Question

\(\sqrt[3]{1+7x}+\sqrt[3]{2x-1}=2\sqrt[3]{x}\)

Answer 1

\(\left(\sqrt[3]{1+7x}+\sqrt[3]{2x-1}\right)^3=\left(2\sqrt[3]{x}\right)^3\)

\(\Leftrightarrow1+7x+2x-1+3\sqrt[3]{\left(1+7x\right)\left(2x-1\right)}\left(\sqrt[3]{1+7x}+\sqrt[3]{2x-1}\right)=8x\)

\(\Leftrightarrow3\sqrt[3]{\left(1+7x\right)\left(2x-1\right)}\left(2\sqrt[3]{x}\right)=-x\)

\(\Leftrightarrow6\sqrt[3]{14x^2-5x-1}.\sqrt[3]{x}=-x\)

\(\Leftrightarrow216x\left(14x^2-5x-1\right)=-x^3\)

\(\Leftrightarrow x\left(3025x^2-1080x-216\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}x=0\\x=\dfrac{108+30\sqrt{42}}{605}\\x=\dfrac{108-30\sqrt{42}}{605}\end{matrix}\right.\)