Find the other root (Vieta's use): One root of the quadratic equation x^2 − 5x + 6 = 0 is 3. Determine the other root.

Introduction / Context:
The classic way to find the second root of a quadratic when one root is known is to use the sum and product of roots. For ax^2 + bx + c = 0 with roots r1 and r2, r1 + r2 = −b/a and r1*r2 = c/a. With r1 given, solve for r2 quickly without factoring from scratch.

Given Data / Assumptions:

• Equation: x^2 − 5x + 6 = 0.
• One root r1 = 3.

Concept / Approach:
Use Vieta: Sum of roots = 5 (since −b/a = −(−5)/1 = 5). Then r2 = 5 − r1. Alternatively, factor the quadratic if convenient.

Step-by-Step Solution:

Verification / Alternative check:
Factorization: x^2 − 5x + 6 = (x − 2)(x − 3) = 0, confirming roots 2 and 3.

Why Other Options Are Wrong:

• −2, 1, −1: Do not satisfy both sum 5 and product 6 with the given root 3.

Common Pitfalls:
Sign mistakes in using −b/a for the sum, or mixing up which coefficient is b. Always write the standard form and read off a, b, c carefully.

Final Answer: