The solutions to x^2 - 6x + 5 = 0 are p and q, where p lt q. The solutions...

1. SIMPLIFY the first quadratic equation

• Given: \(\mathrm{x^2 - 6x + 5 = 0}\)
• Factor: Look for two numbers that multiply to 5 and add to -6
• Those numbers are -1 and -5: \(\mathrm{(x - 1)(x - 5) = 0}\)
• Solutions: \(\mathrm{x = 1}\) and \(\mathrm{x = 5}\)
• Since \(\mathrm{p \lt q}\): \(\mathrm{p = 1}\) and \(\mathrm{q = 5}\)

2. SIMPLIFY the second quadratic equation

• Given: \(\mathrm{x^2 - 4x + 3 = 0}\)
• Factor: Look for two numbers that multiply to 3 and add to -4
• Those numbers are -1 and -3: \(\mathrm{(x - 1)(x - 3) = 0}\)
• Solutions: \(\mathrm{x = 1}\) and \(\mathrm{x = 3}\)
• Since \(\mathrm{r \lt s}\): \(\mathrm{r = 1}\) and \(\mathrm{s = 3}\)

3. INFER what the roots of the third equation are

• We're told the roots of \(\mathrm{x^2 - 10x + k = 0}\) are \(\mathrm{p + r}\) and \(\mathrm{q + s}\)
• Calculate: \(\mathrm{p + r = 1 + 1 = 2}\) and \(\mathrm{q + s = 5 + 3 = 8}\)
• So the third equation has roots 2 and 8

4. INFER how to find k using Vieta's formulas

• For any quadratic \(\mathrm{x^2 - 10x + k = 0}\) with roots 2 and 8:
• Sum of roots = \(\mathrm{2 + 8 = 10}\) (this matches the 10 in -10x ✓)
• Product of roots = \(\mathrm{k = 2 \times 8 = 16}\)

Answer: C) 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making factoring errors on the quadratics, especially confusing signs or not finding the right factor pairs.

For example, incorrectly factoring \(\mathrm{x^2 - 6x + 5}\) as \(\mathrm{(x - 2)(x - 3)}\) or getting confused about which root is p vs q. This leads to wrong values for \(\mathrm{p + r}\) and \(\mathrm{q + s}\), resulting in an incorrect product calculation. This may lead them to select Choice A (8) or Choice B (12).

Second Most Common Error:

Missing INFER connection: Understanding that they need to find individual roots but not recognizing how to use Vieta's formulas to find k from the known roots of the third equation.

Students might correctly find that the roots are 2 and 8, but then get stuck on how these relate to finding k. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can systematically work through a chain of quadratic relationships. Success requires solid factoring skills combined with recognizing when and how to apply Vieta's formulas—it's not enough to just solve quadratics in isolation.