Compare greater roots from two quadratics (definition repaired): Let x be the greater root of 6x^2 + 41x + 63 = 0 and let y be the greater root of 4y^2 + 8y + 3 = 0. Choose the correct relation between x and y.

Introduction / Context:
The original “I/II” style stem lacked the comparison rule. Using the Recovery-First Policy, we define x and y as the greater roots of the two quadratics and compare them. This is a common exam pattern in quantity comparison questions.

Given Data / Assumptions:

• x is the larger root of 6x^2 + 41x + 63 = 0.
• y is the larger root of 4y^2 + 8y + 3 = 0.

Concept / Approach:
Compute roots using the quadratic formula. For ax^2 + bx + c = 0, roots are (−b ± √(b^2 − 4ac)) / (2a). Identify the greater root for each equation, then compare numerically (exact fractions avoid rounding errors).

Step-by-Step Solution:

Verification / Alternative check:
Because both larger roots are negative, the less negative (closer to zero) is the greater number; −0.5 is greater than −2.333…

Why Other Options Are Wrong:
x ≥ y, x > y, equality, or indeterminate are contradicted by the exact computed values.

Common Pitfalls:
Picking the more negative value as “larger” by mistake; always compare on the number line carefully.

Final Answer:
x < y