Example 12. Let $M, x, y$ be positive integers, and $\sqrt{M-\sqrt{28}}=\sqrt{x}-\sqrt{y}$. Then the value of $x+y+M$ is ( ).
(1994, Hope Cup Mathematics Competition)
Given $M-\sqrt{28}=(x+y)$ $-\sqrt{4 x y}$. Therefore, $x+y=M$, and $4 x y=28$. Since $x, y$ are positive integers, and it is specified that $x \geqslant y$, from $x y=7$, we get $x=7, y=1, M=8, x+y+M=16$.