Задача 10. Найдите корень уравнения \(\sqrt {6 + 5x} = x.\) Если уравнение имеет более одного корня, в ответе запишите меньший из корней.
Решение
\(\sqrt {6 + 5x} = x\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{6 + 5x = {x^2}}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{x^2} — 5x — 6 = 0}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge 0}\\{\left[ {\begin{array}{*{20}{c}}{x = — 1}\\{x = 6}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = 6.\)
Ответ: 6.