Explanation

Let α be the common root: α²−pα+q = 0 and α²+qα−p = 0. Subtracting gives (p+q)(1−α) = 0, so α = 1. Substituting into the first equation yields 1−p+q = 0, i.e. p−q = 1, which is p−q−1 = 0, option D. The alternatives p−q = 0, p+q = 2 and p+q = 1 do not follow from eliminating α².