2. (2 points) The numbers $p$ and $q$ are distinct non-zero roots of the quadratic equation $x^{2} - a x + b = 0$, and the numbers $a$ and $b$ are distinct non-zero roots of the quadratic equation $x^{2} - p x - q = 0$. What can these numbers be equal to?
Answer: $a=1, b=-2, p=-1, q=2$.

Applying Vieta's theorem to both equations, we form the system of equations:

Solution:

$$
\left\{\begin{array}{l}
p+q=a \\
a+b=p \\
p q=b \\
a b=-q
\end{array}\right.
$$

Adding the first two equations, after simplification we get: $b+q=0$, i.e., $b=-q$. Substituting $-q$ for $b$ in the third and fourth equations and simplifying by $q \neq 0$, we find that $p=-1$ and $a=1$.

Now from the first two equations, we find $q=2$ and $b=-2$.