Given one root of 3x^2 − 10x + 3 = 0 is 1/3. Find the other root of the quadratic.

Introduction / Context:
For a quadratic ax^2 + bx + c = 0, the sum and product of roots are given by S = −b/a and P = c/a. Knowing one root allows quick computation of the other using the product (or sum).

Given Data / Assumptions:

• Quadratic: 3x^2 − 10x + 3 = 0
• One root r1 = 1/3

Concept / Approach:
Use product of roots P = c/a. If r1 is known, then r2 = P / r1. This avoids solving the quadratic again. Optionally, confirm by substitution or by using the sum of roots formula.

Step-by-Step Solution:
a = 3, b = −10, c = 3 ⇒ P = c/a = 3/3 = 1Given r1 = 1/3, then r2 = P / r1 = 1 / (1/3) = 3

Verification / Alternative check:
Check via substitution: For x = 1/3, LHS = 3*(1/9) − 10*(1/3) + 3 = 1/3 − 10/3 + 3 = 0. For x = 3, LHS = 27 − 30 + 3 = 0. Both satisfy the equation.

Why Other Options Are Wrong:
1/3 is the given root, not the other; −3 and 10/3 do not satisfy the equation as roots; “None of these” is unnecessary as a correct value exists among options.

Common Pitfalls:
Using the sum instead of product incorrectly or mishandling fractions. Product method is clean and fast here.

Final Answer:
3