This is what the steps look like when we complete the square
x^2 - 6x + 6 = 0
x^2 - 6x = -6
x^2 - 6x + 9 = -6+9
(x-3)^2 = 3
We add 9 to both sides so that we can factor x^2-6x+9 into (x-3)(x-3) = (x-3)^2
To get the 9, we cut the -6 (from -6x) in half to get -6/2 = -3. Then we square that to get (-3)^2 = 9
It might help to look at (a+b)^2 = a^2+2ab+b^2 and note the relationship between 2ab and b^2.
