If x^2 + 1/x^2 = 2 for a nonzero real number x, then using algebraic identities, what is the value of the expression x - 1/x?

Introduction / Context:
This question uses a standard technique where a symmetric expression like x^2 + 1/x^2 is related to another symmetric expression like x - 1/x. By recalling and applying the correct identity, you can quickly go from one expression to the other without solving for x explicitly. These transformations are common in algebra and aptitude questions to test recognition of patterns and manipulation of exponents and reciprocals.

Given Data / Assumptions:

• x^2 + 1/x^2 = 2.
• x is a nonzero real number.
• We must find the value of x - 1/x.

Concept / Approach:
We use the identity (x - 1/x)^2 = x^2 + 1/x^2 - 2. This connects the given quantity x^2 + 1/x^2 with the desired quantity x - 1/x. After substituting the given value 2 into this identity, we will find the square of x - 1/x. Then we can deduce the actual value of x - 1/x. Importantly, we must consider which values are possible given the algebraic equation that x satisfies.

Step-by-Step Solution:
Start with the identity: (x - 1/x)^2 = x^2 + 1/x^2 - 2. We are given x^2 + 1/x^2 = 2. Substitute this into the identity: (x - 1/x)^2 = 2 - 2 = 0. So (x - 1/x)^2 = 0. The only real number whose square is 0 is 0 itself. Therefore, x - 1/x = 0. This simplifies to x = 1/x, which implies x^2 = 1. Hence x = 1 or x = -1, but in either case x - 1/x = 0.

Verification / Alternative check:
We can verify by solving for x directly from x^2 + 1/x^2 = 2. Multiply both sides by x^2 (x ≠ 0): x^4 + 1 = 2x^2. Rearrange to x^4 - 2x^2 + 1 = 0, which factors as (x^2 - 1)^2 = 0. So x^2 = 1, giving x = 1 or x = -1. Compute x - 1/x for both values. For x = 1, x - 1/x = 1 - 1 = 0. For x = -1, x - 1/x = -1 - (-1) = -1 + 1 = 0. In both cases, the expression equals 0, confirming our answer.

Why Other Options Are Wrong:
Values like 2, -2, 1, or -1 would give nonzero squares when substituted into (x - 1/x)^2. For example, if x - 1/x were 2, then x^2 + 1/x^2 would be 2 + 2 = 4, not 2. Therefore these other options contradict the given condition x^2 + 1/x^2 = 2.

Common Pitfalls:
Some students mistakenly use the formula for (x + 1/x)^2 instead of (x - 1/x)^2, which leads to x^2 + 1/x^2 + 2 and an incorrect value. Another mistake is to assume x must be a single numeric value like 1 and not consider -1, but here both possible x values give the same result for x - 1/x. Focus on the identity with the correct sign to avoid confusion.

Final Answer:
The value of the expression x - 1/x is 0.