If $$a$$, $$b$$, $$c$$ are integers such that $$c > 0$$. Determine condition(s) on $$a$$, $$b$$, $$c$$ to put $$a + b\sqrt{c}$$ in the form $$(\sqrt{p} \pm \sqrt{q})^2$$ where $$p$$ and $$q$$ are integers.

PLACEHOLDER ANSWER - the answer is a set of conditions. $$(\sqrt{p} \pm \sqrt{q})^2 = (p + q) \pm 2\sqrt{pq}$$, so we need $$p + q = a$$ and $$b^2 c = 4pq$$, i.e. $$p$$ and $$q$$ are the roots of $$t^2 - at + \frac{b^2 c}{4} = 0$$. The conditions are that $$b^2 c$$ is divisible by 4 and that $$a^2 - b^2 c$$ is a perfect square $$k^2$$ (then $$p, q = \frac{a \pm k}{2}$$ are integers, as $$a$$ and $$k$$ have the same parity). For real square roots we also need $$p, q \ge 0$$, i.e. $$a \ge 0$$ and $$a^2 \ge b^2 c$$.