Solve for roots in terms of parameters: Find the roots of 3a^2 x^2 − a b x − 2 b^2 = 0 in terms of a and b (assume a ≠ 0).

Introduction / Context:
This is a parameterized quadratic. You are asked to express the two roots in terms of a and b. The quadratic formula applies directly; careful algebraic simplification of the discriminant reveals simple rational multiples of b/a.

Given Data / Assumptions:

• Equation: 3a^2 x^2 − a b x − 2 b^2 = 0
• Assume a ≠ 0 to avoid division by zero.

Concept / Approach:
Use the quadratic formula x = [ab ± √(a^2 b^2 + 24 a^2 b^2)]/(2*3a^2). Factor a^2 b^2 out of the square root to simplify. Then reduce the fraction to a clean form in terms of b/a.

Step-by-Step Solution:

Verification / Alternative check:
Sum of roots = (b/a) + (−2b/(3a)) = b/(3a) = −(coefficient of x)/(coefficient of x^2) = −(−ab)/(3a^2) = b/(3a). Product = (b/a)*(−2b/(3a)) = −2b^2/(3a^2) = constant term / leading coefficient = (−2b^2)/(3a^2). Checks out.

Why Other Options Are Wrong:
Sign errors or swapped signs do not satisfy Vieta’s relations for this quadratic.

Common Pitfalls:
Missing the negative sign in the constant term or mishandling √(25 a^2 b^2) = 5ab can lead to incorrect roots.

Final Answer:
b/a and −2b/(3a)