1. **State the problem:** Solve the equation $$(x-1)^2 + 2(x-1) + x^2 = 0$$ for $x$. 2. **Expand and simplify:** $$(x-1)^2 = x^2 - 2x + 1$$ So the equation becomes: $$x^2 - 2x + 1 + 2(x-1) + x^2 = 0$$ 3. **Distribute and combine like terms:** $$x^2 - 2x + 1 + 2x - 2 + x^2 = 0$$ Combine like terms: $$x^2 + x^2 - 2x + 2x + 1 - 2 = 0$$ $$2x^2 - 1 = 0$$ 4. **Isolate $x^2$:** $$2x^2 = 1$$ 5. **Divide both sides by 2:** $$\cancel{2}x^2 = \cancel{2} \times \frac{1}{2}$$ $$x^2 = \frac{1}{2}$$ 6. **Take the square root of both sides:** $$x = \pm \sqrt{\frac{1}{2}} = \pm \frac{\sqrt{2}}{2}$$ **Final answer:** $$x = \frac{\sqrt{2}}{2} \text{ or } x = -\frac{\sqrt{2}}{2}$$