1. Let's solve the quadratic equation $x^2 - 6x - 10 = 0$ step by step. 2. The formula to solve any quadratic equation $ax^2 + bx + c = 0$ is the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. Here, $a = 1$, $b = -6$, and $c = -10$. 4. First, calculate the discriminant $\Delta = b^2 - 4ac$: $$\Delta = (-6)^2 - 4 \times 1 \times (-10) = 36 + 40 = 76$$ 5. Since $\Delta > 0$, there are two real solutions. 6. Now plug values into the quadratic formula: $$x = \frac{-(-6) \pm \sqrt{76}}{2 \times 1} = \frac{6 \pm \sqrt{76}}{2}$$ 7. Simplify $\sqrt{76}$: $$\sqrt{76} = \sqrt{4 \times 19} = 2\sqrt{19}$$ 8. Substitute back: $$x = \frac{6 \pm 2\sqrt{19}}{2}$$ 9. Cancel the common factor 2 in numerator and denominator: $$x = \frac{\cancel{2}(3 \pm \sqrt{19})}{\cancel{2}} = 3 \pm \sqrt{19}$$ 10. So the two solutions are: $$x = 3 + \sqrt{19} \quad \text{and} \quad x = 3 - \sqrt{19}$$ 11. These are the exact answers. You can approximate $\sqrt{19} \approx 4.36$ if you want decimal answers: $$x \approx 3 + 4.36 = 7.36$$ $$x \approx 3 - 4.36 = -1.36$$ Final answer: $x = 3 \pm \sqrt{19}$ or approximately $7.36$ and $-1.36$.