AHSME 1962 Problem 11

The difference between the larger root and the smaller root of $x^2 - px + (p^2 - 1)/4 = 0$ is: $\textbf{(A)}\ 0 \textbf{(B)}\ 1 \textbf{(C)}\ 2 \textbf{(D)}\ p \textbf{(E)}\ p+1$