$\begin{array}{l} {x_0} = \sqrt[3]{{1 + \dfrac{{\sqrt {84} }}{9}}} + \sqrt[3]{{1 - \dfrac{{\sqrt {84} }}{9}}}\\ \Rightarrow x_0^3 = 1 + \dfrac{{\sqrt {84} }}{9} + 1 - \dfrac{{\sqrt {84} }}{9} + 3\sqrt[3]{{\left( {1 + \dfrac{{\sqrt {84} }}{9}} \right)\left( {1 - \dfrac{{\sqrt {84} }}{9}} \right)}}\left( {\sqrt[3]{{1 + \dfrac{{\sqrt {84} }}{9}}} + \sqrt[3]{{1 - \dfrac{{\sqrt {84} }}{9}}}} \right)\\ \Rightarrow x_0^3 = 2 + 3.\sqrt[3]{{\left( {1 - \dfrac{{84}}{{81}}} \right)}}.{x_0}\\ \Leftrightarrow x_0^3 = 2 + 3.\dfrac{{ - 1}}{3}.{x_0}\\ \Leftrightarrow x_0^2 + {x_0} - 2 = 0 \end{array}$
