1. **State the problem:** Simplify the expression $x^2 - 2x + 1$. 2. **Recall the formula:** This is a quadratic expression that can be factored using the perfect square trinomial formula: $$a^2 - 2ab + b^2 = (a - b)^2$$ 3. **Identify terms:** Here, $a = x$ and $b = 1$. 4. **Factor the expression:** $$x^2 - 2x + 1 = (x - 1)^2$$ 5. **Explain:** The expression is a perfect square because it fits the pattern of $(a - b)^2$ where $a = x$ and $b = 1$. 6. **Final answer:** $$x^2 - 2x + 1 = (x - 1)^2$$