<s>[INST] $\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$ - 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?
 [/INST] 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
$$\begin{align}
x = u^2 - a^2 
  & = \left(\frac{c^2 + a^2-b^2}{2c}\right)^2 - a^2
  = \left(\frac{(c+a)^2 - b^2}{2c}\right)\left(\frac{(c-a)^2-b^2}{2c}\right)\\
& = \frac{(c+a+b)(c+a-b)(c-a+b)(c-a-b)}{4c^2}\\
& = \frac{2194 \cdot 1914 \cdot 318 \cdot 38}{4\cdot 1116^2}
  = \frac{352392601}{34596} 
  \approx 10185.93481905423
\end{align}
$$
</s>