The real numbers $x$ and $y$ satisfy $x^2 + y^2 = 4x + 4y.$ Find the largest possible value of $x.$
+130511 x^2 + y^2 = 4x + 4y rearrange as
x^2 - 4x + y^2 - 4y = 0 complete the square on x and y
x^2 - 4x + 4 + y^2 - 4y + 4 = 4 + 4 simplify
(x - 2)^2 + (y - 2)^2 = 8
x will be maximzed when y = 2 ....so we have that
(x - 2)^2 + (2 - 2)^2 = 8
(x - 2)^2 = 8 take the positive root
x - 2 = √8
x = √8 + 2
x = 2√2 + 2
x = 2 (√2 + 1)
