\(39\mathrm{x}^2 + (39\mathrm{a} + \mathrm{b})\mathrm{x} + \mathrm{ab} = 0\)In the given equation, a and b are positive constants. The sum...

1. TRANSLATE the problem information

• Given equation: \(39\mathrm{x}^2 + (39\mathrm{a} + \mathrm{b})\mathrm{x} + \mathrm{ab} = 0\)
• The sum of solutions equals \(-(\mathrm{a} + \mathrm{kb})\)
• Need to find: the constant k

2. INFER the approach needed

• This is asking for the sum of solutions of a quadratic equation
• I need to use the sum of solutions formula: For \(\mathrm{Ax}^2 + \mathrm{Bx} + \mathrm{C} = 0\), \(\mathrm{sum} = -\mathrm{B}/\mathrm{A}\)
• Then match this result with the given form \(-(\mathrm{a} + \mathrm{kb})\)

3. TRANSLATE the equation into standard form coefficients

• Compare \(39\mathrm{x}^2 + (39\mathrm{a} + \mathrm{b})\mathrm{x} + \mathrm{ab} = 0\) with \(\mathrm{Ax}^2 + \mathrm{Bx} + \mathrm{C} = 0\)
• \(\mathrm{A} = 39\)
• \(\mathrm{B} = 39\mathrm{a} + \mathrm{b}\)
• \(\mathrm{C} = \mathrm{ab}\)

4. SIMPLIFY using the sum formula

• Sum of solutions \(= -\mathrm{B}/\mathrm{A}\)
• \(= -(39\mathrm{a} + \mathrm{b})/39\)
• Break apart the fraction:
• \(-(39\mathrm{a} + \mathrm{b})/39 = -39\mathrm{a}/39 - \mathrm{b}/39\)
• Simplify:
• \(-39\mathrm{a}/39 - \mathrm{b}/39 = -\mathrm{a} - \mathrm{b}/39\)
• \(= -(\mathrm{a} + \mathrm{b}/39)\)

5. INFER the value of k by comparison

• We have: \(\mathrm{sum} = -(\mathrm{a} + \mathrm{b}/39)\)
• Given form: \(\mathrm{sum} = -(\mathrm{a} + \mathrm{kb})\)
• Setting equal: \(-(\mathrm{a} + \mathrm{b}/39) = -(\mathrm{a} + \mathrm{kb})\)
• Therefore: \(\mathrm{b}/39 = \mathrm{kb}\)
• Solving for k: \(\mathrm{k} = (\mathrm{b}/39)/\mathrm{b} = 1/39\)

Answer: A. 1/39

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't recognize this as a sum of solutions problem, or they incorrectly identify the coefficients in the quadratic equation.

They might confuse which terms are A, B, and C, especially with the compound coefficient (39a + b) for the x term. This leads to using the wrong values in the sum formula and getting an incorrect expression for k.

This may lead them to select Choice C (1) if they mistakenly think k equals the ratio of the leading coefficients.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly apply the sum formula but make algebraic errors when separating \(-(39\mathrm{a} + \mathrm{b})/39\).

They might incorrectly simplify this as just \(-\mathrm{a} - \mathrm{b}\) instead of \(-\mathrm{a} - \mathrm{b}/39\), missing the crucial division of the b term by 39. This leads to comparing \(--(\mathrm{a} + \mathrm{b})\) with \(-(\mathrm{a} + \mathrm{kb})\), concluding that k = 1.

This leads them to select Choice C (1).

The Bottom Line:

This problem tests whether students can systematically apply the sum of solutions formula while carefully tracking algebraic manipulations involving compound expressions.