2. The sum of the squares of all real roots of the equation $x^{2}-31 x+220=2^{x}\left(31-2 x-2^{x}\right)$ is $\qquad$
Answer 25.
Analysis Let $y=x+2^{x}$, then the original equation is equivalent to
$$
y^{2}-31 y+220=0 \Rightarrow y_{1}=11, y_{2}=20 \Rightarrow x_{1}+2^{x_{1}}=11 \text { and } x_{2}+2^{x_{2}}=20
$$

Since $f(x)=x+2^{x}$ is a monotonically increasing function, each equation has at most one real root, which are 3 and 4, respectively. Therefore, the sum of the squares of all real roots of the original equation is 25.