Use x + a/x = b to simplify a rational expression If x + a/x = b, evaluate (x^2 + b x + a) / (b x^2 − x^3).

Introduction / Context:This problem relies on manipulating the relation x + a/x = b. Converting it to a quadratic in x lets us replace x^2 terms and reduce a complicated rational expression to a simple ratio.

Given Data / Assumptions:

• x + a/x = b, with x ≠ 0 and a ≠ 0.
• Expression: E = (x^2 + b x + a) / (b x^2 − x^3).

Concept / Approach:From x + a/x = b, multiply by x to get x^2 + a = b x ⇒ x^2 − b x + a = 0. Use this to replace x^2 and also to simplify b − x in terms of a/x, which collapses the denominator neatly.

Step-by-Step Solution:From x^2 − b x + a = 0, we have x^2 = b x − a.Numerator: x^2 + b x + a = (b x − a) + b x + a = 2 b x.Also from x + a/x = b ⇒ b − x = a/x.Denominator: b x^2 − x^3 = x^2 (b − x) = x^2 * (a/x) = a x.Therefore, E = (2 b x) / (a x) = 2b/a.

Verification / Alternative check:Pick sample values a = 2, x satisfying x + 2/x = b; compute both sides numerically to confirm E = 2b/a regardless of the particular root x.

Why Other Options Are Wrong:a + b, b/a, and ab ignore the key transformations; a/(2b) flips the correct ratio.

Common Pitfalls:Losing track of signs when substituting x^2 = b x − a; not noticing that b − x = a/x.

Final Answer:2b/a