Question:The function p is defined by \(\mathrm{p(x) = (x - 5)(m - 3x)}\), where m is a constant. The quadratic...

1. SIMPLIFY the function into standard form

• Given: \(\mathrm{p(x) = (x - 5)(m - 3x)}\)
• SIMPLIFY by expanding:
  • \(\mathrm{p(x) = x(m - 3x) - 5(m - 3x)}\)
  • \(\mathrm{p(x) = mx - 3x^2 - 5m + 15x}\)
  • \(\mathrm{p(x) = -3x^2 + (m + 15)x - 5m}\)

2. INFER the approach using vertex properties

• Since we have \(\mathrm{p(x) = -3x^2 + (m + 15)x - 5m}\), this is a quadratic where \(\mathrm{a = -3 \lt 0}\)
• The parabola opens downward, so it has a maximum at its vertex
• For \(\mathrm{ax^2 + bx + c}\), vertex occurs at \(\mathrm{x = -\frac{b}{2a}}\)

3. SIMPLIFY to find the vertex location

• Vertex x-coordinate: \(\mathrm{x = -\frac{(m + 15)}{2(-3)} = \frac{(m + 15)}{6}}\)
• We know the maximum occurs at \(\mathrm{x = 2}\), so:
  • \(\mathrm{\frac{(m + 15)}{6} = 2}\)
  • \(\mathrm{m + 15 = 12}\)
  • \(\mathrm{m = -3}\)

4. INFER that we should verify our result

• Check: \(\mathrm{p(2) = (2 - 5)(-3 - 3(2)) = (-3)(-9) = 27}\) ✓
• This confirms our value of m is correct

5. SIMPLIFY to find the final answer

• Now calculate \(\mathrm{p(0)}\): \(\mathrm{p(0) = (0 - 5)(-3 - 3(0)) = (-5)(-3) = 15}\)

Answer: 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to plug \(\mathrm{x = 2}\) directly into the original form \(\mathrm{p(x) = (x - 5)(m - 3x)}\) without recognizing they need to find m first.

They might write \(\mathrm{p(2) = (2 - 5)(m - 6) = -3(m - 6) = 27}\), giving them \(\mathrm{-3m + 18 = 27}\), so \(\mathrm{m = -3}\). While this actually works, they often make sign errors in this approach and don't understand why it works. More problematically, they might try to substitute \(\mathrm{x = 2}\) and set the result equal to 27 without a clear strategy, leading to confusion about how to handle the unknown m.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x - 5)(m - 3x)}\), particularly with the signs.

For example, they might get \(\mathrm{p(x) = -3x^2 + (m - 15)x - 5m}\) instead of the correct \(\mathrm{-3x^2 + (m + 15)x - 5m}\). This leads to an incorrect vertex formula calculation and wrong value for m, ultimately giving an incorrect answer for \(\mathrm{p(0)}\).

The Bottom Line:

This problem requires students to connect the abstract concept of a vertex with concrete algebraic manipulation, then remember to complete the final step of evaluating the function at the requested point.