Example 16. $k$ is a real number, the quadratic equation in $x$ $7 x^{2}-(k+13) x$ $+k^{2}-k-2=0$ has two real roots, one in the open interval $(0,1)$, and one in the open interval $(1,2)$, find the range of values for $k$.
From the problem, we draw a schematic diagram of the quadratic function (Figure 4).
Let the quadratic equation be
$$
\begin{array}{l}
f(x)=0, \\
f(0)>0, \\
f(1)<0 .
\end{array} \text { i.e. }\left\{\begin{array}{l}
k^{2}-k-2>0, \\
k^{2}-2 k-8<0 .
\end{array}\right.
$$

Therefore, $-2<\mathrm{k}<-1$ or $3<\mathrm{k}<4$.