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Консультация № 198675
23.05.2020, 00:41
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Здравствуйте, уважаемые эксперты! Прошу Вашей помощи в решении задачи:
Найти вероятность того, что корни уравнения x^2+2ax+b=0 вещественны, если значения коэффициентов равновероятны в пределах |a| ≤ 2 и |b| ≤ 5.
Найти вероятность того, что корни уравнения x^2+2ax+b=0 вещественны, если значения коэффициентов равновероятны в пределах |a| ≤ 2 и |b| ≤ 5.
Обсуждение
27.05.2020, 17:23
общий
это ответ
Здравствуйте, aleexsmirn94!
Корни данного квадратного уравнения определяются по стандартной формуле:
Они будут вещественными при выполнении условия a[sup]2[/sup] - b [$8805$] 0, то есть при b [$8804$] a[sup]2[/sup]. Вероятность такого события можно определить геометрически: если считать значения a и b координатами на плоскости, то множество заданных в условии задачи значений коэффициентов образует прямоугольник, ограниченный прямыми a = -2, a = 2, b = -5 и b = 5, площадь которого равна 4[$183$]10 = 40; значениям же коэффициентов, удовлетворяющих условию b [$8804$] a[sup]2[/sup], соответствует часть прямоугольника, ограниченная сверху кривой b = a[sup]2[/sup]. Её площадь можно найти с помощью интеграла
Она составляет 76/3:40 = 19/30 от общей площади прямоугольника, что и даёт нам искомую вероятность.
Корни данного квадратного уравнения определяются по стандартной формуле:
Они будут вещественными при выполнении условия a[sup]2[/sup] - b [$8805$] 0, то есть при b [$8804$] a[sup]2[/sup]. Вероятность такого события можно определить геометрически: если считать значения a и b координатами на плоскости, то множество заданных в условии задачи значений коэффициентов образует прямоугольник, ограниченный прямыми a = -2, a = 2, b = -5 и b = 5, площадь которого равна 4[$183$]10 = 40; значениям же коэффициентов, удовлетворяющих условию b [$8804$] a[sup]2[/sup], соответствует часть прямоугольника, ограниченная сверху кривой b = a[sup]2[/sup]. Её площадь можно найти с помощью интеграла
Она составляет 76/3:40 = 19/30 от общей площади прямоугольника, что и даёт нам искомую вероятность.
5
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