If \(x + \frac{1}{x} = 2\), then which one of the following is the value of \(x^{32} + \frac{1}{x^{32}}\) ?

To solve the equation x + 1/x = 2, we first convert it into a standard quadratic form. Multiplying the entire equation by x yields x² + 1 = 2x, which simplifies to x² - 2x + 1 = 0. This is a perfect square identity (x - 1)² = 0, implying that the only root is x = 1. Substituting x = 1 into the expression x³² + 1/x³² gives 1³² + 1/1³², which equals 1 + 1 = 2. This follows the principle that a solution must satisfy the original equation. In algebraic identities, when the sum of a variable and its reciprocal equals 2, the variable must be 1, and any positive integer power of that variable will also result in 1. Thus, 1 raised to any power remains 1, making the final sum 2.

Sources

1. [1] https://en.wikipedia.org/wiki/Quadratic_formula