\(\sqrt[3]{x+1}+\sqrt[3]{x-1}=\sqrt[3]{5x}\\ \Leftrightarrow x+1+3\sqrt[3]{\left(x+1\right)\left(x-1\right)}\left(\sqrt[3]{x+1}+\sqrt[3]{x-1}\right)+x-1=5x\\ \\ \Leftrightarrow3\sqrt[3]{5x\left(x^2-1\right)}=3x\\ \Leftrightarrow\sqrt[3]{5x^3-5x}=x\\ \Leftrightarrow5x^3-5x=x^3\\ \Leftrightarrow4x^3-5x=0\\ \Leftrightarrow x\left(4x^2-5\right)=0\\ \Leftrightarrow x\left(2x+\sqrt{5}\right)\left(2x-\sqrt{5}\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=0\\2x+\sqrt{5}=0\\2x-\sqrt{5}=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-\dfrac{\sqrt{5}}{2}\\x=\dfrac{\sqrt{5}}{2}\end{matrix}\right.\)
√x+1+3√x−1=3√5x⇔x+1+33√(x+1)(x−1)(3√x+1+3√x−1)+x−1=5x⇔33√5x(x2−1)=3x⇔3√5x3−5x=x⇔5x3−5x=x3⇔4x3−5x=0⇔x(4x2−5)=0⇔x(2x+√5)(2x−√5)=0⇔⎡⎢⎣x=02x+√5=02x−√5=0⇔⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣x=0x=−√52x=√52
