Question

What is the vertex form of `y=-3x^2 -2x+5`?

Answer 1

`color(green)(y=-3(x+1/3)^2+5 1/3)`

Explanation:

Out target is the quadratic vertex form: `y=color(magenta)m(x-color(red)a)^2+color(blue)b`

We are given
`color(white)("XXX")y=-3x^2-2x+5`

Extracting the `color(magenta)m`
`color(white)("XXX")y=color(magenta)(-3)(x^2+2/3x)+5`

Completing the square
`color(white)("XXX")y=color(magenta)(-3)(x^2+2/3xcolor(brown)(+(1/3)^2))+5color(brown)(-(color(magenta)(-3))((1/3)^2))`

`color(white)("XXX")y=color(magenta)(x+1/3)^2+5+1/3`

`color(white)("XXX")y=color(magenta)(-3)(x-color(red)(""(-1/3)))^2+color(blue)(5 1/3)`

This is technically a more correct version than I give in the "Answer" since it clearly identifies the vertex as being at `(color(red)(-1/3),color(blue)(5 1/3))`
but I thought the "Answer" version was easier to read.
(Check with your instructor for the preferred form).