A company's profit function is \(\mathrm{P(x) = x^2 - 8x + 3}\) (in thousands of dollars), where x represents months...

1. TRANSLATE the problem information

• Given information:
  • Profit function: \(\mathrm{P(x) = x^2 - 8x + 3}\)
  • Break-even line: \(\mathrm{B(x) = 3x - 12}\)
  • Need to find: number of intersection points

• What this tells us: Intersection points occur where \(\mathrm{P(x) = B(x)}\)

2. INFER the solution strategy

• Key insight: Instead of solving for specific intersection points, we can use the discriminant to determine how many solutions exist
• This is more efficient than finding actual x-values when we only need to count intersections

3. SIMPLIFY by setting up the equation

• Set \(\mathrm{P(x) = B(x)}\):
  \(\mathrm{x^2 - 8x + 3 = 3x - 12}\)

• Rearrange to standard quadratic form:
  \(\mathrm{x^2 - 8x + 3 - 3x + 12 = 0}\)
  \(\mathrm{x^2 - 11x + 15 = 0}\)

4. INFER that discriminant analysis applies

• For quadratic \(\mathrm{ax^2 + bx + c = 0}\), we have: \(\mathrm{a = 1, b = -11, c = 15}\)
• The discriminant \(\mathrm{\Delta = b^2 - 4ac}\) tells us the number of real solutions

5. SIMPLIFY the discriminant calculation

• \(\mathrm{\Delta = (-11)^2 - 4(1)(15)}\)
• \(\mathrm{\Delta = 121 - 60 = 61}\)

6. INFER the final answer from discriminant value

• Since \(\mathrm{\Delta = 61 \gt 0}\), the quadratic has exactly two distinct real solutions
• Therefore, the functions intersect at exactly two points

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to solve the quadratic completely instead of recognizing that only the number of solutions matters. They get bogged down in factoring \(\mathrm{x^2 - 11x + 15 = 0}\) or applying the quadratic formula, potentially making arithmetic errors along the way. Some may incorrectly factor and get wrong values, leading them to count intersection points incorrectly.

This may lead them to select Choice B (Exactly one) if they make an error that suggests the discriminant equals zero, or causes confusion and abandoning the systematic approach.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when rearranging \(\mathrm{x^2 - 8x + 3 = 3x - 12}\) into standard form, getting something like \(\mathrm{x^2 - 5x - 9 = 0}\) instead of \(\mathrm{x^2 - 11x + 15 = 0}\). This leads to a different discriminant value and wrong conclusion about the number of intersections.

This may lead them to select Choice C (Zero) if their incorrect discriminant is negative.

The Bottom Line:

This problem rewards strategic thinking over computational persistence. Students who recognize that discriminant analysis efficiently answers "how many" without requiring "what values" will solve this much more quickly and accurately than those who attempt complete solution.