Question

find the roots and the vertex of the quadratic on a calculator. round all values to 3 decimal places (if necessary).
y = x² - 12x - 85
answer attempt 1 out of 2
roots:
and
vertex: (
,
)

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, \(a = 1\), \(b=-12\), \(c = - 85\).
First, calculate the discriminant \(\Delta=b^2-4ac=(-12)^2-4\times1\times(-85)=144 + 340 = 484\).
Then, \(x=\frac{12\pm\sqrt{484}}{2}=\frac{12\pm22}{2}\).
For the plus sign: \(x_1=\frac{12 + 22}{2}=\frac{34}{2}=17\).
For the minus sign: \(x_2=\frac{12-22}{2}=\frac{-10}{2}=-5\).

Step2: Find the vertex of the parabola

The \(x\)-coordinate of the vertex of a parabola \(y = ax^2+bx + c\) is given by \(x=-\frac{b}{2a}\).
Substituting \(a = 1\), \(b=-12\), we get \(x=-\frac{-12}{2\times1}=6\).
To find the \(y\)-coordinate, substitute \(x = 6\) into the equation \(y=x^2-12x - 85\).
\(y=(6)^2-12\times6-85=36-72 - 85=-121\).

Answer:

Roots: \(17\) and \(-5\)
Vertex: \((6, - 121)\)