Parameter k when x = 3 is a root of 3x^2 + (k − 1)x + 9 = 0. Find the value of k that satisfies this condition.

Introduction / Context:When a number r is a root of a polynomial, substituting x = r makes the polynomial equal zero. Here we are told x = 3 is a root of a quadratic with unknown parameter k, and we must determine k so that the equation holds true.

Given Data / Assumptions:

• Quadratic: 3x^2 + (k − 1)x + 9 = 0
• Condition: x = 3 is a solution

Concept / Approach:Use the root-substitution property: plug x = 3 into the left-hand side, set the result to zero, and solve the resulting linear equation in k.

Step-by-Step Solution:Substitute x = 3: 3*(3^2) + (k − 1)*3 + 9 = 0Compute: 3*9 + 3k − 3 + 9 = 0 ⇒ 27 + 3k − 3 + 9 = 0Combine constants: 27 − 3 + 9 = 33 ⇒ 3k + 33 = 0Solve: 3k = −33 ⇒ k = −11

Verification / Alternative check:Plug k = −11 back: 3x^2 + (−12)x + 9 = 0. For x = 3: 27 − 36 + 9 = 0, verified.

Why Other Options Are Wrong:13, −13, 11 do not satisfy 3k + 33 = 0; only k = −11 works. “−9” is a distractor without algebraic support.

Common Pitfalls:Arithmetic slips when adding constants after substitution. Carefully add 27, −3, and 9 to avoid errors.

Final Answer:−11