Roots Vertex 81Aa6E

1. **State the problem:** We need to find the roots and the vertex of the quadratic function $$y = x^2 - 7x - 109$$. 2. **Formula for roots:** The roots of a quadratic equation $$ax^2 + bx + c = 0$$ are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = 1$$, $$b = -7$$, and $$c = -109$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-7)^2 - 4(1)(-109) = 49 + 436 = 485$$ 5. **Calculate the roots:** $$x = \frac{-(-7) \pm \sqrt{485}}{2(1)} = \frac{7 \pm \sqrt{485}}{2}$$ Calculate $$\sqrt{485} \approx 22.022$$. So, $$x_1 = \frac{7 + 22.022}{2} = \frac{29.022}{2} = 14.511$$ $$x_2 = \frac{7 - 22.022}{2} = \frac{-15.022}{2} = -7.511$$ 6. **Formula for vertex:** The vertex $$ (h, k) $$ of a parabola $$y = ax^2 + bx + c$$ is given by: $$h = -\frac{b}{2a}$$ $$k = f(h) = a h^2 + b h + c$$ 7. **Calculate vertex x-coordinate:** $$h = -\frac{-7}{2(1)} = \frac{7}{2} = 3.5$$ 8. **Calculate vertex y-coordinate:** $$k = (3.5)^2 - 7(3.5) - 109 = 12.25 - 24.5 - 109 = -121.25$$ 9. **Final answers:** Roots: $$14.511$$ and $$-7.511$$ Vertex: $$(3.5, -121.25)$$