Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 505

\[\boxed{\mathbf{505}.}\]

\[x = a\left( \frac{m^{2} + n^{2}}{2mn} \right)^{\frac{1}{2}};\ \ \ n > m > 0.\]

\[x^{2} + a^{2} = a^{2}\left( \frac{m^{2} + n^{2}}{2mn} \right) + a^{2} =\]

\[= a^{2}\left( \frac{m^{2} + n^{2}}{2mn} + 1 \right) =\]

\[= \frac{a^{2}\left( m^{2} + n^{2} + 2mn \right)}{2mn} =\]

\[= \frac{a^{2}(m + n)^{2}}{2mn}\]

\[x^{2} - a^{2} = a^{2}\left( \frac{m^{2} + n^{2}}{2mn} \right) - a^{2} =\]

\[= a^{2}\left( \frac{m^{2} + n^{2}}{2mn} - 1 \right) =\]

\[= \frac{a^{2}\left( m^{2} + n^{2} - 2mn \right)}{2mn} =\]

\[= \frac{a^{2}(n - m)^{2}}{2mn}\]