Question

gpt \(\sqrt[3]{x-2}+\sqrt[3]{x+2}=\sqrt[3]{5x}\)

Answer 1

\(PT\Leftrightarrow x+2+x-2+3\sqrt[3]{\left(x+2\right)\left(x-2\right)}\left(\sqrt[3]{x+2}+\sqrt[3]{x-2}\right)=5x\)

\(\Leftrightarrow\sqrt[3]{\left(x+2\right)\left(x-2\right).5x}=x\)

\(\Leftrightarrow x^3=5x\left(x-2\right)\left(x+2\right)\)

\(\Leftrightarrow x\left(x^2-5x^2+20\right)=0\)

\(\Leftrightarrow4x\left(5-x^2\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=\sqrt{5}\\x=-\sqrt{5}\end{matrix}\right.\)