$2{x^2} - \frac{2}{{{x^2}}} = 2({x^2} - \frac{1}{{{x^2}}}) = 2(x - \frac{1}{x})(x + \frac{1}{x}) = 2 \times 5 \times \sqrt {29}  = 10\sqrt {29} $

$ = 2(x - \frac{1}{x})(x + \frac{1}{x}) = 2 \times 5 \times  - \sqrt {29}  =  - 10\sqrt {29} $

$\begin{gathered}
  {(x - \frac{1}{x})^2} = {5^2} \hfill \\
  {x^2} + \frac{1}{{{x^2}}} - 2 = 25 \Rightarrow {x^2} + \frac{1}{{{x^2}}} = 25 + 2 = 27 \Rightarrow {x^2} + \frac{1}{{{x^2}}} = 27 \hfill \\
  {(x + \frac{1}{x})^2} = {x^2} + \frac{1}{{{x^2}}} + 2 = 27 + 2 = 29 \Rightarrow {(x + \frac{1}{x})^2} = 29 \Rightarrow x + \frac{1}{x} =  \pm \sqrt {29}  \hfill \\ 
\end{gathered} $