Question

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

Answer 1

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

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

Do hàm số \(f\left(t\right)=t^3+2t\) đồng biến

\(\Rightarrow f\left(x\right)=f\left(\sqrt[3]{2x-1}\right)\Leftrightarrow x=\sqrt[3]{2x-1}\)

\(\Leftrightarrow x^3-2x+1=0\)

\(\Leftrightarrow\left(x-1\right)\left(x^2+x-1\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}x=1\\x=\dfrac{-1-\sqrt{5}}{2}\\x=\dfrac{-1+\sqrt{5}}{2}\end{matrix}\right.\)