1. **State the problem:** Solve the quadratic equation $x^2 - 2x - 1 = 0$. 2. **Recall the quadratic formula:** For any quadratic equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients. 3. **Identify coefficients:** Here, $a = 1$, $b = -2$, and $c = -1$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-2)^2 - 4 \times 1 \times (-1) = 4 + 4 = 8$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-2) \pm \sqrt{8}}{2 \times 1} = \frac{2 \pm \sqrt{8}}{2}$$ 6. **Simplify the square root:** $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$ 7. **Substitute back:** $$x = \frac{2 \pm 2\sqrt{2}}{2}$$ 8. **Simplify the fraction by canceling 2:** $$x = \frac{\cancel{2} \pm \cancel{2}\sqrt{2}}{\cancel{2}} = 1 \pm \sqrt{2}$$ 9. **Final solutions:** $$x = 1 + \sqrt{2} \quad \text{or} \quad x = 1 - \sqrt{2}$$