$2 \cdot 59$ The difference between the larger and smaller roots of the equation $x^{2}-p x+\frac{p^{2}-1}{4}=0$ is
(A) 0.
(B) 1.
(C) 2.
(D) $p$.
(E) $p+1$.
(13th American High School Mathematics Examination, 1962)
[Solution] Let the two roots of the equation be $\alpha$ and $\beta$, and assume $\alpha>\beta$.
Then $\alpha-\beta=\sqrt{(\alpha-\beta)^{2}}=\sqrt{(\alpha+\beta)^{2}-4 \alpha \beta}$
$$
=\sqrt{p^{2}-\left(p^{2}-1\right)}=1 \text {. }
$$

Therefore, the answer is $(B)$.