ГДЗ по алгебре 8 класс Макарычев ФГОС Задание 705

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\[\textbf{а)}\ \left\{ \begin{matrix} x + y = 8\ \\ xy = - 20 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = 8 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y(8 - y) + 20 = 0 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = 8 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 8y - y^{2} + 20 = 0 \\ \end{matrix} \right.\ \]

\[y^{2} - 8y - 20 = 0\]

\[D_{1} = 16 + 20 = 36\]

\[y_{1} = 4 - 6 = - 2;\ \ \]

\[y_{2} = 4 + 6 = 10.\]

\[1)\ \left\{ \begin{matrix} y_{1} = - 2 \\ x_{1} = 10 \\ \end{matrix} \right.\ ;\ \ 2)\ \left\{ \begin{matrix} y_{2} = 10 \\ x_{2} = - 2. \\ \end{matrix} \right.\ \]

\[Ответ:( - 2;10);\ \ (10; - 2).\]

\[\textbf{б)}\ \left\{ \begin{matrix} x - y = 0,8 \\ xy = 2,4\ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = 0,8 + y\ \ \ \ \ \ \ \ \\ y(0,8 + y) = 2,4 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = 0,8 + y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y^{2} + 0,8y - 2,4 = 0 \\ \end{matrix} \right.\ \]

\[y^{2} + 0,8y - 2,4 = 0\]

\[5y^{2} + 4y - 12 = 0\]

\[D_{1} = 4 + 12 \cdot 5 = 64\]

\[y_{1} = \frac{- 2 - 8}{5} = - 2;\ \ \]

\[y_{2} = \frac{- 2 + 8}{5} = \frac{6}{5} = 1,2.\]

\[1)\ \left\{ \begin{matrix} y_{1} = - 2\ \ \\ x_{1} = - 1,2 \\ \end{matrix} \right.\ ;\ \ 2)\ \left\{ \begin{matrix} y_{2} = 1,2 \\ x_{2} = 2\ \ \ \ \\ \end{matrix} \right.\ \]

\[Ответ:( - 1,2;\ - 2);\ \ (2;1,2).\]

\[\textbf{в)}\ \left\{ \begin{matrix} x^{2} - y^{2} = 8 \\ x - y = 4\ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} (x - y)(x + y) = 8 \\ x - y = 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} 4 \cdot (x + y) = 8 \\ x - y = 4\ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

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

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

\[\Longrightarrow \left\{ \begin{matrix} x = 2 - y \\ - 2y = 2\ \ \ \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x = 3\ \ \ \\ y = - 1 \\ \end{matrix} \right.\ .\]

\[Ответ:(3; - 1).\]

\[\textbf{г)}\left\{ \begin{matrix} x^{2} + y^{2} = 5 \\ x + y = - 3 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = - y - 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ ( - y - 3)^{2} + y^{2} = 5 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = - y - 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y^{2} + 6y + 9 + y^{2} = 5 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = - y - 3\ \ \ \ \ \ \ \ \ \ \ \\ 2y^{2} + 6y + 4 = 0 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = - y - 3\ \ \ \ \ \ \ \ \\ y^{2} + 3y + 2 = 0 \\ \end{matrix} \right.\ \]

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

\[y_{1} + y_{2} = - 3;\ \ \ y_{1} \cdot y_{2} = 2\]

\[y_{1} = - 2;\ \ y_{2} = - 1.\]

\[1)\ \left\{ \begin{matrix} y_{1} = - 2 \\ x_{1} = - 1 \\ \end{matrix} \right.\ ;\ \ \ \ 2)\ \left\{ \begin{matrix} y_{2} = - 1 \\ x_{2} = - 2. \\ \end{matrix} \right.\ \ \]

\[Ответ:( - 2; - 1);\ \ ( - 1; - 2).\]