Let $x$ and $y$ be real numbers. If $x$ and $y$ satisfy x^2 + y^2 = 4x + 2y then find the largest possible value of $x.$ Give your answer in exact form using radicals, simplified as far as possible.
x^2 - 4x + y^2 - 2y = 0 complete the square on x,y
x^2 - 4x + 4 + y^2 - 2y + 1 = 4 +1
(x- 2)^2 + ( y - 1)^2 = 5
This is a circle centered at ( 2 , 1) with a radius of sqrt 5
The largest value of x = 2 + sqrt 5
+1167 
