$\begin{array}{l} P = \left| Q \right|\\ \dfrac{{12}}{{\sqrt x + 5}} = \left| {\dfrac{{\sqrt x - 5}}{{\sqrt x - 3}}} \right|\\ \Leftrightarrow \left[ \begin{array}{l} \dfrac{{\sqrt x - 5}}{{\sqrt x - 3}} = \dfrac{{12}}{{\sqrt x + 5}}\\ \dfrac{{\sqrt x - 5}}{{\sqrt x - 3}} = - \dfrac{{12}}{{\sqrt x + 5}} \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} x - 25 = 12\left( {\sqrt x - 3} \right)\\ \left( {x - 25} \right) = - 12\left( {\sqrt x - 3} \right) \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x - 12\sqrt x + 11 = 0\\ x + 12\sqrt x - 61 = 0 \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} \left( {\sqrt x - 11} \right)\left( {\sqrt x - 1} \right) = 0\\ \sqrt x = 6 \pm \sqrt {97} \end{array} \right. \Rightarrow \left[ \begin{array}{l} \sqrt x = 11\\ \sqrt x = 1\\ \sqrt x = 6 + \sqrt {97} \end{array} \right.\left( {6 - \sqrt {97} < 0} \right)\\ \Leftrightarrow \left[ \begin{array}{l} x = 121\\ x = 1\\ x = 133 + 12\sqrt {97} \end{array} \right. \end{array}$
