Question

How do you solve by completing the square for `2x^2 + 2x = 0`?

Answer 1

It's a rather strange way to solve it as it is easy to see that `x=0` and `x=-1` are both solutions. However, here's how you would do it...

`0 = 2x^2+2x = 2(x+1/2)^2 - 1/2`

Add 1/2 to both sides to get

`2(x+1/2)^2 = 1/2`

Divide both sides by 2 to get

`(x+1/2)^2 = 1/4`

So

`x+1/2 = +-sqrt(1/4) = +-sqrt(1)/sqrt(2^2) = +-1/2`

Subtract 1/2 from both sides to get

`x=-1/2+-1/2=(-1+-1)/2`

Answer 2

If `2x^2 + 2x = 0`, then `x^2 + x = 0`.

To complete this square, I need to square half of the coefficient of the 'x', which is 1/2.

`(1/2)^2=1/4` , so `x^2 + x + 1/4 = 1/4`

`(x + 1/2)^2=1/4`

`|x + 1/2| = sqrt(1/4)` (square roots can be positive or negative)

`|x + 1/2| = 1/2`

`x = {-1, 0}`