1. **State the problem:** Find the roots and vertex of the quadratic function $$y = x^2 + 14x - 72$$. 2. **Formula for roots:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=14$$, and $$c=-72$$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 14^2 - 4(1)(-72) = 196 + 288 = 484$$. 4. **Find the roots:** $$x = \frac{-14 \pm \sqrt{484}}{2(1)} = \frac{-14 \pm 22}{2}$$ 5. **Calculate each root:** - $$x_1 = \frac{-14 + 22}{2} = \frac{8}{2} = 4$$ - $$x_2 = \frac{-14 - 22}{2} = \frac{-36}{2} = -18$$ 6. **Formula for vertex:** The vertex $$x$$-coordinate is $$x = -\frac{b}{2a} = -\frac{14}{2} = -7$$. 7. **Find the vertex $$y$$-coordinate:** Substitute $$x = -7$$ into the equation: $$y = (-7)^2 + 14(-7) - 72 = 49 - 98 - 72 = -121$$. 8. **Final answers:** - Roots: $$4$$ and $$-18$$ - Vertex: $$(-7, -121)$$