1. (1981 National High School League Question) Discuss the equation $x|x|+p x+q=0$. Among the following conclusions, the incorrect one is ( ).
A. There are at most three real roots
B. There is at least one real root
C. There are real roots only when $p^{2}-4 q \geqslant 0$
D. When $p<0$ and $q<0$, there are three real roots
1. C, D. Reason: (1) For the equation $x|x|+p x+q=0$,

When $x \geqslant 0$, (1) becomes $x^{2}+p x+q=0$.
When $x<0$, (1) becomes $-x^{2}+p x+q=0$.
If $q>0$, (2) and (3) have no positive roots, and if $q<0$, (2) and (3) have no negative roots. If $q=0$, (2) and (3) have exactly one root $x=0$. Therefore, (1) has at most one real root, so A is correct.
If $q>0$, (2) and (3) have no positive roots, and if $q<0$, (2) and (3) have no negative roots. If $q=0$, (2) and (3) have exactly three real roots $x_{1}=0, x_{2}=p, x_{3}=-p$, so (1) has at most three real roots, thus A is correct.
(2) From (2) and (3), for $q>0, q<0$, consider the equation $x|x|-x+1=0$. When $x>0$, $x^{2}-x+1=0$ has no real roots; when $x<0$, $x^{2}-x-1=0$ has two real roots, so the conclusion that $x|x|-x+1=0$ has three real roots does not hold. D is also incorrect.