Geometry

First of all, let's move almost all the variables to one side of the equation and combine all the like terms. 

We get $x^2 - 21x + y^2 + 13y = 25$. 

Now, we complete the square for both x and y on the right side of the equation. 

$x^2 - 21x + 441/4 + y^2 + 13y + 169/4 = 25 + 441/4 + 169/4$

$(x - 21/2)^2 + (y + 13/2)^2 = 355 / 2$

Wait! This looks like the equation for a circle! 

Indeed, we have a circle at center  $(21/2 , -13/2)$ with radius $\sqrt{355 / 2}$. 

The value of the largest possible x is just the radius added onto the x value of the center. 

$x = (21/2) + \sqrt{355 / 2}$

Thanks! :)