Question Find the roots and the vertex of the quadratic on a calculator. Round all values to 3 decimal places (if necessary). \[ y=20 x^{2}+180 x-567 \] Answer stamptrout of 2 Roots: \( \square \) and \( \square \)

To find the roots and the vertex of the quadratic equation: \[ y = 20x^{2} + 180x - 567 \] we'll use the quadratic formula and vertex formulas. All values will be rounded to three decimal places. ### **1. Finding the Roots** The quadratic equation can be solved using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where: - \( a = 20 \) - \( b = 180 \) - \( c = -567 \) **Step 1: Calculate the Discriminant (\( D \))** \[ D = b^2 - 4ac = (180)^2 - 4(20)(-567) = 32400 + 45360 = 77760 \] **Step 2: Calculate the Square Root of the Discriminant** \[ \sqrt{D} = \sqrt{77760} \approx 278.856 \] **Step 3: Find the Roots** \[ x = \frac{-180 \pm 278.856}{40} \] - **First Root:** \[ x_1 = \frac{-180 + 278.856}{40} = \frac{98.856}{40} \approx 2.471 \] - **Second Root:** \[ x_2 = \frac{-180 - 278.856}{40} = \frac{-458.856}{40} \approx -11.471 \] ### **2. Finding the Vertex** The vertex of a quadratic equation \( y = ax^2 + bx + c \) is given by: \[ \text{Vertex} \left( h, k \right) \] where: \[ h = -\frac{b}{2a} \] \[ k = y(h) = 20h^2 + 180h - 567 \] **Calculating \( h \):** \[ h = -\frac{180}{2 \times 20} = -\frac{180}{40} = -4.5 \] **Calculating \( k \):** \[ k = 20(-4.5)^2 + 180(-4.5) - 567 = 20(20.25) - 810 - 567 = 405 - 810 - 567 = -972 \] **Vertex:** \[ (-4.500, -972.000) \] ### **Final Answer** **Roots:** \( 2.471 \) and \( -11.471 \) **Vertex:** \( (-4.500,\ -972.000) \)