Integer Solutions for a Multivariable Quadratic

Question

I'm solving a math puzzle and arrived at a quadratic:$$ \frac{6000n}{x(x-n)}=c $$ I only just graduated from high school and have very limited knowledge. I'm wondering if it's possible to find all integer solutions to this equation where $n$ and $c$ are variables, not constants. I know that when $n=3$ and $c=100$, $x=15$ and $x=-12$. Also, when $n=3$ and $c=1000$, $x=6$ and $x=-3$. Also again, for $n=3$, as $c \to \infty$ there also seem to be roots at $x=0$ and $x=3$. There is no limit on the domain/ range.

Answer 1

Hint

If your equation is $$\frac{6000\,n} {x(x-n)}=c$$ multiply both sides by $x(x-n)$ and expand to get $$6000\,n=c x^2-c n x\implies cx^2-c nx -6000n=0$$ and use the solution of the quadratic equation.