3. When $a$ equals $\qquad$, the equations $x^{2}+a x-6$ $=0$ and $x^{2}-6 x+a=0$ have at least one common root.
3. -6 or 5.

Let the common root of the two equations be $x_{0}$, then
$$
\begin{array}{l}
x_{0}^{2} + a x_{0} - 6 = 0, \\
x_{0}^{2} - 6 x_{0} + a = 0.
\end{array}
$$

Subtracting the two equations,
$$
(a+6) x_{0} = a+6 \text{. }
$$
(i) When $a+6 \neq 0, a \neq -6$, $x_{0} = 1$. Substituting into the equations gives $a = 5$, in which case the two equations have only one common root;
(ii) When $a+6 = 0, a = -6$, the equations have two common roots.
In summary, $a = 5$ or -6.