Example 1 Given that a certain two-digit number, when added to the product of its tens and units digits, is exactly equal to the square of the sum of its tens and units digits. Find all such two-digit numbers.
(2008, Croatian Mathematical Competition)
From the problem, we can derive the equation
$$
10 x+y+x y=(x+y)^{2} \text {. }
$$

Rearranging the above equation into a quadratic equation with $x$ as the main variable, we get
$$
x^{2}+(y-10) x+\left(y^{2}-y\right)=0 \text {. }
$$

If the required $x$ exists, then
$$
\begin{array}{l}
\Delta=(y-10)^{2}-4\left(y^{2}-y\right) \geqslant 0 \\
\Rightarrow 3 y^{2}+16 y-100 \leqslant 0 .
\end{array}
$$

When $y \geqslant 4$, the above inequality does not hold.
Therefore, we only need to discuss the cases $y=0,1,2,3$, and we find that the numbers satisfying the condition are $13,63,91$.