Простейшие иррациональные уравнения. Задача 21

Задача 21. Решите уравнение \(\sqrt {3x + 7} = x + 1.\)

Ответ

ОТВЕТ: 3.

Решение

\(\sqrt {3x + 7} = x + 1.\)

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

\(\sqrt {3x + 7} = x + 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x + 1 \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{3x + 7 = {x^2} + 2x + 1}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge -1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{x^2}-x-6 = 0}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge -1,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{x = -2,}\\{x = 3\,\,\,\,}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = 3.\)

Ответ: 3.