Find the roots and the vertex of the quadratic on a calculator. Round all values to 3 decims places (if necessary). \[ y=x^{2}+10 x-96 \]

To find the **roots** and the **vertex** of the quadratic equation \( y = x^{2} + 10x - 96 \), we'll use the following methods: ### 1. Finding the Roots: The roots of a quadratic equation \( ax^{2} + bx + c = 0 \) can be found using the **Quadratic Formula**: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] For the given equation: - \( a = 1 \) - \( b = 10 \) - \( c = -96 \) Plugging these values into the formula: \[ x = \frac{-10 \pm \sqrt{10^{2} - 4(1)(-96)}}{2(1)} = \frac{-10 \pm \sqrt{100 + 384}}{2} = \frac{-10 \pm \sqrt{484}}{2} \] \[ \sqrt{484} = 22 \] So, \[ x = \frac{-10 \pm 22}{2} \] This gives two solutions: \[ x = \frac{12}{2} = 6 \quad \text{and} \quad x = \frac{-32}{2} = -16 \] **Roots:** \( x = 6 \) and \( x = -16 \) ### 2. Finding the Vertex: The vertex \((h, k)\) of a quadratic equation \( y = ax^{2} + bx + c \) is given by: \[ h = -\frac{b}{2a} \] \[ k = y(h) = a h^{2} + b h + c \] Calculating \( h \): \[ h = -\frac{10}{2(1)} = -5 \] Calculating \( k \): \[ k = (-5)^{2} + 10(-5) - 96 = 25 - 50 - 96 = -121 \] **Vertex:** \( (-5.000, -121.000) \) ### **Summary:** - **Roots:** \( x = 6 \) and \( x = -16 \) - **Vertex:** \( (-5.000,\ -121.000) \)