6 Given two quadratic equations: $a x^{2}+b x+c=0, u x^{2}+v x+w=$ 0 both have real roots, where $a \neq u$. If the coefficients of the quadratic terms of these two equations are swapped, then
$w c>0$ is ( ) for at least one of the two quadratic equations obtained after the coefficient swap to have real roots.
(A) a sufficient but not necessary condition
(B) a necessary but not sufficient condition
(C) a sufficient and necessary condition
(D) neither a sufficient nor a necessary condition
6. A Hint: The discriminants of the original two quadratic equations are $\Delta_{1}=b^{2}-4 a c$; $\Delta_{2}=v^{2}-4 u w$.

After swapping the quadratic coefficients, the discriminants of the two equations become $\Delta_{1}^{\prime}=b^{2}-4 u c$; $\Delta_{2}^{\prime}=v^{2}-4 a w$.
$$
\begin{aligned}
\left(\Delta_{1}-\Delta_{1}^{\prime}\right)\left(\Delta_{2}-\Delta_{2}^{\prime}\right) & =16 \omega c(u-a)(a-u) \\
& =-16 \omega c(u-a)^{2} .
\end{aligned}
$$

If $w c>0$, then $\left(\Delta_{1}-\Delta_{1}^{\prime}\right)\left(\Delta_{2}-\Delta_{2}^{\prime}\right)\Delta_{1} \geqslant 0$, or $\Delta_{2}^{\prime}>\Delta_{2} \geqslant 0$, meaning that after swapping the quadratic coefficients, at least one of the two equations has real roots, so the condition “ $w c>0$ ” is sufficient.

The condition is not necessary. In fact, if $w c=0$, after swapping the quadratic coefficients, at least one of the two equations has a real root $x=0$. Therefore, the answer is A.