$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$ - I keep getting imaginary numbers $$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$$ 
My attempt
$\sqrt{x+938^2} + \sqrt{x + 140^2} = 1116$
$(\sqrt{x+938^2} + \sqrt{x + 140^2})^2 = (1116)^2$
$x+938^2  + 2*\sqrt{x+938^2}*\sqrt{x + 140^2} + x + 140^2 = 1116^2$
$2x + 2*\sqrt{x+938^2}*\sqrt{x + 140^2}  = 1116^2 - 938^2 - 140^2$
$x + \sqrt{x^2 + 2(938^2 + 140^2)x+(938*140)^2} = 1116^2 - 938^2 - 140^2$
At this point trying to solve for x inside the sqrt in the quadratic gives me an imaginary number. How is it possible to solve this?

You received good answers and solutions. However, it seems that you made a few mistakes in your calculations. The line $$2x + 2\sqrt{x+938^2}\sqrt{x + 140^2}  = 1116^2 - 938^2 - 140^2$$ is correct.  What is not correct is in the next line since $$(x+938^2)(x + 140^2)=x^2+(938^2+140^2)x+938^2\times 140^2$$ I do not know how appeared the factor $2$ in the radical and disappeared in other places.
Using this correction, we then have $$2x+2\sqrt{x^2+(938^2+140^2)x+938^2\times 140^2}=346012$$ Dividing both sides by $2$ and moving the $x$ term to the rhs then gives $$\sqrt{x^2+899444 x+17244942400}=173006-x$$ Squaring again, the $x^2$ disappear and what is left is $$1245456 x-12686133636=0\implies x=\frac{352392601}{34596}$$
But, as Achille Hui answered, replace your numbers by letters to avoid these enormous intermediate numbers.
