1. The problem is to solve the quadratic equation $x^2 - 2x + 1 = 0$. 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$. 3. Here, $a=1$, $b=-2$, and $c=1$. 4. We can solve this by factoring or using the quadratic formula. Let's try factoring first. 5. Notice that $x^2 - 2x + 1$ is a perfect square trinomial: it can be written as $(x - 1)^2$. 6. So, the equation becomes $(x - 1)^2 = 0$. 7. To solve, set the factor equal to zero: $x - 1 = 0$. 8. Therefore, $x = 1$. 9. This is the only solution, and it is a repeated root. Final answer: $x = 1$.