I am looking for solution for a problem of finding a perfect square given by

$$
{r}^{2}+u\,r+v
$$

where $u > 0$ and $v > 0$ are integer constants and expected $r >= 0$.

The closest I was able to get to is 

$$
a^2 = r^2+u\,r+v
$$
$$
\sum\_{c=1}^{a}2\,c-1 = v+r\,u+\sum\_{c=1}^{r}2\,c-1
$$
$$
(\sum\_{c=r+1}^{a}2\,c-1)-r\,u-v = 0
$$

and I have no idea where to go from here, except for brute force search for an $r$ which would satisfy that ${r}^{2}+u\,r+v$ matches a perfect square test.

I am not sure whether there is a straightforward answer to this question, any suggestions welcomed!