$6.269 x^{3}-2 a x^{2}+\left(a^{2}+2 \sqrt{3} a-9\right) x-\left(2 a^{2} \sqrt{3}-12 a+6 \sqrt{3}\right)=0$.
Solution. $x^{3}-2 a x^{2}+\left(a^{2}+2 \sqrt{3} a-9\right) x-2 \sqrt{3}(a-\sqrt{3})^{2}=0$.

This equation is a special case of the model equation (1) if we set $v=2 \sqrt{3}, u=a-\sqrt{3}, w=0$.

Answer: $x_{1}=2 \sqrt{3}, x_{2,3}=a-\sqrt{3}$.