D (40) as the "largest positive option."

The Bottom Line:

This problem requires precise understanding of discriminant conditions combined with careful inequality solving. Students must distinguish between boundary cases (equality) and strict inequalities, while also handling absolute value relationships correctly.