Найдите целое число, задаваемое выражением: \( \left( \dfrac{1}{\sqrt{2} - 1} + \dfrac{1}{\sqrt{2} + 1} \right)^2 + \left( \dfrac{1}{\sqrt{2} - 1} - \dfrac{1}{\sqrt{2} + 1} \right)^2 .\)

Решение

\[
\left( \dfrac{1}{\sqrt{2} - 1} + \dfrac{1}{\sqrt{2} + 1} \right)^2 + \left( \dfrac{1}{\sqrt{2} - 1} - \dfrac{1}{\sqrt{2} + 1} \right)^2 =
\]
\[
=\left(\dfrac{\sqrt{2} + 1 + \sqrt{2} - 1}{(\sqrt{2} - 1)(\sqrt{2} + 1)} \right)^2 + \left(\dfrac{\sqrt{2} + 1 - \sqrt{2} + 1}{(\sqrt{2} - 1)(\sqrt{2} + 1)} \right)^2 =
\]
\[
=(2\sqrt{2})^2 + 2^2 = 8 + 4 = 12.
\]

Ответ: 12.