Question

\(x^3+\frac{1}{x^3}=13\left(x+\frac{1}{x}\right)\)

Answer 1

\(x^3+\frac{1}{x^3}=\left(x^2-1+\frac{1}{x^2}\right)\left(x+\frac{1}{x}\right)=13\left(x+\frac{1}{x}\right)\Leftrightarrow x^2-1+\frac{1}{x^2}=13\Leftrightarrow x^2+2+\frac{1}{x^2}=16\Leftrightarrow\left(x+\frac{1}{x}\right)^2=16=\left(\pm4\right)^2\Leftrightarrow\left[{}\begin{matrix}x+\frac{1}{x}=4\\x+\frac{1}{x}=-4\end{matrix}\right.\) \(+,x+\frac{1}{x}=4\Leftrightarrow\frac{x^2+1}{x}=4\Leftrightarrow x^2+1=4x\Leftrightarrow x^2-4x+1=0\Leftrightarrow\left(x-2\right)^2=3\Leftrightarrow x-2=\pm\sqrt{3}\Leftrightarrow x=\pm\sqrt{3}+2\)

\(+,x+\frac{1}{x}=-4\Leftrightarrow\frac{x^2+1}{x}=-4\Leftrightarrow x^2+1=-4x\Leftrightarrow x^2+4x+1=0\Leftrightarrow\left(x+2\right)^2=3\Leftrightarrow x=\pm\sqrt{3}-2\)