Question

What is the vertex of `y= -3x^2-2x-1+(2x-1)^2`?

Answer 1

vertex: `(x,y)=(3,-9)`

Explanation:

First simplify the given equation:
`color(white)("XXX")y=color(orange)(-3x^2-2x-1)+color(brown)((2x-1)^2)`

`color(white)("XXX")y=color(orange)(-3x^2-2x-1)+color(brown)(4x^2-4x+1)`

`color(white)("XXX")y=x^2-6x`

One of the easiest ways of finding the vertex is to convert the equation into "vertex form":
`color(white)("XXX")y=color(green)(m)(x-color(red)(a))^2+color(blue)(b)` with vertex at `(color(red)(a),color(blue)(b))`
by "completing the square"

(Note that in this case we can ignore `color(green)(m)` or write it with its implied value of `color(green)(1)`).

`color(white)("XXXXXX")`Remember `(x+k)^2 = x^2+2kx+k^2`
`color(white)("XXXXXX")`So in this case `k=-3`
`color(white)("XXXXXX")` and we will need to add `(-3)^2` to complete the square

`color(white)("XXX")y=x^2-6xcolor(purple)(+9-9)`

`color(white)("XXX")y=(x-color(red)(3))^2+color(blue)("("-9")")`

which is in vertex form with the vertex at `(color(red)(3),color(blue)("("-9")"))`

Here is a graph of the original equation to help verify our result:
graph{-3x^2-2x-1+(2x-1)^2 [-7.46, 12.54, -10.88, -0.88]}