Compare x and y (unique relation required): I. x^2 − 24x + 144 = 0 II. y^2 − 26y + 169 = 0 Choose the correct relationship between x and y: x > y, x < y, x = y, or cannot be determined.

Introduction / Context:
This is a comparison problem where any root from each quadratic may be chosen. If a single relationship holds for every valid pairing, we select it; otherwise we answer that the relationship cannot be determined.

Given Data / Assumptions:

• I: x^2 − 24x + 144 = 0.
• II: y^2 − 26y + 169 = 0.
• Consider all real roots for each equation.

Concept / Approach:
Factor both quadratics to list all roots. Then compare sets. If ordering varies across choices (or equality occurs in some and not others), the relationship is indeterminate.

Step-by-Step Solution:

Verification / Alternative check:
Compute numerically: 12 versus 13 ⇒ 12 < 13, no ambiguity.

Why Other Options Are Wrong:

• x > y or x = y: Contradicted by 12 and 13.
• Relationship cannot be determined: Here it is determined because each equation has a unique repeated root.

Common Pitfalls:
Misreading perfect squares or thinking “double root” adds variability. It does not; it simply repeats the same value.

Final Answer: