đkxđ \(\left\{{}\begin{matrix}x^2-x-2\ne0\\x^2+x-2\ne0\end{matrix}\right.\)
\(\frac{x}{x^2-x-2}+\frac{x}{x^2+x-2}=0\)
\(\Leftrightarrow\frac{x\left(x^2+x-2\right)+x\left(x^2-x-2\right)}{\left(x^2-x-2\right)\left(x^2+x-2\right)}=0\)
\(\Leftrightarrow\frac{x^3+x^2-2x+x^3-x^2-2x}{\left(x^2-x-1\right)\left(x^2+x-1\right)}=0\)
\(\Leftrightarrow2x^3-4x=0\)
\(\Leftrightarrow2x\left(x^2-2\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}2x=0\\x^2-2=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=0\\x=\sqrt{2}\end{matrix}\right.\)
ĐKXĐ : \(\left\{{}\begin{matrix}x^2-x-2\ne0\\x^2+x-2\ne0\end{matrix}\right.\) => \(\left\{{}\begin{matrix}\left(x-\frac{1}{2}\right)^2\ne\frac{9}{4}\\\left(x+\frac{1}{2}\right)^2\ne\frac{9}{4}\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}x-\frac{1}{2}\ne-\frac{3}{2}\\x-\frac{1}{2}\ne\frac{3}{2}\\x+\frac{1}{2}\ne-\frac{3}{2}\\x+\frac{1}{2}\ne\frac{3}{2}\end{matrix}\right.\) => \(\left\{{}\begin{matrix}x\ne-1\\x\ne2\\x\ne-2\\x\ne1\end{matrix}\right.\)
Ta có : \(\frac{x}{x^2-x-2}+\frac{x}{x^2+x-2}=0\)
=> \(\frac{x\left(x^2+x-2\right)+x\left(x^2-x-2\right)}{\left(x^2+x-2\right)\left(x^2-x-2\right)}=0\)
=> \(x\left(x^2+x-2+x^2-x-2\right)=0\)
=> \(x\left(x^2-2\right)=0\)
=> \(\left[{}\begin{matrix}x=0\\x^2=2\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x=0\\x=\pm\sqrt{2}\end{matrix}\right.\) ( TM )
Vậy phương trình trên có tập nghiệm là \(S=\left\{0,\sqrt{2};-\sqrt{2}\right\}\)
