find the $x$ of quadratic equation, such that the squre root of quadratic equation result is integer 
given this equation
$$S=\sqrt{x^2+1500x-1472}$$
find $x$, such that $x$ is positive integer, and $S$ is positive integer.

I have tried to solve this, and I get that $x = 36$, but I get that $x$ by trial and test, so
if I try $x = 1$, I got $S = 5.3851\dots$ ($S$ is not a positive integer),
if I try $x = 2$, I got $S = 39.140\dots$ ($S$ is not a positive integer),
if I try $x = 3$, I got $S = 55.108\dots$ ($S$ is not a positive integer), and so on
until I try $x = 36$, I got $S = 232$  ($S$ is a positive integer)
Is there alternative to solve this question?
 A: There is only another solution besides that one: $x=140\,244$, in which case $S=140\,992$.
In order to see why, note that\begin{align}x^2+1500x-1472&=(x+750)^2-1472-750^2\\&=(x+750)^2-563\,972.\end{align}Let $Z=x+750$. Then you are after all pairs $(S,Z)$ of natural numbers such that $S^2=Z^2-563\,972$. But this is the same thing as\begin{align}(Z-S)(Z+S)&=563\,972\\&=2^2\times277\times509\end{align}(decomposition into prime factors). So, for each of the twelve divisors $d$ of $563\,972$, if $d'=\frac{563\,972}d$, you should solve the system$$\left\{\begin{array}{l}Z-S=d\\Z+S=d'.\end{array}\right.$$Its solutions are$$Z=\frac{d+d'}2\quad\text{and}\quad S=\frac{-d+d'}2.$$So, you only need to consider those cases in which $d$ and $d'$ have the same parity and $d'>d$. That's when $d=2$ and when $d=554$. If $d=2$, then $Z=140\,994$ (and therefore $x=140\,244$) and $S=140\,992$; and if $d=554$, then $Z=786$ (and therefore $x=36$) and $S=232$.
A: We have $$S^2=(x+750)^2-563972\iff(x+750)^2-S^2=563972$$ Then $$(x+750+S)(x+750-S)=2^2\cdot277\cdot509$$ $S$ and $x$ must have same parity so
$$x+750+S=2M\\x+750-S=2N$$ The only possibilities for $M$ and $N$ are $$(M,N)=(509,277)\\(M,N)=(207\cdot509,1)=(105363,1)$$ from this we have the two systems
$$\begin {cases}x+750+S=2(509)\\x+750-S=2(277)\end{cases}$$
$$\begin {cases}x+750+S=2(105263)\\x+750-S=2\end{cases}$$
The first gives $S=232\Rightarrow x+750=786\Rightarrow\sqrt{x^2+1500x-1472}\in\mathbb N$ GOOD!
The second gives $S=105262\Rightarrow x+750=105264\Rightarrow \sqrt{x^2+1500x-1472}\notin\mathbb N$ BAD.
Consequently the only solution is $(x,S)=(36,232)$.
