1. **State the problem:** Find the roots and the vertex of the quadratic function $$y = x^2 - 8x - 48$$. 2. **Formula for roots:** The roots are the solutions to $$y=0$$, so solve $$x^2 - 8x - 48 = 0$$. 3. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=-8$$, and $$c=-48$$. 4. **Calculate the discriminant:** $$b^2 - 4ac = (-8)^2 - 4(1)(-48) = 64 + 192 = 256$$. 5. **Find the roots:** $$x = \frac{-(-8) \pm \sqrt{256}}{2(1)} = \frac{8 \pm 16}{2}$$ 6. **Calculate each root:** $$x_1 = \frac{8 + 16}{2} = \frac{24}{2} = 12$$ $$x_2 = \frac{8 - 16}{2} = \frac{\cancel{8} - 16}{\cancel{2}} = \frac{-8}{2} = -4$$ 7. **Formula for vertex:** The vertex $$x$$-coordinate is $$x = -\frac{b}{2a}$$. 8. **Calculate vertex $$x$$-coordinate:** $$x = -\frac{-8}{2(1)} = \frac{8}{2} = 4$$ 9. **Calculate vertex $$y$$-coordinate:** Substitute $$x=4$$ into the original equation: $$y = (4)^2 - 8(4) - 48 = 16 - 32 - 48 = -64$$ 10. **Final answers:** Roots: $$12$$ and $$-4$$ Vertex: $$(4, -64)$$