Простейшие иррациональные уравнения. Задача 22

Задача 22. Решите уравнение \(\sqrt {2x + 3} = 6-x.\)

Ответ

ОТВЕТ: 3.

Решение

\(\sqrt {2x + 3} = 6-x.\)

Уравнение вида: \(\sqrt {f\left( x \right)} = g\left( x \right)\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{g\left( x \right) \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,}\\{f\left( x \right) = {g^2}\left( x \right).}\end{array}} \right.\)

\(\sqrt {2x + 3} = 6-x\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{6-x \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{2x + 3 = 36-12x + {x^2}}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \le 6,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{x^2}-14x + 33 = 0}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \le 6,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{x = 3,\,}\\{x = 11}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = 3.\)

Ответ: 3.