If x^2 − 3x + 1 = 0, compute the value of x + 1/x.

Introduction / Context:
When x satisfies a quadratic, expressions like x + 1/x can be simplified by using the quadratic to eliminate x^2. This avoids explicitly solving for x and keeps the algebra light and clean.

Given Data / Assumptions:

• x^2 − 3x + 1 = 0
• x ≠ 0 (since 1/x is used)

Concept / Approach:
Compute x + 1/x = (x^2 + 1)/x. Replace x^2 using the given quadratic: x^2 = 3x − 1. This method is standard for symmetric expressions in x and 1/x.

Step-by-Step Solution:
x + 1/x = (x^2 + 1)/xGiven x^2 = 3x − 1Hence (x^2 + 1)/x = (3x − 1 + 1)/x = 3x/x = 3

Verification / Alternative check:
Alternatively, divide the original equation by x: x − 3 + 1/x = 0 ⇒ x + 1/x = 3, same result.

Why Other Options Are Wrong:
0, 2, 1, −1 do not match the identity derived directly from the quadratic.

Common Pitfalls:
Forgetting to ensure x ≠ 0 (it is not a root here), or mishandling algebra when substituting x^2.

Final Answer:
3