4. Let the equations
$$
a x^{2}+b x+c=0 \text { and } c x^{2}+b x+a=0
$$

have only one common positive real root. Try to find the relationship between $a$, $b$, and $c$.
Given that $a \neq c$, let the common positive real root be $t$. Then
$$
\begin{array}{l}
a t^{2}+b t+c=0, \\
c t^{2}+b t+a=0 .
\end{array}
$$

Subtracting the two equations and rearranging gives
$$
(a-c) t^{2}=(a-c) \Rightarrow t=1(t>0) .
$$

Therefore, $a+b+c=0$.