Pair of Integers

$A$ and $B$ are the solutions of the quadratic equation $x^2 - qx + p = 0.$ By the quadratic formula, $A,B = \frac {q \pm \sqrt{q^2 - 4p}}2.$ If $q^2 - 4p$ is not a perfect square, this expression is irrational. If $q^2 - 4p$ is a perfect square, then $\sqrt{q^2 - 4p}$ has the same parity (odd/even) as $q$ , showing that the numerator is even; therefore $A$ and $B$ will be integers.

Arjen's solution is definitely the most efficient, but just as an alternative way to arrive at the same solutions for $A$ and $B$ :

$\begin{aligned} A + B &= q \qquad &&(1) \\ \\ A^2 + 2AB + B^2 &= q^2 \\ \\ A^2 - 2AB + B^2 &= q^2 - 4p \\ \\ A - B &= \pm \sqrt{q^2 - 4p} \qquad &&(2) \\ \\ A,B &= \dfrac{q \pm \sqrt{q^2 - 4p} }{2} \qquad \qquad &&\small \text{[by first adding (1) + (2), then subtracting (1) - (2)]} \end{aligned}$

and finishing like Arjen did.