\(4\mathrm{x}^2 - (4\mathrm{p} + 3\mathrm{q})\mathrm{x} + \mathrm{pq} = 0\) In the given equation, p and q are positive constants. The...

1. TRANSLATE the problem information

• Given equation: \(4\mathrm{x}^2 - (4\mathrm{p} + 3\mathrm{q})\mathrm{x} + \mathrm{pq} = 0\)
• The sum of solutions equals \(\mathrm{p} + \mathrm{tq}\) (need to find t)
• This tells us we need to find the sum of the roots of this quadratic

2. INFER the approach

• Since we need the sum of roots for a quadratic, use the sum of roots formula
• For \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\), sum of roots = \(-\mathrm{b}/\mathrm{a}\)
• Strategy: Identify coefficients, apply formula, then compare with given form

3. TRANSLATE coefficients from standard form

• In \(4\mathrm{x}^2 - (4\mathrm{p} + 3\mathrm{q})\mathrm{x} + \mathrm{pq} = 0\):
• \(\mathrm{a} = 4\)
• \(\mathrm{b} = -(4\mathrm{p} + 3\mathrm{q})\)
• \(\mathrm{c} = \mathrm{pq}\)

4. Apply sum of roots formula

• Sum = \(-\mathrm{b}/\mathrm{a}\)
• \(= -[-(4\mathrm{p} + 3\mathrm{q})]/4\)
• \(= (4\mathrm{p} + 3\mathrm{q})/4\)

5. SIMPLIFY the expression

• \((4\mathrm{p} + 3\mathrm{q})/4\)
• \(= 4\mathrm{p}/4 + 3\mathrm{q}/4\)
• \(= \mathrm{p} + (3/4)\mathrm{q}\)

6. INFER the value of t

• We have: \(\mathrm{p} + (3/4)\mathrm{q}\)
• Given form: \(\mathrm{p} + \mathrm{tq}\)
• Comparing coefficients: \(\mathrm{t} = 3/4\)

Answer: A (3/4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Incorrectly identifying the coefficient b as \((4\mathrm{p} + 3\mathrm{q})\) instead of \(-(4\mathrm{p} + 3\mathrm{q})\)

Students see "\(-(4\mathrm{p} + 3\mathrm{q})\mathrm{x}\)" and think \(\mathrm{b} = (4\mathrm{p} + 3\mathrm{q})\), missing the negative sign that's part of the coefficient. This leads to sum = \(-(4\mathrm{p} + 3\mathrm{q})/4 = -\mathrm{p} - (3/4)\mathrm{q}\), which doesn't match any answer choice format. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Making algebraic errors when distributing or combining terms

Students correctly find \((4\mathrm{p} + 3\mathrm{q})/4\) but then incorrectly simplify it, such as getting \(\mathrm{p} + 3\mathrm{q}\) instead of \(\mathrm{p} + (3/4)\mathrm{q}\). This might lead them to think \(\mathrm{t} = 3\), but since that's not an option, this causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students can correctly apply the sum of roots formula while carefully handling negative coefficients and fractional simplification - both common stumbling blocks in quadratic problems.