To solve the equation:

(x-1)² - 2/(x-1) = 0

First, let's simplify the equation:

(x² - 2x + 1) - 2/(x-1) = 0
(x² - 2x + 1) - 2/(x-1)

To simplify further, let's find a common denominator for the fraction:

(x² - 2x + 1) - 2(x-1)/(x-1) = 0
(x² - 2x + 1) - 2(x-1)/(x-1)

Now, simplify in the numerator:

(x² - 2x + 1) - 2x + 2 = 0
x² - 4x + 3 = 0

Now, we have a quadratic equation that can be factored:

(x-3)(x-1) = 0

So, x = 3 or x = 1

Therefore, the solutions to the equation are x = 3 or x = 1.