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Консультация № 183122
11.05.2011, 11:17
43.95 руб.
11.05.2011, 11:29
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Решите уравнение 27^x-7кубических корней(7*3^x+6)=6.
Заранее благодарен.
Решите уравнение 27^x-7кубических корней(7*3^x+6)=6.
Заранее благодарен.
Обсуждение
11.05.2011, 13:50
общий
это ответ
Здравствуйте, Тимофеев Алексей Валентинович!
Пусть y = 3[sup]x[/sup]. Запишем уравнение в виде
Функции (y[sup]3[/sup]-6)/7 и [sup]3[/sup][$8730$](7y+6), очевидно, являются взаимно обратными. Кроме того, функция (y[sup]3[/sup]-6)/7 - взаимно-однозначное отображение R[$8594$]R. Следовательно, графики этих двух функций симметричны относительно прямой f(y) = y, и все решения уравнения (точки пересечения графиков) лежат на этой прямой, то есть
Отсюда имеем уравнение y[sup]3[/sup] - 7y - 6 = 0, корни которого - y[sub]1[/sub] = -2, y[sub]2[/sub] = -1, y[sub]3[/sub] = 3. Из них только последний даёт вещественное значение x = log[sub]3[/sub]3 = 1. Следовательно, исходное уравнение имеет единственное решение x = 1.
Пусть y = 3[sup]x[/sup]. Запишем уравнение в виде
Функции (y[sup]3[/sup]-6)/7 и [sup]3[/sup][$8730$](7y+6), очевидно, являются взаимно обратными. Кроме того, функция (y[sup]3[/sup]-6)/7 - взаимно-однозначное отображение R[$8594$]R. Следовательно, графики этих двух функций симметричны относительно прямой f(y) = y, и все решения уравнения (точки пересечения графиков) лежат на этой прямой, то есть
Отсюда имеем уравнение y[sup]3[/sup] - 7y - 6 = 0, корни которого - y[sub]1[/sub] = -2, y[sub]2[/sub] = -1, y[sub]3[/sub] = 3. Из них только последний даёт вещественное значение x = log[sub]3[/sub]3 = 1. Следовательно, исходное уравнение имеет единственное решение x = 1.
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