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21.04.2023 09:34

Раскройте скобки по формуле квадрата разности двух выражений : (х , все задания сделать нужно ​


Раскройте скобки по формуле квадрата разности двух выражений : (х , все задания сделать нужно ​

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Ответ:
asdfdsf01oziqid
16.03.2022 23:12
(a-1)x^2-2x-a\ \textgreater \ 0
Если a=1, то получим линейное неравенство:
-2x-1\ \textgreater \ 0
\\\
x\ \textless \ - \frac{1}{2}
Полученный промежуток не включает в себя заданыый x\ \textgreater \ 3.
Рассматриваем случай, когда a \neq 1 - имеем квадратное неравенство.
Заданное неравенство ">0", в зависимости от знака старшего коэффициента общие решения неравенства можно записать в виде:
 - если старший коэффициент больше 0: x\in(-\infty;x_1)\cup(x_2;+\infty)
 - если старший коэффициент меньше 0: x\in (x_3;x_4)
Вывод: необходимо рассмотреть случай с положительным старшим коэффициентом: a-1\ \textgreater \ 0, тогда a\ \textgreater \ 1
Решаем неравенство. Приравниваем левую часть к нулю:
(a-1)x^2-2x-a=0
\\\
D_1=(-1)^2-(a-1)\cdot(-a)=a^2-a+1
Получившийся дискриминант всегда больше 0, т.к. a^2-a+1=a^2-2\cdot \frac{1}{2} + \frac{1}{4} - \frac{1}{4} +1=(a- \frac{1}{2} )^2+ \frac{3}{4}\ \textgreater \ 0

x= \frac{1\pm \sqrt{a^2-a+1} }{a-1} 
\\\
\Rightarrow x\in(-\infty; \frac{1-\sqrt{a^2-a+1} }{a-1} )\cup( \frac{1+\sqrt{a^2-a+1} }{a-1} ;+\infty)
Чтобы получившийся ответ включал интервал х>3, необходимо потребовать выполнение следующего условия:
\frac{1+\sqrt{a^2-a+1} }{a-1} \leq 3
\\\
 \frac{1+\sqrt{a^2-a+1} -3(a-1)}{a-1} \leq 0
\\\
 \frac{4-3a+\sqrt{a^2-a+1} }{a-1} \leq 0
Так как в рассматриваемом случае a-1\ \textgreater \ 0, то можно перейти к следующему неравенству:
4-3a+\sqrt{a^2-a+1} \leq 0
\\\
\sqrt{a^2-a+1} \leq 3a-4
\\\
\begin{cases} a^2-a+1 \leq (3a-4)^2 \\ 3a-4\ \textgreater \ 0 \right \end{cases}
\\\
\begin{cases} a^2-a+1 \leq 9a^2-24a+16 \\ 3a\ \textgreater \ 4 \right \end{cases}
\\\
\begin{cases} 8a^2-23a+15 \geq 0 \\ a\ \textgreater \ \frac{4}{3} \right \end{cases}
\\\
\begin{cases} a\in(-\infty;1]\cup[ \frac{15}{8} ;+\infty) \\ a\ \textgreater \ \frac{4}{3} \right \end{cases}
Итоговое решение с учетом рассматриваемого ограничения a-1\ \textgreater \ 0: a\in[ \frac{15}{8} ;+\infty)
Искомое минимальное целое значение a_{min; \in Z}=2
ответ: 2
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Ответ:
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