109. In the equation $x^{2}-2 x+a=0$, the square of the difference of the roots is 20. Find $a$.
If $x_{1}$ and $x_{2}$ are the roots of the given quadratic equation, then we have the system of equations:

$$
\left\{\begin{array}{l}
\left(x_{1}-x_{2}\right)^{2}=20 \\
x_{1}+x_{2}=2
\end{array}\right.
$$

Squaring the second equation, we get:

$$
\left\{\begin{array}{c}
x_{1}^{2}-2 x_{1} x_{2}+x_{2}^{2}=20 \\
x_{1}^{2}+2 x_{1} x_{2}+x_{2}^{2}=4
\end{array}\right.
$$

from which $4 x_{1} x_{2}=-16 ; a=x_{1} x_{2}=-4$.

Answer: $a=-4$.