\(64\mathrm{x}^2 - (16\mathrm{a} + 4\mathrm{b})\mathrm{x} + \mathrm{ab} = 0\) In the given equation, a and b are positive constants. The...

1. TRANSLATE the problem information

• Given equation: \(64\mathrm{x}^2 - (16\mathrm{a} + 4\mathrm{b})\mathrm{x} + \mathrm{ab} = 0\)
• Need to find: \(\mathrm{k}\) where sum of solutions = \(\mathrm{k}(4\mathrm{a} + \mathrm{b})\)
• Key insight: This is asking about the sum of roots of a quadratic equation

2. INFER the appropriate approach

• Since we need the sum of roots, we should use the sum of roots formula
• For any quadratic \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\), sum of roots = \(-\mathrm{b}/\mathrm{a}\)
• We need to identify the coefficients carefully in our equation

3. TRANSLATE the equation into standard form

• Our equation: \(64\mathrm{x}^2 - (16\mathrm{a} + 4\mathrm{b})\mathrm{x} + \mathrm{ab} = 0\)
• In standard form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\):
  • Coefficient of \(\mathrm{x}^2\): 64
  • Coefficient of \(\mathrm{x}\): \(-(16\mathrm{a} + 4\mathrm{b})\)
  • Constant term: \(\mathrm{ab}\)

4. Apply the sum of roots formula

• Sum of roots = \(-\mathrm{b}/\mathrm{a}\)
• \(= -[-(16\mathrm{a} + 4\mathrm{b})]/64\)
• \(= (16\mathrm{a} + 4\mathrm{b})/64\)

5. SIMPLIFY the expression

• \((16\mathrm{a} + 4\mathrm{b})/64\)
• \(= 4(4\mathrm{a} + \mathrm{b})/64\)
• \(= (4\mathrm{a} + \mathrm{b})/16\)
• This equals \((1/16)(4\mathrm{a} + \mathrm{b})\)

6. INFER the value of k

• We know: sum of solutions = \(\mathrm{k}(4\mathrm{a} + \mathrm{b})\)
• We found: sum of solutions = \((1/16)(4\mathrm{a} + \mathrm{b})\)
• Therefore: \(\mathrm{k}(4\mathrm{a} + \mathrm{b}) = (1/16)(4\mathrm{a} + \mathrm{b})\)
• This gives us: \(\mathrm{k} = 1/16\)

Answer: 1/16, 0.0625, 0.062, 0.063

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misidentify the coefficient of \(\mathrm{x}\) as \((16\mathrm{a} + 4\mathrm{b})\) instead of \(-(16\mathrm{a} + 4\mathrm{b})\). This happens because they miss the negative sign in front of the parentheses.

Using the wrong coefficient, they calculate sum = \(-(16\mathrm{a} + 4\mathrm{b})/64\), leading to \(\mathrm{k} = -1/16\). This creates confusion since \(\mathrm{k}\) should be positive based on the context, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the sum as \((16\mathrm{a} + 4\mathrm{b})/64\) but struggle with the algebraic manipulation to convert this to the form \(\mathrm{k}(4\mathrm{a} + \mathrm{b})\).

They might leave their answer as \((16\mathrm{a} + 4\mathrm{b})/64\) or make arithmetic errors when simplifying, such as getting \((4\mathrm{a} + \mathrm{b})/4\) instead of \((4\mathrm{a} + \mathrm{b})/16\). This leads to confusion and guessing among the given answer formats.

The Bottom Line:

This problem challenges students to carefully handle negative coefficients and perform multi-step algebraic simplification while maintaining the connection between two equivalent expressions for the same quantity.