\[x^{2} = \left( \sqrt{7 - 2\sqrt{6}} - \sqrt{7 + 2\sqrt{6}} \right)^{2}\]
\[7 - 2\sqrt{6} - 2\sqrt{\left( 7 - 2\sqrt{6} \right)\left( 7 + 2\sqrt{6} \right)} + 7 + 2\sqrt{6} =\]
\[= 14 - 2\sqrt{49 - 24} =\]
\[= 14 - 2 \cdot 5 = 4;\]
\[x^{2} = 4\]
\[x = \pm 2.\]
\[Что\ и\ требовалось\ доказать.\]