1. **State the problem:** Solve the quadratic equation $x^2 - x - 2 = 0$. 2. **Recall the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-1$, and $c=-2$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-1)^2 - 4 \times 1 \times (-2) = 1 + 8 = 9$$ 4. **Find the square root of the discriminant:** $$\sqrt{\Delta} = \sqrt{9} = 3$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-1) \pm 3}{2 \times 1} = \frac{1 \pm 3}{2}$$ 6. **Calculate the two solutions:** - For the plus sign: $$x = \frac{1 + 3}{2} = \frac{4}{2} = 2$$ - For the minus sign: $$x = \frac{1 - 3}{2} = \frac{\cancel{1} - 3}{\cancel{2}} = \frac{-2}{2} = -1$$ 7. **Final answer:** The solutions to the equation $x^2 - x - 2 = 0$ are $$x = 2 \quad \text{and} \quad x = -1$$