Known root in a quadratic, find the other root and coefficient: Given 2x^2 + P x + 4 = 0 has a root x = 2. Find the second root and the value of P.

Introduction / Context:
When one root of a quadratic is known, you can plug it into the equation to determine the missing coefficient. Then use Vieta’s relations (sum and product of roots) to find the second root efficiently without re-solving the entire quadratic from scratch.

Given Data / Assumptions:

• Quadratic: 2x^2 + P x + 4 = 0
• One root r1 = 2.

Concept / Approach:
Substitute r1 into the polynomial to determine P. Then apply sum of roots = −P/2 and product of roots = 4/2 = 2 to deduce the second root r2. Cross-check both sum and product to avoid slips.

Step-by-Step Solution:

Verification / Alternative check:
Product = c/a = 4/2 = 2; indeed 2 * 1 = 2. With P = −6, the quadratic is 2x^2 − 6x + 4 = 0 = 2(x^2 − 3x + 2) whose roots are 1 and 2.

Why Other Options Are Wrong:
Only (1, −6) matches both the coefficient value and the other root consistent with Vieta’s relations.

Common Pitfalls:
Forgetting to divide by the leading coefficient when using Vieta’s formulas or sign mistakes when computing −P/2.

Final Answer:
1, −6