\(\mathrm{x^2 + (a + b)x + 57ab = 0}\) In the given equation, a and b are positive constants. The...

1. TRANSLATE the problem information

• Given information:
  • Quadratic equation: \(\mathrm{x^2 + (a + b)x + 57ab = 0}\)
  • a and b are positive constants
  • The product of solutions equals kab (where k is what we need to find)

2. INFER the mathematical relationship needed

• For any quadratic equation in the form \(\mathrm{x^2 + px + q = 0}\), the product of the roots equals q
• Our equation fits this form with \(\mathrm{q = 57ab}\)
• Therefore, the actual product of solutions = \(\mathrm{57ab}\)

3. TRANSLATE the given constraint

• We know: product of solutions = \(\mathrm{kab}\)
• We found: product of solutions = \(\mathrm{57ab}\)
• This means: \(\mathrm{kab = 57ab}\)

4. SIMPLIFY to solve for k

• From \(\mathrm{kab = 57ab}\)
• Divide both sides by ab: \(\mathrm{k = 57}\)

Answer: D (57)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual gap: Doesn't remember the relationship between quadratic coefficients and roots

Students may not recall that for \(\mathrm{x^2 + px + q = 0}\), the product of roots equals q. Without this key relationship, they cannot connect the given equation to the constraint about kab. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Weak TRANSLATE reasoning: Confusing sum and product relationships

Some students might remember that quadratics have relationships with roots, but confuse the sum of roots (which equals \(\mathrm{-p}\)) with the product of roots (which equals \(\mathrm{q}\)). This could lead them to incorrectly use (a + b) instead of 57ab, potentially selecting Choice A (1/57) if they think \(\mathrm{k = 1/(a+b)}\) when \(\mathrm{a+b = 57}\).

The Bottom Line:

This problem tests whether you know the fundamental relationship between quadratic coefficients and their roots. Once you know that the product of roots equals the constant term, the algebra is straightforward.