Radical simplification leading to a golden-ratio identity: If x = √( (√5 + 1)/(√5 − 1) ), evaluate x^2 − x − 1.

Introduction / Context:
This expression is designed to connect to the golden ratio φ, which satisfies φ^2 − φ − 1 = 0. By rationalizing the fraction under the square root, we can recognize x^2 as φ^2, thereby evaluating x^2 − x − 1 exactly without approximation.

Given Data / Assumptions:

• x = √( (√5 + 1)/(√5 − 1) ), principal square root
• All radicals are the positive branches

Concept / Approach:
First compute x^2 by removing the outer square root. Then rationalize the denominator inside x^2 by multiplying numerator and denominator by (√5 + 1). Simplify the resulting expression and compare with the known values for the golden ratio.

Step-by-Step Solution:

Verification / Alternative check:
Numerically, √5 ≈ 2.236; x ≈ √( (3.236)/(1.236) ) ≈ √(2.618) ≈ 1.618, which is φ. Then x^2 − x − 1 ≈ 2.618 − 1.618 − 1 ≈ 0.

Why Other Options Are Wrong:

• 1, 2, 5, −1: Contradict the defining identity φ^2 − φ − 1 = 0 satisfied by this x.

Common Pitfalls:
Not rationalizing properly or assuming x itself equals (√5 + 1)/2 without justification. Work via x^2 first for clarity.

Final Answer:
0