Question

What is the vertex form of `y=2x^2 +2x-8`?

Answer 1

`2(x+1/2)^2-17/2`

Explanation:

The vertex form of a quadratic equation looks like this:
`y=a(x-h)^2+k`

To get our equation into this form, we need to complete the square, but first I want to make the `x^2` term have a coefficient of `1` (you'll notice that the `x` inside the vertex form has this):
`2x^2+2x-8=2(x^2+x-4)`

To complete the square, we can use the following formula:
`x^2+px+q=(x+p/2)^2-(p/2)^2+q`

Applying this to `x^2+x-4`, we get:
`x^2+x-4=(x+1/2)^2-(1/2)^2-4=(x+1/2)^2-17/4`

Now we put this back into our original expression:
`2((x+1/2)^2-17/4)=2(x+1/2)^2-17/2`

And this is in vertex form, so it is our answer.

Answer 2

`y=2(x+1/2)^2-17/2`

Explanation:

`"the equation of a parabola in "color(blue)"vertex form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(y=a(x-h)^2+k)color(white)(2/2)|)))`

`"where "(h,k)" are the coordinates of the vertex and a"`
`"is a multiplier"`

`"to express in this form use "color(blue)"completing the square"`

`• " ensure the coefficient of the "x^2" term is 1"`

`rArry=2(x^2+x-4)`

`• " add/subtract "(1/2"coefficient of x-term")^2" to"`
`x^2+x`

`y=2(x^2+2(1/2)x color(red)(+1/4)color(red)(-1/4)-4)`

`color(white)(y)=2(x+1/2)^ 2+2xx-17/4`

`rArry=2(x+1/2)^2-17/2larrcolor(red)"in vertex form"`