Parameter from a known root: If one root of x^2 − 6kx + 5 = 0 is 5, find the value of k.

Introduction / Context:
When a root of a polynomial is known, substituting it into the equation yields a direct relation among parameters. Here, plugging x = 5 into the quadratic allows us to solve for k in one step.

Given Data / Assumptions:

• Equation: x^2 − 6kx + 5 = 0
• One root is x = 5

Concept / Approach:
If r is a root, then r satisfies the equation identically. Substitute r = 5 and solve the resulting linear equation for k. No need for the quadratic formula here.

Step-by-Step Solution:

Verification / Alternative check:
With k = 1, equation becomes x^2 − 6x + 5 = 0, which factors as (x − 1)(x − 5) = 0, confirming 5 is indeed a root.

Why Other Options Are Wrong:

• −1/2, −1, 2, 0: Substitution does not result in zero; they fail the root condition for x = 5.

Common Pitfalls:
Forgetting to multiply 6k by x or mishandling algebraic signs when combining constants.

Final Answer:
1