x^2 - 40x - 10 = 0 What is the sum of the solutions to the given equation?...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{x^2 - 40x - 10 = 0}\)
• Question asks for: sum of the solutions (not individual solutions)
• Standard form: \(\mathrm{ax^2 + bx + c = 0}\) where \(\mathrm{a = 1, b = -40, c = -10}\)

2. INFER the most efficient approach

• Since we only need the sum (not individual solutions), Vieta's formulas provide the quickest path
• Key insight: For \(\mathrm{ax^2 + bx + c = 0}\), sum of roots = \(\mathrm{-b/a}\)
• This avoids the work of finding each root separately

3. SIMPLIFY using Vieta's formula

• Sum of solutions = \(\mathrm{-b/a}\)
• Sum = \(\mathrm{-(-40)/1}\)
• \(\mathrm{= 40/1}\)
• \(\mathrm{= 40}\)

Answer: D. 40

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they can find the sum directly using Vieta's formulas. Instead, they attempt to solve for individual roots using the quadratic formula or completing the square, leading to complex calculations with \(\mathrm{\sqrt{410}}\) terms. Even if they complete this correctly, they may make arithmetic errors when adding \(\mathrm{(20 + \sqrt{410}) + (20 - \sqrt{410})}\), potentially forgetting that the radical terms cancel out.

This computational complexity increases chances of errors and may lead them to select Choice A (0) if they incorrectly think the radicals don't cancel.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misidentify the coefficients, particularly confusing the sign of b. They might think \(\mathrm{b = 40}\) instead of \(\mathrm{b = -40}\), leading to sum = \(\mathrm{-40/1 = -40}\). Since -40 isn't among the choices, this leads to confusion and guessing.

The Bottom Line:

This problem rewards recognizing that you don't always need to solve completely - sometimes the relationship between coefficients and solutions (Vieta's formulas) provides a much more direct path than brute-force calculation.