How to solve this quadratic system of equations? The equations are:
$$ \begin{aligned} b + d &= c^2 - 6 \\ b - d &= -\frac{1}{c} \\ b d &= 6 \end{aligned} $$
and I want integer solutions for this.
I tried using various methods such as using $(a+b)^2-(a-b)^2=4ab$ or trying to solve for $b+d$ using the last two equations and substituting into the first. None of these methods worked. How do I do this?

If $b, c, d$ are integers,
then $c$ can only be
$1$ or $-1$.
In either case,
$b+d=-5$.
From the last,
$(b,d)$ is one of
$(1,6), (2,3),
(3,2), (6,1)$,
or these replaced by their negatives.
The only ones that work are
$(-2,-3)$ and $(-3,-2)$.
If $c=1$, then $b-d=-1$ so
$b=-3$, $d=-2$.
If $c=-1$, then $b-d=1$ so
$b=-2$, $d=-3$.
