Задача 7А. Решите уравнение \(\left| {{x^2} + 3x} \right| = 4\)
ОТВЕТ: -4; 1.
\(\left| {{x^2} + 3x} \right| = 4.\) Уравнение вида \(\left| {f\left( x \right)} \right| = a\), где \(a \ge 0\), равносильно совокупности: \(\left[ {\begin{array}{*{20}{c}}{f\left( x \right) = a,\,\,\,}\\{f\left( x \right) = -a.}\end{array}} \right.\) \(\left| {{x^2} + 3x} \right| = 4\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{x^2} + 3x = 4,}\\{{x^2} + 3x = -4}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{x^2} + 3x-4 = 0,}\\{{x^2} + 3x + 4\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 1,\,\,\,\,}\\{x = -4.}\end{array}} \right.\) Ответ: \(-4;\;\;1.\)