$$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$$
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 $$\begin{align} x = u^2 - a^2 & = \left(\frac{c^2 + a^2-b^2}{2c}\right)^2 - a^2 = \left(\frac{(c+a)^2 - b^2}{2c}\right)\left(\frac{(c-a)^2-b^2}{2c}\right)\\ & = \frac{(c+a+b)(c+a-b)(c-a+b)(c-a-b)}{4c^2}\\ & = \frac{2194 \cdot 1914 \cdot 318 \cdot 38}{4\cdot 1116^2} = \frac{352392601}{34596} \approx 10185.93481905423 \end{align} $$
Repeated squaring is not necessary. Let $y > -140$ be such that $x = y^2 + 280y$, so that $$\sqrt{x+140^2} - 140 = \sqrt{(y + 140)^2} - 140 = y.$$ Then $$\begin{align*} 38 &= \sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 \\ &= \sqrt{y^2 + 280y + 938^2} - 938 + y, \end{align*}$$ or equivalently, $$976-y = \sqrt{y^2 + 280y + 938^2}.$$ Now we square only once, giving $$y^2 - 2(976)y + 976^2 = y^2 + 280y + 938^2.$$ The quadratic terms cancel, and we get $$2(140+976)y = 976^2 - 938^2 = (976-938)(976+938) = 38(1914),$$ or $$y = \frac{6061}{186}.$$ Then substitute back to find the solution for $x$.
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.
After the first line of your attempt, multiply both sides by $\sqrt{x+938^2}-\sqrt{x+140^2}$
$$x+938^2-(x+140^2)=1116(\sqrt{x+938^2}-\sqrt{x+140^2})$$
Divide both sides by $1116$.
$$\dfrac{938^2-140^2}{1116}=\sqrt{x+938^2}-\sqrt{x+140^2}$$
Taken with the previous equation
$$\sqrt{x+938^2}+\sqrt{x+140^2}=1116$$
we can add both equations to solve for $\sqrt{x+938^2}$ or subtract to solve for $\sqrt{x+140^2}$. Then simply square both sides and solve for $x$.
