Question

giải phương trình \(\frac{2x}{x^2-x+1}-\frac{x}{x^2+x+1}=\frac{5}{3}\)

Answer 1

\(x=0\) không phải nghiệm, phương trình tương đương:

\(\frac{2}{x+\frac{1}{x}-1}-\frac{1}{x+\frac{1}{x}+1}=\frac{5}{3}\)\(\Rightarrow x+\frac{1}{x}=a\)

\(\frac{2}{a-1}-\frac{1}{a+1}=\frac{5}{3}\Leftrightarrow2\left(a+1\right)-\left(a-1\right)=\frac{5}{3}\left(a^2-1\right)\)

\(\Leftrightarrow5a^2-3a-14=0\) \(\Rightarrow\left[{}\begin{matrix}a=2\\a=-\frac{7}{5}\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x+\frac{1}{x}=2\\x+\frac{1}{x}=-\frac{7}{5}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2-2x+1=0\\x^2+\frac{7}{5}x+1=0\left(vn\right)\end{matrix}\right.\) \(\Rightarrow x=1\)