Difficulty: Medium
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:
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.
Final Answer:−4a and 3b/a
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