Question

How do you find the roots, real and imaginary, of `y=-2(x +3)^2-5x + 6` using the quadratic formula?

Answer 1

Roots: `x in {-7.723, -0.777}`

Explanation:

First convert the given equation:
`y=-2(x+3)^2-5x+6` into standard quadratic form:

`y=-2(x^2+6x+9)-5x+6`

`y=-2x^2-12x-18-5x+6`

`y=-2x^2-17x-12`

Apply the quadratic formula:
`color(white)("XXX")x=(-b+-sqrt(b^2-4ac))/(2a)`
with `a=-2`, `b=-17`, and `c=-12`

`x=(17+-sqrt(17^2-4(2)(12)))/(2(-2)`
which (with a bit of arithmetic) gives:
`color(white)("XXX")x~~-7.723` or `x~~-0.777`