Let x be the greater real root of x^2 + x − 20 = 0 and y be the greater real root of y^2 − y − 30 = 0. Compare x and y and choose the correct relation.
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AIf x > y
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BIf x ≥ y
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CIf x < y
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DIf x ≤ y
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EIf x = y
Answer
Correct Answer: If x < y
Explanation
Introduction / Context: The original statement was ambiguous because each quadratic has two roots. Applying the Recovery-First Policy, we define x and y explicitly as the greater real roots. We then compare these values by solving each quadratic via factoring.
Given Data / Assumptions:
- x solves x^2 + x − 20 = 0 and is the greater root.
- y solves y^2 − y − 30 = 0 and is the greater root.
- We compare x to y.
Concept / Approach: Factor each quadratic to get roots quickly. For ax^2 + bx + c with small integers, try integer factor pairs of c that sum to b. Identify the larger root in each case and compare.
Step-by-Step Solution:
x^2 + x − 20 = 0 ⇒ (x + 5)(x − 4) = 0 ⇒ roots: −5, 4 ⇒ greater x = 4.y^2 − y − 30 = 0 ⇒ (y − 6)(y + 5) = 0 ⇒ roots: 6, −5 ⇒ greater y = 6.Therefore x = 4 and y = 6 ⇒ x < y.Verification / Alternative check: Quadratic formula gives identical results; factoring is sufficient here.
Why Other Options Are Wrong: x > y or x ≥ y contradict the computed values; equality does not hold.
Common Pitfalls: Not clarifying which root is used; comparing mismatched (e.g., larger with smaller) roots gives inconsistent relations.
Final Answer: If x < y