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

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Solve this multiple-choice question and choose the correct option.

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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: I factors as (x − 12)^2 = 0 ⇒ x = 12 (double root).II factors as (y − 13)^2 = 0 ⇒ y = 13 (double root).Every valid x equals 12; every valid y equals 13. Hence for all choices, x = 12 and y = 13, so x < y.However, since only single values exist, the comparison is actually consistent: x < y. 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: x < y

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.

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