9.6. First, Betya writes the numbers $1, 2$ in her notebook, and Nik writes $3, 4$ in his notebook. Then, at the beginning of each minute, Nik and Betya each write down a quadratic polynomial with real coefficients, whose roots are the two numbers in their notebooks, denoted as $f(x)$ and $g(x)$, respectively. If the equation $f(x)=g(x)$ has two distinct real roots $x_{1}$ and $x_{2}$, one of them immediately changes the numbers in their notebook to $x_{1}$ and $x_{2}$; if the equation $f(x)=g(x)$ does not have two distinct real roots $x_{1}$ and $x_{2}$, they do nothing and wait for the next minute to start. After some time, one of the numbers in Betya's notebook is 5. Find all possible values of the other number in her notebook.
9.6. It is not hard to see that the two numbers in each person's exercise book are always the roots of the equation
$$
\alpha(x-1)(x-2)+\beta(x-3)(x-4)=0
$$

where $\alpha, \beta$ are parameters that depend on time.
At a certain moment, 5 is a root of this equation.
Substituting 5, we get $6 \alpha+\beta=0$.
Thus, $\alpha(x-1)(x-2)+\beta(x-3)(x-4)$
$=\alpha\left(-5 x^{2}+39 x-70\right)$
$=-\alpha(x-5)(5 x-14)$.

The other root is $\frac{14}{5}$.