Difficulty: Medium
Correct Answer: 4/11
Explanation:
Introduction / Context:This algebra simplification hinges on interpreting the condition x + 1/x = 2. For real numbers, this equality forces x = 1 (since x + 1/x ≥ 2 with equality at x = 1). Substituting x = 1 into the given rational expression yields an exact fraction.
Given Data / Assumptions:
Concept / Approach:From the AM ≥ GM inequality or by direct algebra, x + 1/x = 2 implies x = 1. Plug this into the rational expression and simplify carefully to obtain the final fraction.
Step-by-Step Solution:
Since x + 1/x = 2, conclude x = 1.Compute numerator: 2x^2 + 2 → 2(1)^2 + 2 = 4.Compute denominator: 3x^2 + 5x + 3 → 3(1)^2 + 5(1) + 3 = 11.Hence value = 4 / 11.Verification / Alternative check:Try any x ≠ 1 and the condition fails; thus no other real x satisfies the premise. Therefore the direct substitution is valid and unique.
Why Other Options Are Wrong:1/2, 13/4, 7, and 2/11 are distractors that result from mis-substitution or arithmetic slips; none match the exact evaluation at x = 1.
Common Pitfalls:Attempting to manipulate x symbolically without first using the key implication x = 1, or miscomputing the denominator 3 + 5 + 3.
Final Answer:4/11
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