Question

Question #abef5

Answer 1

Min: `(-3.5, -3.25)`
Roots: `x~=-5.3028` and `x~=-1.6972`

Explanation:

This polynomial is in the form

`y=ax^2+bx+c`

Where
`a=1`
`b=7`
`c=9`

The `x`-value of the minimum point of the parabola is given by

`x_min=-b/(2a)=-7/2~=-3.5`

The minimum `y`-value occurs when you plug that `x_min` into `y`

`y_min=(-7/2)^2+7(-7/2)+9=-13/4~=-3.25`

So the minimum point of the parabola is at the point `(-3.5, -3.25)`

Finally, you find where the parabola crosses the `x`-axis by plugging the given equation into the quadratic formula using `a`, `b`, and `c` from above.

`x=(-b+-sqrt(b^2-4ac))/(2a)`
`x=(-7+-sqrt((-7)^2-4(1)(9)))/(2(1))`
`x=-7+-1/2sqrt(13)`
So, `x~=-5.3028` and `x~=-1.6972`

Graphing gives:
graph{x^2+7x+9 [-6.4, 0.1, -3.49, 0.408]}