Moment & Equilibrium

You do realize that your 20+10sqrt(2) is longer than the 10 base which is obviously physically impossible as you can't remove what doesn't exist.
Will look through the numbers now.

I get that we have a restriction that the square base must be of 10 size. Therefore, x = 20 - 10√2. But what are you trying to imply when you say that:
"you can't remove what doesn't exist."

Apart from that, I totally understand what you are saying and have arrived at the right answer.

x is the amount you're removing from the base. You're trying to remove a length of 34 from a base of length 10. I can't see how you'd do that?
20-10sqrt(2) ~ 6 which sounds plausible. Simple geometry for balancing a rectangle area (right) and a trapezium area (left) would show that.