ГДЗ по алгебре 9 класс Макарычев ФГОС Задание 223

\[\boxed{\text{224.}\text{\ }\text{ОК\ ГДЗ\ -\ домашка\ на\ 5}}\]

\[\textbf{а)}\ x^{4} - 25x^{2} + 144 = 0\]

\[Пусть\ x^{2} = t,\ \ x^{4} = t^{2},\]

\[\ \ t \geq 0,\ тогда:\]

\[t^{2} - 25t + 144 = 0\]

\[D = 25^{2} - 4 \cdot 144 =\]

\[= 625 - 576 = 49\]

\[t_{1,2} = \frac{25 \pm 7}{2},\ \ t_{1} = 16,\ \ \]

\[t_{2} = 9;\]

\[\left\{ \begin{matrix} x² = 16 \\ x² = 9\ \ \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x = \pm 4 \\ x = \pm 3 \\ \end{matrix} \right.\ \]

\[Ответ:x = \pm 3;\ \ x = \pm 4.\]

\[\textbf{б)}\ y^{4} + 14y² + 48 = 0\]

\[Пусть\ \ y^{2} = t,\ \ y^{4} = t^{2},\ \]

\[\ t \geq 0,\ тогда:\]

\[t^{2} + 14t + 48 = 0\]

\[D = 7^{2} - 48 = 49 - 48 = 1\]

\[t_{1,2} = - 7 \pm 1,\ \ \]

\[t_{1,2} = - 6; - 8 \Longrightarrow корней\ нет.\]

\[Ответ:исходное\ уравнение\ \]

\[не\ имеет\ корней.\]

\[\textbf{в)}\ x^{4} - 4x^{2} + 4 = 0\]

\[Пусть\ x^{2} = t,\ \ \ t \geq 0,\ тогда:\]

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

\[(t - 2)^{2} = 0,\ \ t = 2;\]

\[x^{2} = 2 \Longrightarrow x = \pm \sqrt{2}\]

\[Ответ:x = \pm \sqrt{2}.\]

\[\textbf{г)}\ t^{4} - 2t^{2} - 3 = 0\]

\[Пусть\ t^{2} = a,\ \ a \geq 0,\ тогда:\]

\[a^{2} - 2a - 3 = 0\]

\[D = 1 + 3 = 4\]

\(a_{1,2} = 1 \pm 2 = - 1; - 3;\)

\[t² = 3 \Longrightarrow t = \pm \sqrt{3}.\]

\[Ответ:t = \pm \sqrt{3}.\]

\[\textbf{д)}\ 2x^{4} - 9x^{2} + 4 = 0\]

\[Пусть\ x^{2} = t,\ \ t \geq 0,\ тогда:\]

\[2t^{2} - 9t + 4 = 0\]

\[D = 81 - 2 \cdot 4 \cdot 4 = 49\]

\[t_{1,2} = \frac{9 \pm 7}{4} = \frac{1}{2};4;\]

\[\left\{ \begin{matrix} x^{2} = \frac{1}{2} \\ x^{2} = 4 \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x = \pm \sqrt{\frac{1}{2}} \\ x = \pm 2\ \ \ \ \\ \end{matrix} \right.\ \]

\[Ответ:x = \pm \sqrt{\frac{1}{2}};\ \ x = \pm 2.\]

\[\textbf{е)}\ 5y^{4} - 5y^{2} + 2 = 0\]

\[Пусть\ y^{2} = t,\ \ t \geq 0,\ тогда:\]

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

\[D = 25 - 4 \cdot 5 \cdot 2 = - 15 < 0 \Longrightarrow\]

\[\Longrightarrow корней\ нет.\ \]

\[Ответ:нет\ корней.\]