Задание
Найдите значение выражения:
\(\displaystyle (\sqrt{3} - \sqrt{2})^{2}+(\sqrt{3} + \sqrt{2})^{2}=\)
Решение
Применим формулы возведения в квадрат разности и суммы:
\(\displaystyle \color{blue}{(\sqrt{3} - \sqrt{2})^{2}}+\color{green}{(\sqrt{3} + \sqrt{2})^{2}}=\)
\(\displaystyle \color{blue}{(\sqrt{3})^2 -2\cdot \sqrt{3}\cdot\sqrt{2}+(\sqrt{2})^{2}}+\color{green}{(\sqrt{3})^2 +2\cdot \sqrt{3}\cdot\sqrt{2}+(\sqrt{2})^{2}}{\small.}\)
Имеем:
- \(\displaystyle (\sqrt{3})^2=3{ \small ,}\)
- \(\displaystyle \color{blue}{ -2\cdot \sqrt{3}\cdot\sqrt{2}}+\color{green}{ 2\cdot \sqrt{3}\cdot\sqrt{2}=\color{red}{ 0}}{ \small ,}\)
- \(\displaystyle (\sqrt{2})^2=2{ \small .}\)
Значит,
\(\displaystyle \begin{aligned}\color{blue}{(\sqrt{3})^2 -\cancel{2\cdot \sqrt{3}\cdot\sqrt{2}}+(\sqrt{2})^{2}}+\color{green}{(\sqrt{3})^2 +\cancel{2\cdot \sqrt{3}\cdot\sqrt{2}}+(\sqrt{2})^{2}}=\\=3+2+3+2=10{\small.}\end{aligned}\)
Ответ: \(\displaystyle 10 {\small.} \)