Compare roots from two quadratics (check all combinations): I) 24x^2 + 38x + 15 = 0 II) 12y^2 + 28y + 15 = 0 Decide the correct relationship that always holds between any root x of I and any root y of II.

Introduction / Context:When comparing two quadratics, each with two real roots, we must confirm that a proposed inequality holds across all four pairings (each x with each y). A single counterexample invalidates a universal statement like x ≥ y. Here we solve both quadratics and compare the root sets carefully.

Given Data / Assumptions:

• I) 24x^2 + 38x + 15 = 0
• II) 12y^2 + 28y + 15 = 0
• All comparisons are between any root x of I and any root y of II.

Concept / Approach:Compute exact roots via discriminants. Then order the roots numerically and check whether the smallest x is still ≥ the largest y (for x ≥ y), or similar tests for other relations. If equality can occur for some pairing and strict inequality for others, then ≥ (not >) may be the correct consistent relation.

Step-by-Step Solution:

Verification / Alternative check:Test all pairings: (−0.75 ≥ −0.833…), (−0.75 ≥ −1.5), (−0.833… ≥ −0.833…), (−0.833… ≥ −1.5). All hold; at least one pairing yields equality, so x ≥ y is the strongest always-true statement.

Why Other Options Are Wrong:

• x > y: Fails at equality case (−5/6 vs −5/6).
• x ≤ y or x < y: Contradicted by −0.75 ≥ −0.833…
• Relationship cannot be determined: We just established a universal relation.

Common Pitfalls:Overlooking equal roots across equations or assuming strict inequality without testing equality. Always check extremes to decide between > and ≥.

Final Answer:x ≥ y