Pinko wrote: For which values of the real parameter $r$ the equation $r^2 x^2+2rx+4=28r^2$ has two distinct integer roots? $r=0$ does not fit. If $r\ne 0$, considering equation as a quadratic in $x$ : a) we need $r^2>\frac 3{28}$ b) sum of roots is $-\frac 2r$ and so $r=\frac 2n$ for some integer $n$ such that $n^2<\frac{112}3$ (using a) above) So $1\le|n|\le 6$ $|n|=1$ implies equation $x^2\pm x-27=0$ which does not fit $|n|=2$ implies equation $x^2\pm 2x-24=0$ which indeed fits $|n|=3$ implies equation $x^2\pm 3x-19=0$ which does not fit $|n|=4$ implies equation $x^2\pm 4x-12=0$ which indeed fits $|n|=5$ implies equation $x^2\pm 5x-3=0$ which does not fit $|n|=6$ implies equation $x^2\pm 6x+8=0$ which indeed fits And so $n\in\{\pm 2,\pm 4,\pm6\}$ And so $\boxed{r\in\left\{-1,-\frac 12,-\frac 13,+\frac 13,+\frac 12,+1\right\}}$