Question

find the roots and the vertex of the quadratic on a calculator. round all values to 3 decimal places (if necessary).
y = x² + 6x - 91
answer attempt 1 out of 2
roots:
and
vertex: ( , )
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Explanation:

Step1: Find roots using quadratic formula

For quadratic \(ax^2 + bx + c = 0\), roots are \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = 6\), \(c=-91\).
Discriminant: \(b^2 - 4ac = 6^2 - 4(1)(-91)=36 + 364 = 400\).
Roots: \(x=\frac{-6\pm\sqrt{400}}{2(1)}=\frac{-6\pm20}{2}\).
First root: \(\frac{-6 + 20}{2}=\frac{14}{2}=7\).
Second root: \(\frac{-6 - 20}{2}=\frac{-26}{2}=-13\).

Step2: Find vertex using formula \(x = -\frac{b}{2a}\)

\(x = -\frac{6}{2(1)}=-3\).
Substitute \(x = -3\) into \(y = x^2 + 6x - 91\):
\(y=(-3)^2 + 6(-3)-91 = 9 - 18 - 91=-100\).

Answer:

Roots: \(7\) and \(-13\)
Vertex: \((-3, -100)\)