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Возвращаясь к исходной задаче, замечаем, что при
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAJAAgAhAAAAAAAAAAAHQAAMx0AMgAcSAAzWwAzWgAyWh1GbDMAADNbWzNbgEgcAFozAFszAEZGM1lZf1l/WV1/f2xGHX9ZM39/XXtqRHd3XXd3ZmaIiIBdM4iIZgECAwECAwECAwUfICCOpEgFAlCMjSEqo1My4kAS48GOz6gEpEDAprABQgA7)
=0 уравнение (1) имеет корень
![](data:image/gif;base64,R0lGODlhLQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAoAAwAhAAAAAAAAAAAHR0AAAAcSAAdSB0dSAAzWgAzWx1GbB1IbjMAADQ0HTNbgEgcAEgdAFszAFozAERqe0huf2xGHX9ZM25GM39/XX9uSHd3ZmZ3d2aIiIBbM4iIZgECAwECAwVKICCOZGmeaKquK8G+7BK4cJ3Sdj5SOBcIo4CuZToAECVIYMkMIGG8EmaQGCqbyycAN3IMWVGSBWA0YZc57hcWwQlFj4BibZL9JiEAOw==)
, который требованиям задачи не удовлетворяет.
Рассмотрим случай
![](data:image/gif;base64,R0lGODlhJQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAhAAwAhQAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSAAdSAAzWwAzWgAyWh1GbDMAADMAMzNbWzNZfzNbgEgcAEgdHUgdAFozAEZGM1lZf1l/WV1/f0huf2xGHW5IHX9ZM39/XX9uSHtqRHd3XXd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZjQIBwSCwaj8ikssgJBJbQY0HoiFqFkuHhSqQcBUTwsDkAbJGco2k6rAonCmqx46w740S2HGApZpF/RWtEBEJ6AAZJCEliQmILQxNHdnVHEUMJfG1PSHhJZ24ATlMOh1x0nABBADs=)
. При таких
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAJAAgAhAAAAAAAAAAAHQAAMx0AMgAcSAAzWwAzWgAyWh1GbDMAADNbWzNbgEgcAFozAFszAEZGM1lZf1l/WV1/f2xGHX9ZM39/XXtqRHd3XXd3ZmaIiIBdM4iIZgECAwECAwECAwUfICCOpEgFAlCMjSEqo1My4kAS48GOz6gEpEDAprABQgA7)
условия (3) запишутся в виде
![](data:image/gif;base64,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)
Решая эту систему, находим, что
![](data:image/gif;base64,R0lGODlhYwA1AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgBfAC8AhgAAAAAAAB0AHR0AAAAAHQAAMxQLMx0AMhUMMx0dSQAcSAAdSAAzWgAzWx0zWwAyWggufwszbR1JSR1GbBVAgDMAADMeRzMzWzNbWzNGbjNZfzNbgEgcAEgdAEkdHUceM0geNFszAFozAFozHVoyAFszM0YzRklJHUZGM0hZf1lZf1l/WV1/f0huf113d0Rqe38uCGxGHW5IHW5GM39ZM39ZSH9/XXtqRH9uSHd3XX9/QHd3ZmZ3d2aIiIBAFYBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwf/gACCg4SFhoeIiYqLjI2Oj4UFkJOUhiEBmAGVjA2bnp8Xn6KjiJmmmaSLDqmfmI+hqR+srbO1toear7eQFa6kvoMBkruDspQgwb+5ABXDibDEjxwMiMaNyxWd0YQZoiKJ1ozLzorQ24wco74VzMLnABuiCeCOuT8B84LkheaeP6IjLJ3CpK3UIHIB4kVTWCnGonDBgAGwR44don7vUmnyBcyiIAHnKFSiYWpfMUW5lukbVNCQgYyHZJxqSQhZIlcqAVwKQHMXQ0LpzqGiZIFYLwVCJT4qSurnIIfvci6FNCATOR+JkMIUFSMAAQBaB/1TJNIQ1EELtjZiCiAos0lYshGFVeuI6TdCTnuVLFS20Fm6dVcOOgA3a6mBiBMrXjz0EDVBbgl5WNSX0F/AjIqKeCCoF4ZDJgfFNURiLmZFTDFJanZYNV9Der/iGBD6NAC2n0bbtl0ZUTZSNFjh9gRjawXTnhCk6h3tBCuMlXS/Q76b+a1Ql0GRkr6rxCDqtiNsu/H0+W5IA53ahr4ruyf25xeBPz0Dpgzz8fPrzwh/v/9E9/1XTi0hCLhVCLXlZ0ongQAAOw==)
.
Очевидно, что этот же результат мы получили бы и решая неравенство
![](data:image/gif;base64,R0lGODlhJwAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAiABAAhQAAAAAAAAAAHR0AAAAeHgAAMwAcSAAdSB0dSAAzWgAzWx1GbB1IbjMAADQ0HTNbgEgcAEgdAEceM1oyAFszAEhZf0Rqe11/f0Rubkhuf113d2xGHW5IHW5GM39/XX9uSHtqRHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZeQIBwSCwaj8ikUmlYOpGNQPNJLU6rw8RycyUFBMPAUsw0ahVP7+HILX4GCyxgUDBehxD5nFBsEzsAWkJ1SgNJd3IBa0gTV2QAEQEMABJHJEtRX0qVelScnaChRY+hQQA7)
, где
![](data:image/gif;base64,R0lGODlhEAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAMAAwAhAAAAAAAAAAAHR0AAAAAMwAdSB0dSAAzWgAzWx1GbB1IbjQ0HTNZf0gcAEgdAEceM1szAERqe11/f1VmZkRubl13d2xGHW5IHX9ZM25GM39uSHd3ZmaIiIBbM4iIZgECAwUmICCO5NgFwhiU4wEgrKgNSTw2tpgBrkjkrJXIEVAAHsBjEplMCkMAOw==)
- меньший корень уравнения (1)
Ответ:
![](data:image/gif;base64,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)
.
Рациональные неравенства с параметрами
Пример. Найти все значения параметра
, при которых неравенство ![](data:image/gif;base64,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)
выполняется при всех
. Решение. Исходное неравенство является однородным неравенством второй степени относительно функции
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
. Если разделить его на
![](data:image/gif;base64,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)
, то получится равносильное неравенство
![](data:image/gif;base64,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)
которое после замены
![](data:image/gif;base64,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)
становится квадратным неравенством относительно переменной
![](data:image/gif;base64,R0lGODlhCQAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAGAAoAhAAAAAAAAB0AHQAAHR0AMgAdSB1IbjIAHTNGbjNdgEgcAEgeNFszAFozAERqe0huf1V3al1/f2xGHXtqRHd3XX9uSGaIiIBdM4iIZgECAwECAwECAwECAwECAwECAwECAwUZICCO40QCwsgEbCI2J/CQBamcBLCMwTADIQA7)
с параметром
![](data:image/gif;base64,R0lGODlhEAARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAKAAsAhAAAAAAAAB0AAB4eAB0AHQAAMwAdMx0dSAAdSAAzWx1GbDIAHTIAMjNZfzNbgEgcAEgdHVozAFszAF1/f1luf0huf2xGHX9uSH9/XXtqRGZ3d2aIiIBbM4iIZgECAwECAwUpICCO5CgEQBQUJTCMCmmMDxmNGVGNzBglpMOINbIEjsDSrgUQMo+IVggAOw==)
:
![](data:image/gif;base64,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)
(*)
Найдем множество значений функции
![](data:image/gif;base64,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)
при
![](data:image/gif;base64,R0lGODlhJwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAiAAwAhQAAAAAAAB0AAAAAHR0AHQAAMwAdSAAcSB0dSAAyWgAzWgAzWx1GbB1IbjMAADMeRzQ0HTNbgEgcAEkdHUgdAFszAFozAEYzRl1/f0Rqe113d0huf2xGHW5IHW5GM39/XXtqRH9uSHd3XXd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZiQIBwSCwaj8ik0lgJOAPL6NFSEEKlWKFC6GBkBc/q0TEUCE2BwfA6NElNYolxu8gan3ViSOA1Op5OYkVycEdyRw9SBFxGHgBbRhNSB0OQWIJIZENqVkMUAQ1HgJgAT0N/C0EAOw==)
. Имеем:
![](data:image/gif;base64,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)
, то есть
![](data:image/gif;base64,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)
Отсюда
![](data:image/gif;base64,R0lGODlhIQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAdAAwAhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAdSAAzWx1GbDMAADIAHTNZfzNGbjNbgEgcAEgdAEgeNFszAFozAF1/f0Rqe0huf1V7amxGHW5IHX9ZM39/XXtqRH9uSHd3XWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZUQIBwSCwaj8jkMRMIKJ/FgnBR7ECPj+GBKLgWu0PwpBnIekXSIVVIOZLfUeLCDLh4h2jiQHvn+gEQR2NvTQlFDUMIQgYAEn18AGsAAQR2jwAaTUNBADs=)
при
![](data:image/gif;base64,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)
; другие значения
![](data:image/gif;base64,R0lGODlhCQAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAGAAoAhAAAAAAAAB0AHQAAHR0AMgAcSB1GbB1IbjIAHTNGbjNdgEgcAEgeNFszAFozAERqe0huf1V7al1/f2xGHXtqRHd3XX9uSGaIiIBdM4iIZgECAwECAwECAwECAwECAwECAwUZICCOI0UCwtgErCI6JwCRBbmcBMCMwTADIQA7)
(отличные от нуля) найдем из условия неотрицательности дискриминанта этого квадратного уравнения:
![](data:image/gif;base64,R0lGODlhMQEpAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAsASMAhgAAAAAAAB0AAB0AHR4AHgEAAAAAHQAAMx0AMx0AMgAcSAAdSB0dSB0dSQAcRwAdRwEdSAAdSQAzWgAzWwAyWgAzWR0zWx1Ibh1GbB1GajMAADMAHTIAHTIAMjMAMzMzWzNdXTNZfzFZfTNGbjVIbjNbgDNdgEgcAEccAEkdAEgeNEceM1szAFozAFoyAFozHVkzAFszHVszM0lJHVtISEhZf0hZfV1/f0huf1V3d0Rqe113d1VmZlV3amxGHW5IHWpGHWpIHX9ZM25ISH9ZSH9/XXtqRHd3VXd3XWZmVX9uSGZ3d2aIiIBbM4BdM4BzSIBuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwf/gACCg4SFhoeIiYqLjI2Oj5CRkpOUlZaXmIMsAZycmZ+goaKjpKWmpwAHgxqora6vsLGyoS0ThAGzubq7vL2fGiWEGhK+xcbHyLOqhCfEyc/Q0dKPLYbD09jZ2sknF4UC2+Hi468DhS7LoBq45JjrGO3xpQuEQuyjHfKUQ4Ie+v+ZNMAT1CKACVM4AE56obBhpB+dOtG7tO4eIRqZ/FWq2E8ax08KLGlEEZKSRkMnLZVMKQmIJ1MqBlkEICRTOks/lhFwdRDSyZsbKwlZtq7kpCcEdBQCSomVIKRKCQVpNFSQU1TNRLG0tGEQAlQUuDZaoUgGAKORWKKd9HXQVkRk/xPJmIG2rdQAFRSdILSWVDVBEDn1hDRA6d9BRswdivvIGSjFADpVskCo8OJETQD4qDRwUF9GlgEPOjwodCHGhjJvJvQj0UxCOwclxYppkycOhGy/BoD6EINCCUJJxvRbEIwCLyV9ZoSbkI/liFogtxgcQHOC0ymtrY5IANObVyNFHH6oQabDQEmfThQYfCHdEW056myJUzr1hHobWl0IxnhOeQnSBFPQRWdNCKkUgt8g+hXC3yoIIrJOBksJE2EpMTQCXyfyCdKcD0BdJ9N/rglDn4b/BWCLiK6RaAhEA7Ho338BGlJgIgQSUhCNhHRVCCsgFuLjaCnWWIgDP3bHFP8ATtwEToopTvLgJRMlSEiVhjRoSGfhXTIlJnsJguV6hswY0UM5QjImAAJZOciaDB6y45mrnLgbbLAlFMmGnHQ4SpiC3AQoJaR1yaeKjXRDiyCDTvJZlIQA9RwkjVoVKDOXrNWlkYikcGUp9nSy5CNKcGfdIEosoiViwIESXEyfmJeqIqsK8qUjG0Q1CJKOlGqIEm0N6WsitWpWyKyROLGKKeP5OUkABiACrSQSFeJsfdFeUq0g0yICqyIRTMKCMJxk20i3hMjHQmfoGvJtIg8UwmmRhyTXjq4OYTPkJ2+N0m++icSmDsCkCOyOIQb/ckpNBDcsy78+uQJxQDc6bPE9xbrMgPHGHPtScccgh3zJB8aKbPLJmZglyMcot+yyIEYMcuvLNNcc2Xg256zzzDr3fDLLPgfdcGtCF31xIAA7)
, то есть
![](data:image/gif;base64,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)
.
Итак, исходное неравенство выполняется для всех
![](data:image/gif;base64,R0lGODlhJwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAiAAwAhQAAAAAAAB0AAAAAHR0AHQAAMwAdSAAcSB0dSAAyWgAzWgAzWx1GbB1IbjMAADMeRzQ0HTNbgEgcAEkdHUgdAFszAFozAEYzRl1/f0Rqe113d0huf2xGHW5IHW5GM39/XXtqRH9uSHd3XXd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZiQIBwSCwaj8ik0lgJOAPL6NFSEEKlWKFC6GBkBc/q0TEUCE2BwfA6NElNYolxu8gan3ViSOA1Op5OYkVycEdyRw9SBFxGHgBbRhNSB0OQWIJIZENqVkMUAQ1HgJgAT0N/C0EAOw==)
тогда и только тогда, когда неравенство (*) выполняется для всех
![](data:image/gif;base64,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)
.
Рассмотрим квадратный трехчлен
![](data:image/gif;base64,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)
с абсциссой вершины
![](data:image/gif;base64,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)
и дискриминантом
![](data:image/gif;base64,R0lGODlh1QAZAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgDQABMAhgAAAAAAAB4AHh0AAB4eAB0AHQAAHQAAMx0AMx0AMgAdMwAcSAAdSB0dSR0dSAAzWgAzWwAyWh1GbB1IbjMAADIAHTMAHTIAMjMAMzQeSDMzWzNZfzNdgDNbgEgcAEgdAEkdAEgdHUgeNEceM1ozHVszAFozAFszHVoyAFszM0lJHVtISEhZf1l/WV1/f0huf113d0Rqe1V3d1luf1VmZmxGHW5IHX9ZM25ISH9ZSH9/XXd3XX9uSHd3VXtqRGZmVXd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwf/gACCg4SFhoeIiYqLjI2Oj5CRkpOUlZUUARKWm5ydiJieoaKTOKOmp5EXqKusgyStsLGykiUBtrYPs5QLipgBG6O8sCuRJrarwoQ1B4MUybqOzoXMgiKDv56YyQLQiSYMgiWn0oYQg0PYqBGVKQDPhS+DHpMjie+mN5OaguCi7fcUCgVEdatSjWQm9h0y5+kgIRujbBR89EFYjVMOCd2gRiiXJR4DAgCoxZFTsgqJ5hEKCSxASUL1EL1LMAmkSJKbBhyyOdJlpXcm3lHwaKjWrVsMDVUgMCrjkJeEGigl0AGASkMxDV0UWDXcUaSIlobyOVWQB4WQMsozNGBCJwVr+BV9PSZzENRB8TQWUmj0a1JBQivBNbuo79Gk9ShAHWy10FyRdQkVKITiriQTki0ZPrbsEeZBk9MKReuZELdKn0c+Sw3gtKPNkAE7tnvocaO/m5LdVchikCpBrAtlNRSYkG1GuCNRACbo98KfhAYCj83JAXDLk9SidHzrrgnswwmhmEnJOoDvmyoKutFPkHkA2BmpndteUPi0YEW1F2f8KHfqh1hjiC8G5AVAco3UkN8mINA1iIK2IAhJfYvc1w0jFjQSwigYXPjIht1Y6Mh7sLiGiA+jSCcJiauw6OEjtlDISoeIGBUKjZPEyIpRMsIC4ItABinkIIEAADs=)
. Тогда имеем следующие необходимые и достаточные условия для нахождения искомых значений параметра
![](data:image/gif;base64,R0lGODlhEAARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAKAAsAhQAAAAAAAB0AAB4eAB0AHQAAMwAdMx0dSAAdSAAzWx1GbDIAHTIAMjNZfzNdgDNbgEgcAEgdHVszAFozAF13d1luf0huf11/f1VmZmxGHX9ZM39uSH9/XXtqRGZ3d2aIiIiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwYtQIBwSBwKAgBJoFAEDIYKomEIIU6GHYJlyBhOEsTDkDnMBM7g4rYJELPPiGYQADs=)
:
![](data:image/gif;base64,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)
Последовательно преобразуя, получаем:
![](data:image/gif;base64,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)
Объединяя решения систем (1)-(3), получаем ответ.
Ответ:
![](data:image/gif;base64,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)
Иррациональные уравнения с параметрами
Существует несколько способов решения иррациональных уравнений с параметрами. Познакомимся с ними, разобрав следующий пример.
Пример. В зависимости от значений параметра
решить уравнение
(1) Решение. Решим уравнение (1) пятью способами, которые необходимо знать, ибо наряду с другими подходами они могут быть использованы и при решении иных типов уравнений.
Способ 1. Уравнение (1) равносильно системе
![](data:image/gif;base64,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)
или системе
(2)Решая уравнение из системы (2), находим
(3)откуда следует, что при
уравнение (1) имеет одно решение
. Если
, то
, и тогда уравнение (1) будет иметь два решения при тех значениях параметра
, при которых совместна система
,т.е. при
![](data:image/gif;base64,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)
Уравнение (1) будет иметь только один корень
, если
, а
. В этом случае решая систему ![](data:image/gif;base64,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)
приходим к выводу, что
.Замечая теперь, что при
дискриминант уравнения системы (2) отрицателен, получаемОтвет: если
, то решений нет; если
, то
; если
, то
; если
, то
.Способ 2. Возведя обе части уравнения (1) в квадрат, получим уравнение из системы (2), корни которого задаются формулами (3). Но здесь надо иметь в виду, что при возведении обеих частей уравнения (1) в квадрат могли появиться посторонние корни.
Поэтому при данном способе решения необходимо произвести проверку. Так, подставляя корень
в исходное уравнение, придем к соотношению
,откуда
.Если же подставить корень
в уравнение (1), то придем уже к отношению
, и, таким образом,
.Учитывая теперь, что при
корней нет, а при
имеем
, получаем тот же ответ, что и при первом способе решения.