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МатематикаРешебник по алгебре и начала математического анализа 10 класс Колягин Задание 401
Задание 401
\[\boxed{\mathbf{401}.}\]
\[\left\{ \begin{matrix} 9x^{2} - 4y^{2} + 5x + 10 = 0 \\ 8x^{2} - 3y^{2} + 5y + 10 = 0 \\ \end{matrix} \right.\ ( - )\]
\[x^{2} - y^{2} + 5x - 5y = 0\]
\[(x - y)(x + y) + 5 \cdot (x - y) = 0\]
\[(x - y)(x + y + 5) = 0\]
\[\left\{ \begin{matrix} (x - y)(x + y + 5) = 0\ \ \ \ \\ 8x^{2} - 3y^{2} + 5y + 10 = 0 \\ \end{matrix} \right.\ \]
\[\left\lbrack \begin{matrix} \left\{ \begin{matrix} x - y = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 8x^{2} - 3y^{2} + 5y + 10 = 0 \\ \end{matrix} \right.\ \\ \left\{ \begin{matrix} x + y + 5 = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 8x^{2} - 3y^{2} + 5y + 10 = 0 \\ \end{matrix} \right.\ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\]
\[\left\lbrack \begin{matrix} \left\{ \begin{matrix} x = y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 8y^{2} - 3y^{2} + 5y + 10 = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \left\{ \begin{matrix} x = - 5 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 8 \cdot ( - 5 - y)^{2} - 3y^{2} + 5y + 10 = 0 \\ \end{matrix} \right.\ \\ \end{matrix} \right.\ \]
\[\left\lbrack \begin{matrix} \left\{ \begin{matrix} x = y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 5y^{2} + 5y + 10 = 0\ (1) \\ \end{matrix} \right.\ \text{\ \ \ \ \ } \\ \left\{ \begin{matrix} x = - 5 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 5y^{2} + 85y + 210 = 0\ (2) \\ \end{matrix} \right.\ \\ \end{matrix} \right.\ \text{\ \ \ \ }\]
\[1)\ 5y^{2} + 5y + 10 = 0\ \ \ \ |\ :5\]
\[y^{2} + y + 2 = 0\]
\[D_{1} = 1 - 2 = - 1 < 0\]
\[нет\ корней.\]
\[2)\ 5y^{2} + 85y + 210 = 0\ \ \ |\ :5\]
\[y^{2} + 17y + 42 = 0\]
\[y_{1} + y_{2} = - 17;\ \ \ y_{1} \cdot y_{2} = 42\]
\[y_{1} = - 14;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y_{2} = - 3.\]
\[x_{1} = - 5 + 14 = 9;\ \ \ \]
\[x_{2} = - 5 + 3 = - 2.\]
\[Ответ:(9;\ - 14);( - 2;\ - 3).\]
