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

1. TRANSLATE the problem information

• Given equation: \(57\mathrm{x}^2 + (57\mathrm{b} + \mathrm{a})\mathrm{x} + \mathrm{ab} = 0\)
• The equation is already in standard quadratic form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\)
• We need to find k where the product of solutions equals kab

2. INFER the most efficient approach

• Since we only need the product of roots (not the individual roots), we can use the direct relationship
• For any quadratic \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\), the product of roots = \(\frac{\mathrm{c}}{\mathrm{a}}\)
• This avoids the need to find individual solutions

3. TRANSLATE the coefficients from our equation

• Coefficient of \(\mathrm{x}^2\) (the 'a' in standard form): 57
• Coefficient of x (the 'b' in standard form): \((57\mathrm{b} + \mathrm{a})\)
• Constant term (the 'c' in standard form): ab

4. SIMPLIFY using the product relationship

• Product of roots = \(\frac{\mathrm{c}}{\mathrm{a}}\) = \(\frac{\mathrm{ab}}{57}\)
• We're told this product equals kab
• Set up equation: \(\frac{\mathrm{ab}}{57} = \mathrm{kab}\)

5. SIMPLIFY to solve for k

• Divide both sides by ab (valid since a, b are positive constants)
• \(\frac{\mathrm{ab}}{57} \div \mathrm{ab} = \mathrm{kab} \div \mathrm{ab}\)
• \(\frac{1}{57} = \mathrm{k}\)

Answer: A. \(\frac{1}{57}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to factor the quadratic or use the quadratic formula to find individual roots, making the problem much more complex than necessary.

They might struggle with factoring \(57\mathrm{x}^2 + (57\mathrm{b} + \mathrm{a})\mathrm{x} + \mathrm{ab}\) or get overwhelmed by the quadratic formula with multiple variables. This leads to confusion and abandoning the systematic approach, causing them to guess rather than recognize the simpler coefficient relationship.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misidentify the coefficients when setting up the product relationship.

They might incorrectly think the constant term is just 'a' or 'b' instead of 'ab', or confuse which coefficient goes where in the \(\frac{\mathrm{c}}{\mathrm{a}}\) formula. This could lead them to calculate products like \(\frac{\mathrm{a}}{57}\) or \(\frac{\mathrm{b}}{57}\), potentially selecting Choice B \((\frac{1}{19})\) if they somehow arrive at that through incorrect coefficient identification.

The Bottom Line:

This problem tests whether students recognize that finding the product of roots directly through coefficient relationships is more efficient than finding individual roots first. The key insight is connecting the abstract relationship (product = \(\frac{\mathrm{c}}{\mathrm{a}}\)) to the specific coefficients in this equation.