Đặt \(\left\{{}\begin{matrix}\sqrt[3]{13+2x}=a\\\sqrt[3]{13-2x}=b\end{matrix}\right.\) ta được:
\(\left\{{}\begin{matrix}a+b-2ab=8\\a^3+b^3=26\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a+b-2ab=8\\\left(a+b\right)^3-3ab\left(a+b\right)=26\end{matrix}\right.\)
Đặt \(\left\{{}\begin{matrix}a+b=u\\ab=v\end{matrix}\right.\) với \(u^2\ge4v\)
\(\left\{{}\begin{matrix}u-2v=8\\u^3-3uv=26\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}v=\frac{u-8}{2}\\u^3-3uv=26\end{matrix}\right.\)
\(\Leftrightarrow u^3-3u\left(\frac{u-8}{2}\right)=26\)
\(\Leftrightarrow2u^3-3u^2+24u-52=0\)
\(\Rightarrow u=2\Rightarrow v=-3\Rightarrow\left\{{}\begin{matrix}a+b=2\\ab=-3\end{matrix}\right.\)
\(\Rightarrow\left(a;b\right)=\left(3;-1\right);\left(-1;3\right)\)
\(\Rightarrow\left[{}\begin{matrix}\sqrt[3]{13-2x}=3\\\sqrt[3]{13-2x}=-1\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}13-2x=27\\13-2x=-1\end{matrix}\right.\)
