Đặt \(\sqrt[3]{81x-8}=3y-2\)
\(\Leftrightarrow81x-8=27y^3-54y^2+36y-8\)
\(\Leftrightarrow27y^3-54y^2+36y=81x\)
\(\Leftrightarrow3y^3-6y^2+4y=9x\)
Phương trình đã cho tương đương:
\(3\sqrt[3]{81x-8}=3x^3-6x^2+4x-6\)
\(\Leftrightarrow3\left(3y-2\right)=3x^3-6x^2+4x-6\)
\(\Leftrightarrow3x^3-6x^2+4x=9y\)
Ta có hệ phương trình \(\left\{{}\begin{matrix}3y^3-6y^2+4y=9x\left(1\right)\\3x^3-6x^2+4x=9y\left(2\right)\end{matrix}\right.\)
Trừ vế theo vế \(\left(1\right)\) cho \(\left(2\right)\) ta được
\(3\left(y^3-x^3\right)-6\left(y^2-x^2\right)+4\left(y-x\right)=9\left(x-y\right)\)
\(\Leftrightarrow3\left(y-x\right)\left(y^2+x^2+xy\right)-6\left(y-x\right)\left(x+y\right)+13\left(y-x\right)=0\)
\(\Leftrightarrow\left(3y^2+3x^2+3xy-6x-6y+13\right)\left(y-x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}3y^2+3x^2+3xy-6x-6y+13=0\left(3\right)\\y-x=0\end{matrix}\right.\)
Phương trình \(3y^2+3y\left(x-2\right)+3x^2-6x+13=0\)
\(\Delta=9\left(x-2\right)^2-12\left(3x^2-6x+13\right)=-27x^2+36x-120< 0\)
\(\Rightarrow\) Phương trình \(\left(3\right)\) vô nghiệm
\(\Rightarrow y=x\)
Khi đó \(\sqrt[3]{81x-8}=3x-2\)
\(\Leftrightarrow27x^3-54x^2-33x=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=\frac{3\pm2\sqrt{5}}{3}\end{matrix}\right.\)
