Compare greater roots (definition repaired): Let x be the greater root of x^2 − 20x + 91 = 0 and y be the greater root of y^2 − 32y + 247 = 0. Choose the correct relation between x and y.

Introduction / Context:
To resolve ambiguity, we define x, y as the greater roots of their respective quadratics. We then factor or use the quadratic formula to find the larger root for each and compare the results directly.

Given Data / Assumptions:

• x satisfies x^2 − 20x + 91 = 0 (greater root).
• y satisfies y^2 − 32y + 247 = 0 (greater root).

Concept / Approach:
Factorization by integers helps: look for pairs with the correct sum and product. Once each pair of roots is found, pick the larger one and compare numerically.

Step-by-Step Solution:

Verification / Alternative check:
Cross-check products: 7*13 = 91 and 13*19 = 247; sums 20 and 32, confirming correct factorization.

Why Other Options Are Wrong:
x > y and x ≥ y contradict the actual numbers; “cannot be established” is false because both larger roots are uniquely known; equality is false since 13 ≠ 19.

Common Pitfalls:
Swapping smaller and larger roots; forgetting that both equations have positive distinct roots here.

Final Answer:
x ≤ y