2.97 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) There are three real roots when $p>0$.
(China High School Mathematics League, 1981)
[Solution] We analyze the given equations by assigning special values to $p, q$ that satisfy the conditions:
(1) Consider the equation
$$
x|x|+x+1=0 \text {, }
$$

When $x \geqslant 0$, (1) becomes $x^{2}+x+1=0$, which has no real roots;
When $x \leqslant 0$, (1) becomes $-x^{2}+x+1=0$, which has one negative root.
In summary, equation (1) has one negative root, but $p^{2}-4 q=-3<0$. Therefore, (C) is incorrect.
(2) Consider the equation $\quad x|x|-2 x+1=0$
When $x \geqslant 0$, (2) becomes $x^{2}-2 x+1=(x-1)^{2}=0$. So $x=1$ is
a root of (2);

When $x \leqslant 0$, (2) becomes $-x^{2}-2 x+1=0$, which has one positive root and one negative root, of which only the negative root is a root of (2).
In summary, (2) has only two roots, and it can be proven that (C) and (D) are incorrect. Therefore, the answer is (C) or (D).