6. Given two quadratic equations
$$
a x^{2}+b x+c=0, u x^{2}+v x+w=0(a \neq u)
$$

both have real roots. If the coefficients of the quadratic terms of these two equations are swapped, then $w c>0$ is the ( ) condition for at least one of the two quadratic equations obtained after the swap to have real roots.
(A) Sufficient but not necessary
(B) Necessary but not sufficient
(C) Sufficient and necessary
(D) Neither sufficient nor necessary
6. A.

Given the discriminants of two quadratic equations are
$$
\Delta_{1}=b^{2}-4 a c, \Delta_{2}=v^{2}-4 u w \text {. }
$$

After swapping the quadratic coefficients, the discriminants of the equations become $\Delta_{1}^{\prime}=b^{2}-4 u c, \Delta_{2}^{\prime}=v^{2}-4 a w$.
Then $\left(\Delta_{1}-\Delta_{1}^{\prime}\right)\left(\Delta_{2}-\Delta_{2}^{\prime}\right)=16 u c(u-a)(a-u)$.
If $w c>0$, then $\left(\Delta_{1}-\Delta_{1}^{\prime}\right)\left(\Delta_{2}-\Delta_{2}^{\prime}\right)\Delta_{1} \geqslant 0$ or $\Delta_{2}^{\prime}>\Delta_{2} \geqslant 0$.
Therefore, after swapping the quadratic coefficients, at least one of the two equations has real roots.
Thus, 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 $x=0$.