Задача 11А. Решите уравнение \(\left| {{x^2}-6} \right| = \left| x \right|\)
ОТВЕТ: \( \pm 2;\;\; \pm 3.\)
\(\left| {{x^2}-6} \right| = \left| x \right|.\) Уравнение вида \(\left| {f\left( x \right)} \right| = \left| {g\left( x \right)} \right|\) равносильно совокупности: \(\left[ {\begin{array}{*{20}{c}}{f\left( x \right) = g\left( x \right),\,\,\,}\\{f\left( x \right) = -g\left( x \right).}\end{array}} \right.\) \(\left| {{x^2}-6} \right| = \left| x \right|\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{x^2}-6 = x,\,}\\{{x^2}-6 = -x}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{x^2}-x-6 = 0,}\\{{x^2} + x-6 = 0\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = -2,}\\{x = 3,\,\,\,}\\{x = 2,\,\,\,}\\{x = -3.}\end{array}} \right.\) Ответ: \( \pm 2;\;\; \pm 3.\)