x^2 + y^2 = 2x + 2y
x^2 - 2x + y^2 - 2y = 0
Complete the square on x and y and we have that
x^2 - 2x + 1 + y^2 -2y + 1 = 2
( x - 1)^2 + (y - 1)^2 = 2
x will be maximized when y = 1...so....
(x - 1)^2 = 2 take the positive square root
x - 1 = √2
x = √2 + 1