3.9 Given numbers $\alpha$ and $\beta$ satisfy the following two equations:
$$
\alpha^{3}-3 \alpha^{2}+5 \alpha=1, \beta^{3}-3 \beta^{2}+5 \beta=5 \text {, find } \alpha+\beta \text {. }
$$
[Solution] Let $f(x)=x^{3}-3 x^{2}+5 x$
$$
=(x-1)^{3}+2(x-1)+3 \text {. }
$$

Then the left sides of the given equations are $f(\alpha)$ and $f(\beta)$, respectively. Let
$$
g(y)=y^{3}+2 y,
$$

then
$$
\begin{array}{l}
g(\alpha-1)=f(\alpha)-3=-2, \\
g(\beta-1)=f(\beta)-3=2 .
\end{array}
$$

Since $g(y)$ is an odd function and also an increasing function, we have
$$
\alpha-1=-(\beta-1) \text {. }
$$

That is,
$$
\alpha+\beta=2
$$