Question:px^2 + 10x - 5p = 0In the given equation, p is a positive constant. The sum of the reciprocals...

1. TRANSLATE the problem information

• Given information:
  • Quadratic equation: \(\mathrm{px^2 + 10x - 5p = 0}\)
  • p is a positive constant
  • Sum of reciprocals of solutions = \(\mathrm{k/p}\)
• Need to find: Value of k

2. TRANSLATE "sum of reciprocals" to math notation

• If the solutions are \(\mathrm{r_1}\) and \(\mathrm{r_2}\), then:
• Sum of reciprocals = \(\mathrm{1/r_1 + 1/r_2}\)
• To add these fractions: \(\mathrm{1/r_1 + 1/r_2 = (r_1 + r_2)/(r_1r_2)}\)

3. INFER the approach using Vieta's formulas

• We need both \(\mathrm{(r_1 + r_2)}\) and \(\mathrm{(r_1r_2)}\) to find the sum of reciprocals
• Vieta's formulas give us these directly from the coefficients
• For \(\mathrm{ax^2 + bx + c = 0}\): sum = \(\mathrm{-b/a}\), product = \(\mathrm{c/a}\)

4. TRANSLATE the equation coefficients

• From \(\mathrm{px^2 + 10x - 5p = 0}\):
  • \(\mathrm{a = p}\)
  • \(\mathrm{b = 10}\)
  • \(\mathrm{c = -5p}\)

5. SIMPLIFY using Vieta's formulas

• Sum of roots: \(\mathrm{r_1 + r_2 = -b/a = -10/p}\)
• Product of roots: \(\mathrm{r_1r_2 = c/a = -5p/p = -5}\)

6. SIMPLIFY the sum of reciprocals

• Sum of reciprocals = \(\mathrm{(r_1 + r_2)/(r_1r_2) = (-10/p)/(-5)}\)
• Division by -5 means multiply by -1/5:
• \(\mathrm{(-10/p) \times (-1/5) = 10/(5p) = 2/p}\)

7. INFER the final answer

• Sum of reciprocals = \(\mathrm{2/p}\)
• Problem states this equals \(\mathrm{k/p}\)
• Therefore: \(\mathrm{k = 2}\)

Answer: C (2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert "sum of reciprocals" into the mathematical expression \(\mathrm{1/r_1 + 1/r_2 = (r_1 + r_2)/(r_1r_2)}\). They might try to work directly with the quadratic formula or attempt to actually solve for \(\mathrm{r_1}\) and \(\mathrm{r_2}\), making the problem much harder than necessary.

This leads to confusion and abandoning the systematic approach, resulting in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the problem and use Vieta's formulas, but make sign errors when calculating \(\mathrm{(-10/p)/(-5)}\). They might get \(\mathrm{-2/p}\) instead of \(\mathrm{2/p}\), or forget to simplify \(\mathrm{10/(5p)}\) to \(\mathrm{2/p}\).

This may lead them to select Choice A (-10) if they get confused with signs, or get stuck without a clean answer.

The Bottom Line:

This problem tests whether students can recognize that reciprocal problems often have elegant solutions using Vieta's formulas, rather than requiring the messy work of actually finding the roots first.