Identify roots from coefficients: For ax^2 + (4a^2 − 3b)x − 12ab = 0 (a ≠ 0), choose the correct pair of roots.

Difficulty: Medium

4a and 3b/a
4a and −3b/a
−4a and 3b/a
−4a and −3b/a
3a and −4b/a

Correct Answer: −4a and 3b/a

Explanation:

Introduction / Context:
Instead of applying the quadratic formula, you can test candidate root pairs with Vieta’s relations. The sum of roots equals −B/A and the product equals C/A. Matching both conditions uniquely determines the correct pair and avoids lengthy algebra.

Given Data / Assumptions:

Quadratic: ax^2 + (4a^2 − 3b)x − 12ab = 0
Leading coefficient a ≠ 0.

Concept / Approach:
Let roots be r1 and r2. Then r1 + r2 = −(4a^2 − 3b)/a = −4a + 3b/a and r1*r2 = (−12ab)/a = −12b. Test the options to see which pair has the correct sum and product simultaneously.

Step-by-Step Solution:

Option C proposes r1 = −4a and r2 = 3b/a. Sum = −4a + 3b/a which matches −(4a^2 − 3b)/a. Product = (−4a)*(3b/a) = −12b which matches C/A. Therefore, Option C satisfies both Vieta conditions exactly.

Verification / Alternative check:
Other options either flip a sign in the sum or product. Checking them quickly shows inconsistency with at least one of Vieta’s relations.

Why Other Options Are Wrong:
Wrong sign on either 4a or 3b/a changes the sum and/or product, violating the equation’s coefficients.

Common Pitfalls:
Forgetting the minus in the constant term or miscomputing −B/A when B = 4a^2 − 3b.

Identify roots from coefficients: For ax^2 + (4a^2 − 3b)x − 12ab = 0 (a ≠ 0), choose the correct pair of roots.

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