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Консультация № 191654
02.11.2017, 18:54
0.00 руб.
0
6
2
Здравствуйте! Прошу помощи в следующем вопросе:
Как меняется количество корней уравнения x^2+p=√(x-p) в зависимости от значения параметра р.
Как меняется количество корней уравнения x^2+p=√(x-p) в зависимости от значения параметра р.
Обсуждение
02.11.2017, 22:05
общий
это ответ
Здравствуйте, more_1112!
Кривые, соответствующие функциям y = x2 + p и y = [$8730$](x - p) показаны на графике, где взято p = 1.
При уменьшении p до некоторого значения p0 кривые будут касаться друг друга и уравнение будет иметь единственное решение, при p < p0 уравнение имеет два решения, при p >p0 - ни одного. Чтобы определить p0, заметим, что графики симметричны относительно прямой y = x. Поэтому прямая y = x пройдет через точку касания и будет касаться обеих кривых. Это дает два уравнения (для кривой y = x2 + p):
2x = 1 и x2 + p = x.
Отсюда находим для точки касания x = 1/2 и p = 1/4, т. е., p0 = 1/4.
Таким образом, при p > 1/4 уравнение не имеет решений, при p = 1/4 решение одно, и при p < 1/4 два решения.
Кривые, соответствующие функциям y = x2 + p и y = [$8730$](x - p) показаны на графике, где взято p = 1.
При уменьшении p до некоторого значения p0 кривые будут касаться друг друга и уравнение будет иметь единственное решение, при p < p0 уравнение имеет два решения, при p >p0 - ни одного. Чтобы определить p0, заметим, что графики симметричны относительно прямой y = x. Поэтому прямая y = x пройдет через точку касания и будет касаться обеих кривых. Это дает два уравнения (для кривой y = x2 + p):
2x = 1 и x2 + p = x.
Отсюда находим для точки касания x = 1/2 и p = 1/4, т. е., p0 = 1/4.
Таким образом, при p > 1/4 уравнение не имеет решений, при p = 1/4 решение одно, и при p < 1/4 два решения.
5
04.11.2017, 01:53
общий
При р=-1/4 корень у меня получился - единственный. При р€(-1;0) у меня получился единственный корень. Остальное получилось так. Завтра попробую выложить решение
04.11.2017, 07:07
общий
04.11.2017, 07:08
Похоже, что так. При -1 < p < 0 вершина p полупараболы [$8730$](x - p) попадает между точками пересечения параболы x2 + p c осью x (x1,2 = [$177$][$8730$](-p)), и в этом интервале корень один. Остальное как в ответе.
04.11.2017, 12:54
общий
это ответ
Здравствуйте, more_1112!
После преобразований решал квадратное уравнение относительно р
После преобразований решал квадратное уравнение относительно р
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04.11.2017, 12:56
общий
Продолжение ответа (не знаю, как вставлять сразу две картинки
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