AHSME 1970 Problem 14

Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals $\text{(A) } \sqrt{4q+1} \text{(B) } q-1 \text{(C) } -\sqrt{4q+1} \text{(D) } q+1 \text{(E) } \sqrt{4q-1}$