ГДЗ по алгебре 9 класс Макарычев Задание 277

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Год:2020-2021-2022
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Задание 277

\[\boxed{\text{277}\text{\ (277)}\text{.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[\textbf{а)}\ \left( x^{2} + 3 \right)^{2} - 11 \cdot \left( x^{2} + 3 \right) +\]

\[+ 28 = 0\]

\[Пусть\ x^{2} + 3 = t:\ \]

\[t^{2} - 11t + 28 = 0\]

\[t_{1} + t_{2} = 11;\ \ \ t_{1} \cdot t_{2} = 28\]

\[\text{\ \ }t_{1} = 7;\ \ \ \ \ t_{2} = 4\]

\[\left\{ \begin{matrix} x^{2} + 3 = 7 \\ x^{2} + 3 = 4 \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x^{2} = 4 \\ x^{2} = 1 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x_{1,2} = \pm 2 \\ x_{3,4} = \pm 1 \\ \end{matrix} \right.\ \]

\[Ответ:x = \pm 2;x = \pm 1.\]

\[\textbf{б)}\ \left( x^{2} - 4x \right)^{2} + 9 \cdot \left( x^{2} - 4x \right) +\]

\[+ 20 = 0\]

\[Пусть\ x^{2} - 4x = t:\ \]

\[t^{2} + 9t + 20 = 0\]

\[t_{1} + t_{2} = - 9;\ \ \ t_{1} \cdot t_{2} = 20\]

\[t_{1} = - 5;\ \ \ t_{2} = - 4;\]

\[\left\{ \begin{matrix} x^{2} - 4x = - 5 \\ x^{2} - 4x = - 4 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x^{2} - 4x + 5 = 0\ \ (1) \\ x^{2} - 4x + 4 = 0\ \ (2) \\ \end{matrix} \right.\ \]

\[(1)\ x^{2} - 4x + 5 = 0\]

\[D_{1} = 4 - 5 = - 1 < 0 \Longrightarrow\]

\[\Longrightarrow корней\ нет;\]

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

\[(x - 2)^{2} = 0\]

\[x - 2 = 0\]

\[x = 2\]

\[Ответ:x = 2.\]

\[\textbf{в)}\ \left( x^{2} + 1 \right)\left( x^{2} + x - 5 \right) = 84\]

\[Пусть\ \ x^{2} + x = t:\]

\[t(t - 5) = 84\]

\[t^{2} - 5t - 84 = 0\]

\[t_{1} + t_{2} = 5;\ \ \ t_{1} \cdot t_{2} = - 84\]

\[t_{1} = 12;\ \ \ \ t_{2} = - 7;\]

\[\left\{ \begin{matrix} x^{2} + x = 12 \\ x^{2} + x = - 7 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x^{2} + x - 12 = 0\ \ (1) \\ x^{2} + x + 7 = 0\ \ \ \ (2) \\ \end{matrix} \right.\ \]

\[(1)\ x^{2} + x - 12 = 0\]

\[x_{1} + x_{2} = - 1;\ \ \ x_{1} \cdot x_{2} = - 12\]

\[x_{1} = - 4;\ \ \ x_{2} = 3.\]

\[(2)\ x^{2} + x + 7 = 0\]

\[D = 1 - 4 \cdot 7 = - 27 < 0 \Longrightarrow\]

\[\Longrightarrow корней\ нет.\]

\[Ответ:x = 3;\ \ x = - 4.\]

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