6.349 Solve the equations $2 x^{3}-5 x^{2}+6 x-2=0$ and $6 x^{3}-3 x^{2}-2 x+1=0$, given that they have a common root.
Solution. Let $f(x)=6 x^{3}-3 x^{2}-2 x+1, g(x)=2 x^{3}-5 x^{2}+6 x-2$.

Represent $f(x)$ in the form (3). Then $r_{1}(x)=12 x^{2}-20 x+7$.

Next, $g(x)=r_{1}(x) \cdot q_{2}(x)+r_{2}(x)$, where $r_{2}(\dot{x})=\frac{37}{18} x-\frac{37}{36} \Rightarrow r_{2}(x)=0$ when $x_{0}=\frac{1}{2}$.

Thus, $g(x)=(2 x-1)\left(x^{2}-2 x+2\right), f(x)=(2 x-1)(\sqrt{3} x-1)(\sqrt{3} x+1)$.

Answer: The root of the first equation: $x=+\frac{1}{2}$;

$$
\text { roots of the second equation: } x_{1}=\frac{1}{2}, x_{2,3}= \pm \frac{1}{\sqrt{3}} \text { : }
$$