II. (25 points) Given 15 quadratic equations $x^{2}-p_{i} x+q_{i}=0(i=1,2, \cdots, 15)$ with coefficients $p_{i} 、 q_{i}$ taking values from $1,2, \cdots, 30$, and these coefficients are all distinct. If an equation has a root greater than 20, it is called a "good equation." Find the maximum number of good equations.
Second, if there exists an equation $x^{2}-p_{i} x+q_{i}=0$ with two roots $x_{1}, x_{2}$, then from $x_{1}+x_{2}=p_{i}>0, x_{1} x_{2}=q_{i}>0$, we know that both roots are positive.

Next, if a certain equation has a root greater than 20, then $p_{i}=x_{1}+x_{2}>20$. However, among the numbers $1,2, \cdots, 30$, there are only 10 numbers greater than 20, so the number of good equations is at most 10.
Finally, consider
$$
x^{2}-(20+k) x+k=0(k=1,2, \cdots, 10) \text {, }
$$

these ten equations. Their larger root
$$
\begin{array}{l}
x=\frac{20+k+\sqrt{(20+k)^{2}-4 k}}{2} \\
=\frac{20+k+\sqrt{k^{2}+36 k+400}}{2} \\
>\frac{20+k+k+18}{2}=k+19 \geqslant 20 .
\end{array}
$$

Therefore, the maximum number of good equations is 10.