Example 7 Given that $n$ is a positive integer, and $n^{2}-71$ is divisible by $7 n+55$. Try to find the value of $n$.

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Example 7 Given that $n$ is a positive integer, and $n^{2}-71$ is divisible by $7 n+55$. Try to find the value of $n$.
Analysis: Let $\frac{n^{2}-71}{7 n+55}=k$ ( $k$ is an integer), then $n^{2}$ $-7 k n-(55 k+71)=0$, and $\Delta=49 k^{2}+4$ (5jk $+71)=49 k^{2}+220 k+284$ should be a perfect square.
$$
\begin{array}{l}
\text { and }(7 k+15)^{?}=49 k^{2}+210 k+225 \\
<49 k^{2}+220 k+284 \\
<49 k^{2}+238 k+289 \\
=(7 k+17)^{2}, \\
\therefore \Delta=(7 k+16)^{2}=49 k^{2}+220 k+284 .
\end{array}
$$

Thus $k=7$. Therefore, $n^{2}-49 n-456=0$.
Solving gives $n=-8$ or 57. Therefore, $n=57$.