Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 284

\[\boxed{\mathbf{284}.}\]

\[1)\ x^{2} = y^{2} + 4y + 8\]

\[x^{2} - y^{2} - 4y = 8\]

\[x^{2} + xy + 2x - y^{2} - xy - 2y -\]

\[- 2x - 2y - 4 + 4 = 8\]

\[x(x + y + 2) - y(x + y + 2) -\]

\[- 2 \cdot (x + y + 2) + 4 = 8\]

\[(x + y + 2)(x - y - 2) = 4\]

\[Делители:\ \pm 1;\ \pm 2;\ \pm 4.\]

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

\[2x = 5\]

\[x = 2,5\]

\[x \notin Z.\]

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

\[2x = 5\]

\[x = 2,5\]

\[x \notin Z.\]

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

\[2x = - 5\]

\[x = - 2,5\]

\[x \notin Z.\]

\[\left\{ \begin{matrix} x + y + 2 = - 4 \\ x - y - 2 = - 1 \\ \end{matrix} \right.\ \]

\[2x = - 5\]

\[x = - 2,5.\]

\[x \notin Z.\]

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

\[2x = 4\]

\[x = 2.\]

\[y = - x = - 2.\]

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

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

\[2x = - 4\]

\[x = - 2.\]

\[y = x = - 2.\]

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

\[Ответ:(2; - 2);( - 2; - 2).\]

\[2)\ x^{3} - 3x^{2} - xy - 8x -\]

\[- 2y + 27 = 0\]

\[\left( x^{3} + 2x^{2} \right) - xy - 2y - 5x^{2} -\]

\[- 10x + 2x + 4 - 4 + 27 = 0\]

\[x^{2}(x + 2) - y(x + 2) -\]

\[- 5x(x + 2) + 2 \cdot (x + 2) -\]

\[- 4 + 27 = 0\]

\[(x + 2)\left( x^{2} - y - 5x + 2 \right) = - 23\]

\[Делители:\ \pm 1;\ \pm 23.\]

\[\left\{ \begin{matrix} x + 2 = - 23\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - y - 5x + 2 = 1 \\ \end{matrix} \right.\ \text{\ \ }\]

\[\ \left\{ \begin{matrix} x = - 25\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = x^{2} - 5x + 1 \\ \end{matrix} \right.\ \text{\ \ \ \ }\left\{ \begin{matrix} x = - 25 \\ y = 751\ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x + 2 = 23\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - y - 5x + 2 = 1 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\left\{ \begin{matrix} x = 21\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = x^{2} - 5x + 3 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left\{ \begin{matrix} x = 21\ \ \\ y = 339 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x + 2 = - 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - y - 5x + 2 = 23 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\left\{ \begin{matrix} x = - 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = x^{2} - 2x - 21 \\ \end{matrix} \right.\ \text{\ \ \ }\left\{ \begin{matrix} x = - 3 \\ y = 3\ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x + 2 = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - y - 5x + 2 = - 23 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} x = - 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = x^{2} - 5x + 25 \\ \end{matrix} \right.\ \text{\ \ \ \ }\left\{ \begin{matrix} x = - 1 \\ y = 31\ \\ \end{matrix} \right.\ \]

\[Ответ:( - 1;31);( - 3;3);\]

\[(21;339);( - 25;751).\]

\[3)\ 3xy + 16x + 13y + 61 =\]

\[= 0\ \ \ | \cdot 3\]

\[9xy + 48x + 39y + 183 = 0\]

\[3y(3x + 13) + 16 \cdot (3x + 13) -\]

\[- 25 = 0\]

\[(3x + 13)(3y + 16) = 25\]

\[Делители:\ \pm 1;\ \pm 5;\ \pm 25.\]

\[\left\{ \begin{matrix} 3x + 13 = 1\ \ \ \\ 3y + 16 = 25 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 12 \\ 3y = 9\ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = - 4 \\ y = 3\ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 3x + 13 = - 1\ \ \\ 3y + 16 = - 25 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 14 \\ 3y = - 41 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = - \frac{14}{3} \\ y = - \frac{41}{3} \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 3x + 13 = 25 \\ 3y + 16 = 1\ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} 3x = 12\ \ \ \ \\ 3y = - 15 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

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

\[\left\{ \begin{matrix} 3x + 13 = - 25 \\ 3y + 16 = - 1\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 38 \\ 3y = - 17 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = - \frac{38}{3} \\ y = - \frac{17}{3} \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 3x + 13 = 5 \\ 3y + 16 = 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 8\ \ \\ 3y = - 11 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = - \frac{8}{3}\text{\ \ \ } \\ y = - \frac{11}{3} \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 3x + 13 = - 5 \\ 3y + 16 = - 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 18 \\ 3y = - 21 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = - 6 \\ y = - 7 \\ \end{matrix} \right.\ \]

\[Ответ:( - 4;3);(4; - 5);\]

\[( - 6; - 7).\]

\[4)\ 3xy - 10x + 16y - 45 =\]

\[= 0\ \ \ | \cdot 3\]

\[9xy - 30x + 48y - 135 = 0\]

\[3x(3y - 10) + 16 \cdot (3y - 10) +\]

\[+ 25 = 0\]

\[(3x + 16)(3y - 10) = - 25\]

\[Делители:\ \pm 1;\ \pm 5;\ \pm 25.\]

\[\left\{ \begin{matrix} 3x + 16 = 1\ \ \ \ \ \ \\ 3y - 10 = - 25 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 15 \\ 3y = - 15 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\text{\ \ }\left\{ \begin{matrix} x = - 5 \\ y = - 5 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 3x + 16 = - 25\ \\ 3y - 10 = 1\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 41 \\ 3y = 11\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = - \frac{41}{3} \\ y = \frac{11}{3}\text{\ \ \ } \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 3x + 16 = - 1\ \\ 3y - 10 = 25\ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 17 \\ 3y = 35\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = - \frac{17}{3} \\ y = \frac{35}{3}\text{\ \ \ } \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 3x + 16 = 25\ \\ 3y - 10 = - 1 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} 3x = 9 \\ 3y = 9 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\text{\ \ }\left\{ \begin{matrix} x = 3 \\ y = 3 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 3x + 16 = 5\ \ \ \ \\ 3y - 10 = - 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 11 \\ 3y = 5\ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\text{\ \ }\left\{ \begin{matrix} x = - \frac{11}{3} \\ y = \frac{5}{3}\text{\ \ \ \ \ \ } \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 3x + 16 = - 5 \\ 3y - 10 = 5\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 21 \\ 3y = 15\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = - 7 \\ y = 5\ \ \ \\ \end{matrix} \right.\ \]

\[Ответ:( - 5;\ - 5);(3;3);( - 7;5).\]