Question

\(\left(\sqrt[3]{3x+1}\right)^2+\left(\sqrt[3]{3x-1}\right)^2+\sqrt[3]{9x^2-1}=1\)

Answer 1

\(\left\{{}\begin{matrix}\sqrt[3]{3x+1}=a\\\sqrt[3]{3x-1}=b\end{matrix}\right.\) \(\Rightarrow a^3-b^3=2\) (1)

\(a^2+b^2+ab=1\Leftrightarrow\left(a-b\right)\left(a^2+b^2+ab\right)=a-b\)

\(\Leftrightarrow a^3-b^3=a-b\) (2)

Từ (1) và (2) \(\Rightarrow a-b=2\Rightarrow b=a-2\)

\(\Rightarrow a^2+\left(a-2\right)^2+a\left(a-2\right)=1\Leftrightarrow3a^2-6a+3=0\Rightarrow a=1\)

\(\Rightarrow\sqrt[3]{3x+1}=1\Rightarrow3x+1=1\Rightarrow x=0\)