The quadratic equation 3x^2 - bx + 192 = 0, where b is a positive integer, has no real solutions....

1. TRANSLATE the condition "no real solutions"

• Given: \(3\mathrm{x}^2 - \mathrm{b}\mathrm{x} + 192 = 0\) has no real solutions
• This means: The discriminant must be negative

2. INFER the approach using discriminant

• For any quadratic \(\mathrm{a}\mathrm{x}^2 + \mathrm{b}\mathrm{x} + \mathrm{c} = 0\), discriminant = \(\mathrm{b}^2 - 4\mathrm{a}\mathrm{c}\)
• For no real solutions: \(\mathrm{discriminant} \lt 0\)
• Our strategy: Set up the discriminant inequality and solve for b

3. SIMPLIFY by identifying coefficients and setting up the inequality

• In \(3\mathrm{x}^2 - \mathrm{b}\mathrm{x} + 192 = 0\): a = 3, coefficient of x = -b, c = 192
• Discriminant = \((-\mathrm{b})^2 - 4(3)(192) = \mathrm{b}^2 - 2304\)
• For no real solutions: \(\mathrm{b}^2 - 2304 \lt 0\)

4. SIMPLIFY the inequality

• \(\mathrm{b}^2 - 2304 \lt 0\)
• \(\mathrm{b}^2 \lt 2304\)
• Taking square root: \(\mathrm{b} \lt \sqrt{2304}\)
• Calculate: \(\sqrt{2304} = 48\) (use calculator)
• Therefore: \(\mathrm{b} \lt 48\)

5. APPLY CONSTRAINTS to find the maximum integer value

• b must be a positive integer
• Since \(\mathrm{b} \lt 48\) and b is a positive integer
• The greatest possible value is \(\mathrm{b} = 47\)

Answer: 47

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students know the discriminant formula but don't connect "no real solutions" to the discriminant being negative. They might try to solve the quadratic directly or get confused about what the problem is asking.

This leads to confusion and abandoning systematic solution, causing them to guess.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly find \(\mathrm{b} \lt 48\) but then conclude the answer is 48, not recognizing that b must be strictly less than 48. Since b is an integer, the maximum is 47, not 48.

This leads them to incorrectly answer 48 instead of the correct 47.

The Bottom Line:

This problem requires connecting the abstract condition "no real solutions" to the concrete mathematical tool of the discriminant, then carefully handling the boundary condition for integer constraints.