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).

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Solve this multiple-choice question and choose the correct option.

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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: Discriminant Δ = (−ab)^2 − 4*(3a^2)*(−2b^2) = a^2 b^2 + 24 a^2 b^2 = 25 a^2 b^2 √Δ = 5 a b x = [ab ± 5ab] / (6a^2) = (ab/(6a^2)) * (1 ± 5) = (b/(6a)) * (1 ± 5) Roots: x1 = (b/(6a))*6 = b/a, x2 = (b/(6a))*(−4) = −2b/(3a) 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)

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).

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