Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 213

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\[1)\ 9^{x} - 4 \bullet 3^{x} + 3 = 0\]

\[3^{2x} - 4 \bullet 3^{x} + 3 = 0\]

\[Пусть\ y = 3^{x}:\]

\[y^{2} - 4y + 3 = 0\]

\[D = 4^{2} - 4 \bullet 3 = 16 - 12 = 4\]

\[y_{1} = \frac{4 - 2}{2} = 1;\ \ \ y_{2} = \frac{4 + 2}{2} = 3.\]

\[1)\ 3^{x} = 1\]

\[3^{x} = 3^{0}\]

\[x = 0.\]

\[2)\ 3^{x} = 3\]

\[3^{x} = 3^{1}\]

\[x = 1\]

\[Ответ:\ \ x_{1} = 0;\ \ \ x_{2} = 1.\]

\[2)\ 16^{x} - 17 \bullet 4^{x} + 16 = 0\]

\[4^{2x} - 17 \bullet 4^{x} + 16 = 0\]

\[Пусть\ y = 4^{x}:\]

\[y^{2} - 17y + 16 = 0\]

\[D = 17^{2} - 4 \bullet 16 =\]

\[= 289 - 64 = 225\]

\[y_{1} = \frac{17 - 15}{2} = 1;\ \]

\[= y_{2} = \frac{17 + 15}{2} = 16.\]

\[1)\ 4^{x} = 1\]

\[4^{x} = 4^{0}\]

\[x = 0.\]

\[2)\ 4^{x} = 16\]

\[4^{x} = 4^{2}\]

\[x = 2.\]

\[Ответ:\ \ x_{1} = 0;\ \ \ x_{2} = 2.\]

\[3)\ 25^{x} - 6 \bullet 5^{x} + 5 = 0\]

\[5^{2x} - 6 \bullet 5^{x} + 5 = 0\]

\[Пусть\ y = 5^{x}:\]

\[y^{2} - 6y + 5 = 0\]

\[D = 6^{2} - 4 \bullet 5 = 36 - 20 = 16\]

\[y_{1} = \frac{6 - 4}{2} = 1;\ \text{\ \ }y_{2} = \frac{6 + 4}{2} = 5.\]

\[1)\ 5^{x} = 1\]

\[5^{x} = 5^{0}\]

\[x = 0.\]

\[2)\ 5^{x} = 5\]

\[5^{x} = 5^{1}\]

\[x = 1.\]

\[Ответ:\ \ x_{1} = 0;\ \ \ x_{2} = 1.\]

\[4)\ 64^{x} - 8^{x} - 56 = 0\]

\[8^{2x} - 8^{x} - 56 = 0\]

\[Пусть\ y = 8^{x}:\]

\[y^{2} - y - 56 = 0\ \]

\[D = 1^{2} + 4 \bullet 56 = 1 + 224 = 225\]

\[y_{1} = \frac{1 - 15}{2} = - 7;\ \ \text{\ \ }\]

\[y_{2} = \frac{1 + 15}{2} = 8.\]

\[1)\ 8^{x} = - 7\]

\[нет\ корней.\]

\[2)\ 8^{x} = 8\]

\[8^{x} = 8^{1}\]

\[\ x = 1.\]

\[Ответ:\ \ x = 1.\]