Question

find the roots and the vertex of the quadratic on a calculator. round all values to 3 decimal places (if necessary).
y = x² + 20x + 84
answer attempt 1 out of 2
roots:
and
vertex: (
,
)

Explanation:

Step1: Find the roots using quadratic formula

For a quadratic equation \(y = ax^2 + bx + c\), the roots are given by \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Here, \(a = 1\), \(b = 20\), \(c = 84\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(20)^{2}-4\times1\times84 = 400 - 336=64\).
Then, the roots are \(x=\frac{-20\pm\sqrt{64}}{2\times1}=\frac{-20\pm8}{2}\).
For the plus sign: \(x=\frac{-20 + 8}{2}=\frac{-12}{2}=-6\).
For the minus sign: \(x=\frac{-20-8}{2}=\frac{-28}{2}=-14\).

Step2: Find the vertex of the quadratic

The x - coordinate of the vertex of a quadratic \(y = ax^2+bx + c\) is given by \(x=-\frac{b}{2a}\).
Substitute \(a = 1\), \(b = 20\) into the formula: \(x=-\frac{20}{2\times1}=- 10\).
To find the y - coordinate, substitute \(x=-10\) into the quadratic equation \(y=x^{2}+20x + 84\).
\(y=(-10)^{2}+20\times(-10)+84=100-200 + 84=-16\).

Answer:

Roots: \(-6\) and \(-14\)
Vertex: \((-10,-16)\)