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.
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Ax > y
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BRelationship between x and y cannot be established
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Cx ≥ y
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Dx ≤ y
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Ex = y
Answer
Correct Answer: x ≤ y
Explanation
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:
x^2 − 20x + 91 = 0 ⇒ (x − 7)(x − 13) = 0 ⇒ roots 7, 13 ⇒ greater root x = 13 y^2 − 32y + 247 = 0 ⇒ (y − 13)(y − 19) = 0 ⇒ roots 13, 19 ⇒ greater root y = 19 Hence 13 < 19 ⇒ x < y ⇒ x ≤ yVerification / 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