Question

find the roots and the vertex of the quadratic on a calculator. round all values to 3 decimal places (if necessary).
y = -3x² - 21x + 206
answer attempt 1 out of 2
roots:
and
vertex:
submit answer

Explanation:

Step1: Find the roots using quadratic formula

For a quadratic equation \(ax^2 + bx + c = 0\), the roots are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a=-3\), \(b = - 21\), \(c = 206\).
First, calculate the discriminant \(\Delta=b^2-4ac=(-21)^2-4\times(-3)\times206=441 + 2472=2913\)
Then, \(x=\frac{-(-21)\pm\sqrt{2913}}{2\times(-3)}=\frac{21\pm\sqrt{2913}}{-6}\)
\(\sqrt{2913}\approx53.972\)
So, \(x_1=\frac{21 + 53.972}{-6}=\frac{74.972}{-6}\approx - 12.495\)
\(x_2=\frac{21-53.972}{-6}=\frac{-32.972}{-6}\approx5.495\)

Step2: Find the vertex of the quadratic

The x - coordinate of the vertex of a quadratic \(y = ax^2+bx + c\) is given by \(x=-\frac{b}{2a}\)
For \(y=-3x^2-21x + 206\), \(a=-3\), \(b=-21\)
\(x=-\frac{-21}{2\times(-3)}=\frac{21}{-6}=-3.5\)
To find the y - coordinate, substitute \(x = - 3.5\) into the equation:
\(y=-3\times(-3.5)^2-21\times(-3.5)+206=-3\times12.25 + 73.5+206=-36.75+73.5 + 206=242.75\)

Answer:

Roots: \(-12.495\) and \(5.495\)
Vertex: \((-3.5,242.75)\)