9. Let $a$ and $b$ be two distinct real numbers. It is known that the two equations

$$
\begin{aligned}
& x^{2}+a x+3 b=0 \\
& x^{2}+b x+3 a=0
\end{aligned}
$$

have a common solution: what are the possible values for the sum $a+b$?
(A) 0 or -3
(B) 0 or 3
(C) Only 0
(D) Only -3
(E) There are infinitely many possible values
9. The answer is (D). Each solution of both equations must also be a solution of their difference, which is

$$
\left(x^{2}+a x+3 b\right)-\left(x^{2}+b x+3 a\right)=(a-b) x+3(b-a)=(a-b)(x-3)=0 .
$$

Since $a$ and $b$ are distinct, the only possibility is $x=3$, which is therefore the only possible common solution of the two original equations. In particular, 3 is a solution of $x^{2}+a x+3 b=0$; substituting gives $9+3 a+3 b=0$, from which $3(a+b)=-9$ and thus $a+b=-3$.