1. The problem is to find the quadratic equation for the expression $x^2 - 2x - 8 = 0$. 2. The quadratic equation is generally written as $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. 3. Here, $a = 1$, $b = -2$, and $c = -8$. 4. To solve this quadratic equation, we can use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. Substitute the values of $a$, $b$, and $c$: $$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \times 1 \times (-8)}}{2 \times 1}$$ 6. Simplify inside the square root: $$x = \frac{2 \pm \sqrt{4 + 32}}{2}$$ 7. Calculate the square root: $$x = \frac{2 \pm \sqrt{36}}{2}$$ 8. Since $\sqrt{36} = 6$, we have two solutions: $$x = \frac{2 + 6}{2} = \frac{8}{2} = 4$$ $$x = \frac{2 - 6}{2} = \frac{-4}{2} = -2$$ 9. Therefore, the solutions to the quadratic equation $x^2 - 2x - 8 = 0$ are $x = 4$ and $x = -2$. 10. The quadratic equation itself is already given as $x^2 - 2x - 8 = 0$.