[Solution] Let the common root be $\alpha$. By Vieta's formulas, the distinct roots are $\frac{b}{\alpha}$ and $\frac{q}{\alpha}$. Since $\alpha$ is the common root, we have $\left\{\begin{array}{l}\alpha^{2}+a \alpha+b=0, \\ a^{2}+p \alpha+q=0 .\end{array}\right.$ Subtracting the two equations, we get $\quad \alpha=\frac{q-b}{\alpha-p}$. Therefore, the two distinct roots are $\frac{b(a-p)}{q-b}$ and $\frac{q(a-p)}{q-b}$. Hence, the required equation is
$$x^{2}-\frac{(b+q)(a-p)}{q-b} x+\frac{b q(a-p)^{2}}{(q-b)^{2}}=0 .$$