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МатематикаРешебник по алгебре 9 класс Макарычев Задание 283
Задание 283
\[\boxed{\text{283\ (283).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[\textbf{а)}\ x^{5} + x^{4} - 6x^{3} - 6x^{2} +\]
\[+ 5x + 5 = 0\]
\[x^{4} \cdot (x + 1) - 6x^{2}(x + 1) +\]
\[+ 5 \cdot (x + 1) = 0\]
\[(x + 1)\left( x^{4} - 6x^{2} + 5 \right) = 0\]
\[1)\ x + 1 = 0\ \ \]
\[x_{1} = - 1;\]
\[2)\ x^{4} - 6x^{2} + 5 = 0\]
\[Пусть\ x^{2} = t;\ \ t \geq 0:\]
\[t^{2} - 6t + 5 = 0\]
\[D_{1} = 3^{2} - 5 = 4\]
\[t_{1} = 3 + 2 = 5;\ \ \ t_{2} = 3 - 2 = 1.\]
\[\Longrightarrow \left\{ \begin{matrix} x^{2} = 1 \\ x^{2} = 5 \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x_{2,3} = \pm 1\ \ \ \ \\ x_{4,5} = \pm \sqrt{5.} \\ \end{matrix} \right.\ \]
\[Ответ:x = \pm 1;\ \ x = \pm \sqrt{5}.\]
\[\textbf{б)}\ x^{5} - x^{4} - 2x^{3} + 2x^{2} -\]
\[- 3x + 3 = 0\]
\[x^{4} \cdot (x - 1) - 2x^{2}(x - 1) -\]
\[- 3 \cdot (x - 1) = 0\]
\[(x - 1)\left( x^{4} - 2x^{2} - 3 \right) = 0\]
\[1)\ x - 1 = 0\]
\[x_{1} = 1.\]
\[2)\ \ x^{4} - 2x^{2} - 3 = 0\]
\[Пусть\ x^{2} = t;\ \ \ t \geq 0:\]
\[t^{2} - 2t - 3 = 0\]
\[D_{1} = 1 + 3 = 4\]
\[t_{1} = 1 + 2 = 3;\ \ \ t_{2} = 1 - 2 = - 1.\]
\[Так\ как\ t \geq 0,\ то\ t = 3:\]
\[x^{2} = 3\]
\[x_{2,3} = \pm \sqrt{3}.\]
\[Ответ:x = 1;\ \ x = \pm \sqrt{3}.\]
\[\boxed{\text{283.}\text{\ }\text{ОК\ ГДЗ\ -\ домашка\ на\ 5}}\]
\[\textbf{а)}\ y^{4} - 24y^{2} - 25 = 0\]
\[Пусть\ t = y^{2},\ \ t^{2} = y^{4},\]
\[\ \ t > 0;\]
\[t^{2} - 24t - 25 = 0\]
\[D = 144 + 25 = 169\]
\[t_{1,2} = 12 \pm 13 = 25; - 1.\]
\[Так\ как\ t > 0,\ то\ t = 25 \Longrightarrow\]
\[\Longrightarrow y^{2} = 25 \Longrightarrow y = \pm 5.\]
\[Ответ:y = \pm 5.\]
\[\textbf{б)}\ x^{4} - 9x^{2} + 18 = 0\]
\[Пусть\ t = x^{2},\ t^{2} = x^{4},\ t > 0;\]
\[t^{2} - 9t + 18 = 0\]
\[D = 81 - 4 \cdot 18 = 9\]
\[t_{1,2} = \frac{9 \pm 3}{2} = 3;6;\]
\[\left\{ \begin{matrix} x² = 3 \\ x² = 6 \\ \end{matrix} \Longrightarrow \right.\ \left\{ \begin{matrix} x_{1,2} = \pm \sqrt{3} \\ x_{3,4} = \pm \sqrt{6}. \\ \end{matrix} \right.\ \]
\(Ответ:x = \pm \sqrt{3};\ \ x = \pm \sqrt{6}.\)
