To solve the equation x^3 - 2x^2 + x = 0, we can factor out an x from each term:

x(x^2 - 2x + 1) = 0

Now, we can factor the quadratic expression inside the parentheses:

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

This equation is satisfied when any of the factors are equal to zero. Therefore, the solutions are:

x = 0, x = 1