1962 AHSME 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 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ p \qquad \textbf{(E)}\ p+1

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.