Question

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

Answer 1

`y = -2(x-1/2 - sqrt(5)/2)(x-1/2 + sqrt(5)/2)`

Explanation:

Separate out the scalar factor `-2`, complete the square, then use the difference of squares identity.

The difference of squares identity can be written:

`a^2-b^2 = (a-b)(a+b)`

We use this with `a=(x-1/2)` and `b=sqrt(5)/2` as follows:

`y = -2x^2+2x+2`

`color(white)(y) = -2(x^2-x-1)`

`color(white)(y) = -2(x^2-x+1/4 - 5/4)`

`color(white)(y) = -2((x-1/2)^2 - (sqrt(5)/2)^2)`

`color(white)(y) = -2((x-1/2) - sqrt(5)/2)((x-1/2) + sqrt(5)/2)`

`color(white)(y) = -2(x-1/2 - sqrt(5)/2)(x-1/2 + sqrt(5)/2)`