Áp dụng phương pháp tập thể dục
\(2-\frac{x-1}{x}=\left(\frac{\sqrt[3]{2x^2+x^3}+x+2}{2x+1}\right)^2\)
\(\Leftrightarrow\frac{x+1}{x}=\frac{\sqrt[3]{\left(2x^2+x^3\right)^2}+2\left(x+2\right)\sqrt[3]{2x^2+x^3}+\left(x+2\right)^2}{\left(2x+1\right)^2}\)
\(\Leftrightarrow\sqrt[3]{\left(2x^2+x^3\right)^2}+2\left(x+2\right)\sqrt[3]{2x^2+x^3}+\left(x+2\right)^2-\frac{\left(x+1\right)\left(2x+1\right)^2}{x}=0\)
\(\Leftrightarrow\left(\sqrt[3]{\left(2x^2+x^3\right)^2}-1\right)+2\left(x+2\right)\left(\sqrt[3]{2x^2+x^3}-1\right)+1+2\left(x+2\right)+\left(x+2\right)^2-\frac{\left(x+1\right)\left(2x+1\right)^2}{x}=0\)
\(\Leftrightarrow\frac{\left(x^2+x-1\right)\left(x^4+3x^3+2x^2+x+1\right)}{\sqrt[3]{\left(2x^2+x^3\right)^4}+\sqrt[3]{\left(2x^2+x^3\right)^2}+1}+\frac{2\left(x+2\right)\left(x+1\right)\left(x^2+x-1\right)}{\sqrt[3]{\left(2x^2+x^3\right)^2}+\sqrt[3]{2x^2+x^3}+1}+\frac{\left(1-3x\right)\left(x^2+x-1\right)}{x}=0\)
\(\Leftrightarrow\left(x^2+x-1\right)\left(\frac{\left(x^4+3x^3+2x^2+x+1\right)}{\sqrt[3]{\left(2x^2+x^3\right)^4}+\sqrt[3]{\left(2x^2+x^3\right)^2}+1}+\frac{2\left(x+2\right)\left(x+1\right)}{\sqrt[3]{\left(2x^2+x^3\right)^2}+\sqrt[3]{2x^2+x^3}+1}+\frac{\left(1-3x\right)}{x}\right)=0\)
\(\Leftrightarrow x^2+x-1=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{-1+\sqrt{5}}{2}\\x=\frac{-1-\sqrt{5}}{2}\end{cases}}\)
