Задача 4. Найдите корень уравнения \(\sqrt {27 + 6x} = x.\) Если уравнение имеет более одного корня, укажите меньший из них.
Решение
\(\sqrt {27 + 6x} = x\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{27 + 6x = {x^2}}\end{array}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{x^2} — 6x — 27 = 0}\end{array}\,\,\,\,\,\,\, \Leftrightarrow } \right.} \right.\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge 0\,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{x = — 3}\\{x = 9}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,x = 9.\)
Ответ: 9.