Задача 5. Найдите корень уравнения \(\sqrt {x + 2} = — x.\) Если уравнение имеет более одного корня, укажите больший из них.
Решение
\(\sqrt {x + 2} = — x\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{ — x \ge 0\,\,\,\,\,\,\,\,\,}\\{x + 2 = {x^2}}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \le 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{x^2} — x — 2 = 0}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \le 0\,\,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{x = — 1}\\{x = 2}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = — 1.\)
Ответ: – 1.