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

1. TRANSLATE the problem information

• Given information:
  • Quadratic equation: \(57\mathrm{x}^2 + (57\mathrm{b} + \mathrm{a})\mathrm{x} + \mathrm{ab} = 0\)
  • a and b are positive constants
  • Product of solutions = \(\mathrm{kab}\)
• What we need: Find the value of k

2. INFER the most efficient approach

• To find the product of solutions, I can use Vieta's formulas rather than solving for individual roots
• For any quadratic \(\mathrm{Ax}^2 + \mathrm{Bx} + \mathrm{C} = 0\), the product of roots equals \(\mathrm{C}/\mathrm{A}\)
• This will directly give me the product without factoring

3. SIMPLIFY using Vieta's formulas

• In our equation \(57\mathrm{x}^2 + (57\mathrm{b} + \mathrm{a})\mathrm{x} + \mathrm{ab} = 0\):
  • \(\mathrm{A} = 57\) (coefficient of x²)
  • \(\mathrm{B} = (57\mathrm{b} + \mathrm{a})\) (coefficient of x)
  • \(\mathrm{C} = \mathrm{ab}\) (constant term)
• Product of roots = \(\mathrm{C}/\mathrm{A} = \mathrm{ab}/57\)

4. SIMPLIFY to find k

• Set up the equation: \(\mathrm{ab}/57 = \mathrm{kab}\)
• Since a and b are positive, I can divide both sides by ab:
  • \(\mathrm{ab}/57 \div \mathrm{ab} = \mathrm{kab} \div \mathrm{ab}\)
  • \(1/57 = \mathrm{k}\)

Answer: A. \(1/57\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to solve the quadratic completely by finding individual roots first instead of recognizing they can use Vieta's formulas directly.

They might try to use the quadratic formula with \(\mathrm{A} = 57\), \(\mathrm{B} = (57\mathrm{b} + \mathrm{a})\), \(\mathrm{C} = \mathrm{ab}\), leading to messy expressions involving square roots of \((57\mathrm{b} + \mathrm{a})^2 - 4(57)(\mathrm{ab})\). This creates unnecessary complexity and increases chances of algebraic errors, potentially leading them to select Choice D (57) if they confuse which coefficient goes where in Vieta's formulas.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need the product of roots but make errors when applying Vieta's formulas.

They might incorrectly think the product of roots is \(\mathrm{A}/\mathrm{C} = 57/\mathrm{ab}\) instead of \(\mathrm{C}/\mathrm{A} = \mathrm{ab}/57\), or make sign errors during the algebraic manipulation. This could lead them to select Choice C (1) if they set up \(\mathrm{ab} = \mathrm{kab}\) and conclude \(\mathrm{k} = 1\).

The Bottom Line:

This problem rewards students who recognize that Vieta's formulas provide a direct path to the product of roots without solving for individual solutions. The key insight is connecting the abstract form "\(\mathrm{kab}\)" to the concrete result from Vieta's formulas.