If x + 1/x = 2 (with x ≠ 0), compute the value of x − 1/x. Justify your reasoning clearly.

Introduction / Context:This is a quick identity-based question. When x + 1/x equals a particular constant, it often pins down x uniquely (for reals), enabling immediate evaluation of related expressions like x − 1/x without heavy algebra.

Given Data / Assumptions:

• x + 1/x = 2.
• x ≠ 0.

Concept / Approach:Observe that among real numbers, x + 1/x ≥ 2 with equality if and only if x = 1 (by AM ≥ GM or by completing the square). Thus x must be 1. Then compute x − 1/x directly. Alternatively, square x − 1/x and use the identity (x − 1/x)^2 = (x + 1/x)^2 − 4 to reach the same result.

Step-by-Step Solution:Since x + 1/x = 2, for real x we get x = 1.Hence x − 1/x = 1 − 1 = 0.Identity check: (x − 1/x)^2 = (2)^2 − 4 = 0 ⇒ x − 1/x = 0.

Verification / Alternative check:Plug x = 1 back into x + 1/x to verify it equals 2; then x − 1/x clearly vanishes.

Why Other Options Are Wrong:

• 1, 2, −2, −1: These disregard the equality case of the AM ≥ GM condition or misuse the identity for (x − 1/x)^2.

Common Pitfalls:Assuming multiple real solutions; forgetting that the minimum value of x + 1/x for real x is 2 at x = 1.

Final Answer:0