hpt <=> \(\left\{{}\begin{matrix}u+v=2\\\left(u-v\right)\left(u+v\right)\left(u^2+v^2\right)=16\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}u=2-v\\\left(u-v\right)\left(u^2+v^2\right)=8\end{matrix}\right.\)
Thay vô gpt bậc 3
\(\left\{{}\begin{matrix}u+v=2\\u^4-v^4=16\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}u+v=2\\\left(u-v\right)\left(u+v\right)\left(u^2+v^2\right)=16\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}u+v=2\\\left(u-v\right)\left(u^2+v^2\right)=8\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}u+v=2\\\left(u-v\right)\left[\left(u+v\right)^2-2uv\right]=8\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}u=2-v\\\left(u-v\right)\left(4-2uv\right)=8\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}u=2-v\\\left(2-2v\right)\left(4-2uv\right)=8\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}u=2-v\\\left(1-v\right)\left(2-uv\right)=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}u=2-v\\2-uv-2v+uv^2=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}u=2-v\\-uv-2v+uv^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}u=2-v\\\left[{}\begin{matrix}v=0\\-u-2+uv=0\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}v=0\\u=2\end{matrix}\right.\\\left\{{}\begin{matrix}u=2-v\\-\left(2-v\right)-2+\left(2-v\right).v=0\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}v=0\\u=2\end{matrix}\right.\\\left\{{}\begin{matrix}u=2-v\\-v^2+4v-4=0\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}u=2\\v=0\end{matrix}\right.\\\left\{{}\begin{matrix}u=0\\v=2\end{matrix}\right.\end{matrix}\right.\) (loại u=0; v=2)
vậy u=2;v=0
