quadratic equations

[Soumyadeep mandal]

Find the number of pairs of all the possible natural numbers p, q such that the equation x^2 – pqx + (p + q) = 0 has two integer roots.

[Nitin Prasad]

Given eqn-
$$x^2-pq x+(p+q)=0$$
Consider discriminant of the above equation
$$D=(pq)^2-(p+q)^2=-(p^2+q^2+pq)=-(p+\frac{q}{2})^2-\frac{3q^2}{4}\leq 0$$

$$Hence D\leq 0$$

But for roots to be real, D>=0 and hence  D=0 which is possible only when p=q=0

 

Hence no pair of natural number (p,q) exists which satisfy the given condition.

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