Question:\(45\mathrm{x}^2 + (15\mathrm{a} + 9\mathrm{b})\mathrm{x} + 3\mathrm{ab} = 0\)In the given equation, a and b are positive constants. The prod...

1. TRANSLATE the problem information

• Given equation: \(45\mathrm{x}^2 + (15\mathrm{a} + 9\mathrm{b})\mathrm{x} + 3\mathrm{ab} = 0\)
• \(\mathrm{a}\) and \(\mathrm{b}\) are positive constants
• Product of solutions equals \(\mathrm{kab}\), where \(\mathrm{k}\) is unknown

2. INFER the approach needed

• This is asking about the product of roots of a quadratic equation
• For any quadratic in standard form \(\mathrm{Ax}^2 + \mathrm{Bx} + \mathrm{C} = 0\), Vieta's formulas tell us the product of roots equals \(\mathrm{C}/\mathrm{A}\)
• I need to identify \(\mathrm{A}\) and \(\mathrm{C}\), then set up an equation to find \(\mathrm{k}\)

3. TRANSLATE the quadratic into standard form

• Standard form: \(\mathrm{Ax}^2 + \mathrm{Bx} + \mathrm{C} = 0\)
• From \(45\mathrm{x}^2 + (15\mathrm{a} + 9\mathrm{b})\mathrm{x} + 3\mathrm{ab} = 0\):
  • \(\mathrm{A} = 45\) (coefficient of \(\mathrm{x}^2\))
  • \(\mathrm{B} = (15\mathrm{a} + 9\mathrm{b})\) (coefficient of \(\mathrm{x}\))
  • \(\mathrm{C} = 3\mathrm{ab}\) (constant term)

4. INFER the product of solutions using Vieta's formulas

• Product of solutions = \(\mathrm{C}/\mathrm{A} = 3\mathrm{ab}/45\)

5. SIMPLIFY the fraction

• \(3\mathrm{ab}/45 = \mathrm{ab}/15\) (dividing numerator and denominator by 3)

6. TRANSLATE the given condition into an equation

• Problem states: product of solutions = \(\mathrm{kab}\)
• From our work: product of solutions = \(\mathrm{ab}/15\)
• Therefore: \(\mathrm{kab} = \mathrm{ab}/15\)

7. SIMPLIFY to solve for k

• \(\mathrm{kab} = \mathrm{ab}/15\)
• Divide both sides by \(\mathrm{ab}\) (valid since \(\mathrm{a}, \mathrm{b} > 0\)): \(\mathrm{k} = 1/15\)

Answer: (B) 1/15

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Students don't remember or recognize the need for Vieta's formulas.

Without knowing that the product of roots equals \(\mathrm{C}/\mathrm{A}\), students might try to actually solve the quadratic equation using the quadratic formula, which becomes very messy with the parameters \(\mathrm{a}\) and \(\mathrm{b}\). This leads to getting overwhelmed by complex algebra and ultimately guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly identify that product = \(3\mathrm{ab}/45\) but make arithmetic errors when simplifying.

Some students might incorrectly simplify \(3\mathrm{ab}/45\), perhaps getting \(1/5\) instead of \(1/15\) by incorrectly reducing the fraction. This could lead them to select Choice (D) (1/3) if they make multiple calculation mistakes.

The Bottom Line:

This problem tests whether students recognize the connection between the abstract statement "product of solutions is \(\mathrm{kab}\)" and the concrete application of Vieta's formulas. The key insight is that you don't need to solve the quadratic explicitly - Vieta's formulas provide a direct shortcut to the product of roots.