-{x^2 + bx - 676 = 0} In the given equation, b is a positive integer. The equation has no...

1. INFER the key relationship

• Given information:
  • Equation: \(-\mathrm{x}^2 + \mathrm{bx} - 676 = 0\)
  • b is a positive integer
  • The equation has no real solutions

• Key insight: "No real solutions" means we need to use the discriminant condition. A quadratic has no real solutions when its discriminant is negative.

2. INFER the discriminant setup

• For the general form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\), discriminant = \(\mathrm{b}^2 - 4\mathrm{ac}\)
• In our equation: \(\mathrm{a} = -1, \mathrm{b} = \mathrm{b}, \mathrm{c} = -676\)
• Discriminant = \(\mathrm{b}^2 - 4(-1)(-676) = \mathrm{b}^2 - 2704\)

3. SIMPLIFY the inequality

• For no real solutions: discriminant \(\lt 0\)
• So: \(\mathrm{b}^2 - 2704 \lt 0\)
• Therefore: \(\mathrm{b}^2 \lt 2704\)
• Taking the positive square root: \(\mathrm{b} \lt \sqrt{2704}\)

4. SIMPLIFY the square root

• \(\sqrt{2704} = 52\) (use calculator or recognize that \(52^2 = 2704\))
• So: \(\mathrm{b} \lt 52\)

5. APPLY CONSTRAINTS to find the answer

• Since b must be a positive integer
• The greatest integer less than 52 is 51

Answer: 51

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "no real solutions" to the discriminant condition. They might try to solve the quadratic directly or get confused about what "no real solution" actually means mathematically.

This leads to confusion and guessing since they can't establish the fundamental relationship needed to start the problem.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students set up the discriminant correctly but make algebraic errors, particularly with the negative signs. For example, they might calculate \(\mathrm{b}^2 - 4(-1)(-676)\) incorrectly as \(\mathrm{b}^2 + 2704\) instead of \(\mathrm{b}^2 - 2704\).

This leads them away from the correct inequality and toward incorrect constraint boundaries.

The Bottom Line:

This problem requires students to bridge abstract quadratic theory (discriminant conditions) with concrete integer constraints. The challenge isn't just knowing the discriminant formula—it's recognizing when and how to apply it to answer questions about solution existence.