Parameter k from a root-sum constraint: For x^2 − 6x + k = 0 with roots α and β, given 3α + 2β = 20, find the value of k.

Introduction / Context:This problem uses elementary symmetric relationships of quadratic roots: for x^2 − 6x + k = 0, the sum of roots is α + β = 6 and the product is αβ = k. A linear constraint involving the roots (3α + 2β = 20) lets us determine the individual roots and hence the parameter k.

Given Data / Assumptions:

• Equation: x^2 − 6x + k = 0
• Roots: α, β
• Condition: 3α + 2β = 20
• Sum α + β = 6; product αβ = k

Concept / Approach:Express β from the sum (β = 6 − α), substitute into 3α + 2β = 20, and solve for α. Then get β and compute k = αβ. This is a direct application of Vieta’s formulas combined with a linear relation.

Step-by-Step Solution:

Verification / Alternative check:Check the given condition: 3*8 + 2*(−2) = 24 − 4 = 20 (satisfied). Check sum: 8 + (−2) = 6 (matches Vieta). Everything is consistent.

Why Other Options Are Wrong:

• 8, −8, 16, 0: None equal αβ with (α, β) = (8, −2).

Common Pitfalls:Forgetting that α + β is fixed by the coefficient of x. Do not attempt to solve via quadratic formula first; the linear relation makes it faster.

Final Answer:−16