Roots Vertex 441D96

1. **State the problem:** Find the roots and vertex of the quadratic function $$y = -x^2 - 9x + 79$$. 2. **Recall the formulas:** - Roots are found by solving $$-x^2 - 9x + 79 = 0$$ using the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a = -1$$, $$b = -9$$, and $$c = 79$$. - The vertex $$ (h, k) $$ is given by $$h = -\frac{b}{2a}$$ and $$k = f(h)$$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-9)^2 - 4(-1)(79) = 81 + 316 = 397$$ 4. **Calculate the roots:** $$x = \frac{-(-9) \pm \sqrt{397}}{2(-1)} = \frac{9 \pm \sqrt{397}}{-2}$$ 5. **Simplify roots:** $$x_1 = \frac{9 + 19.924}{-2} = \frac{28.924}{-2} = -14.462$$ $$x_2 = \frac{9 - 19.924}{-2} = \frac{-10.924}{-2} = 5.462$$ 6. **Calculate the vertex:** $$h = -\frac{b}{2a} = -\frac{-9}{2(-1)} = -\frac{-9}{-2} = -4.5$$ 7. **Find $$k$$ by substituting $$h$$ into the function:** $$k = -(-4.5)^2 - 9(-4.5) + 79 = -20.25 + 40.5 + 79 = 99.25$$ **Final answers:** - Roots: $$-14.462$$ and $$5.462$$ - Vertex: $$(-4.5, 99.25)$$