integer solutions for $d^2 + 4kT=S^2$ for a given $T$ Consider the following equation: 
$d^2 + 4kT=S^2$
We are interested in nonzero integers $d,k,T,S$ that satisfy the above equation. Specifically, we are interested in some values of $T$, for which there exist multiple solutions for $d,k,S$.  For example does there exist $d_1,d_2,k_1,k_2,S_1,S_2,T$ such that $d_1^2 + 4k_1 T=S_1^2$ and $d_2^2 + 4k_2 T=S_2^2$ ?

The discriminant of the equation 
$$kX^2-dX-T=0$$
is $d^2+4kT$.
Since $k,d,T$ are integers, this discriminant is the square of an integer if and only if the equation has rational solutions, let them be $u$ and $v$.
We know that $uv=-T/k$ and $u+v=d/k$. Then, given $T$, you can just give values to $k$ and choose a pair of rational numbers $u,v$ such that $uv=-T/k$, only having in mind that the denominator of $u+v$ must be a divisor of $k$.
Example: Pick $T=12$, $k=10$. Choose $u,v$ such that $uv=-T/k=-6/5$. For example: $u=3$, $v=-2/5$. Then $u+v=13/5$, so $d=26$. Then $26^2+4\cdot12\cdot10=676+480=1156=34^2$. The denominators of $u$ and $v$ should be both divisors of $k=10$: if we picked $u=42$, $v=-1/35$ then the denominator of $u+v$ is $35$ and $d$ would not be integer.
