Đặt \(\left\{{}\begin{matrix}\sqrt[3]{3x-1}=a\\\sqrt[3]{x+1}=b\\\sqrt[3]{-2x}=c\end{matrix}\right.\) ta được hệ:
\(\left\{{}\begin{matrix}a+b=c\\a^3+b^3=-2c^3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a+b=c\\\left(a+b\right)^3-3ab\left(a+b\right)=-2c^3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a+b=c\\c^3-3abc=-2c^3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a+b=c\\c\left(c^2-ab\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a+b=c\\c\left[\left(a+b\right)^2-ab\right]=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a+b=c\\c\left[\left(a+\dfrac{b}{2}\right)^2+\dfrac{3b^2}{4}\right]=0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}c=0\\a=b=0\left(vn\right)\end{matrix}\right.\)
\(\Rightarrow\sqrt[3]{-2x}=0\Leftrightarrow x=0\)
