$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$ - 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?

For this sort of problem, sometimes it will be useful to replace 
explicit numbers by variables to avoid distractions.
Let $a = 938$, $b = 140$, $c = 938 + 140 + 38 = 1116$.
Let $u = \sqrt{x+a^2}$ and $v = \sqrt{x+b^2}$, the equation at hand becomes 
$$u + v = c\tag{*1}$$
Notice $u^2 - v^2 = (x+a^2) - (x+b^2) = a^2-b^2$, we have
$$u - v = \frac{u^2 - v^2}{u+v} = \frac{a^2-b^2}{c}\tag{*2}$$
Combine $(*1)$ and $(*2)$, we have
$\displaystyle\;u = \frac{c^2 + a^2-b^2}{2c}$ and hence
$$\begin{align}
x = u^2 - a^2 
  & = \left(\frac{c^2 + a^2-b^2}{2c}\right)^2 - a^2
  = \left(\frac{(c+a)^2 - b^2}{2c}\right)\left(\frac{(c-a)^2-b^2}{2c}\right)\\
& = \frac{(c+a+b)(c+a-b)(c-a+b)(c-a-b)}{4c^2}\\
& = \frac{2194 \cdot 1914 \cdot 318 \cdot 38}{4\cdot 1116^2}
  = \frac{352392601}{34596} 
  \approx 10185.93481905423
\end{align}
$$
