Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 621

\[\boxed{\mathbf{621}.}\]

\[\left\{ \begin{matrix} \sqrt{7 \cdot (x - y)} - \sqrt{x + y} = \frac{8}{3}\sqrt{\frac{y}{x}} \\ \sqrt{7 \cdot (x - y)} + \sqrt{x + y} = 9\sqrt{\frac{x}{y}} \\ \end{matrix} \right.\ \ \ \ (*)\]

\[7(x - y) - x - y = 24\]

\[6x - 8y = 24\ \ \ |\ :2\]

\[3x - 4y = 12\]

\[4y = 3x - 12\]

\[y = \frac{3}{4}x - 3\]

\[\left\{ \begin{matrix} y = \frac{3}{4}x - 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \sqrt{7 \cdot (x - y)} - \sqrt{x + y} = \frac{8}{3}\sqrt{\frac{y}{x}} \\ \end{matrix} \right.\ \]

\[\sqrt{7 \cdot \left( x - \frac{3}{4}x + 3 \right)} -\]

\[- \sqrt{x + \frac{3}{4}x + 3} = \frac{8}{3}\sqrt{\frac{\frac{3}{4}x - 3}{x}}\]

\[\frac{1}{2}\sqrt{7x + 84} - \frac{1}{2}\sqrt{7x - 12} =\]

\[= \frac{8}{3}\sqrt{\frac{x - 4}{x}} \cdot \frac{\sqrt{3}}{2}\ \ \ | \cdot 2\]

\[\left( \sqrt{7x + 84} - \sqrt{7x - 12} \right)^{2} =\]

\[= \left( \frac{8\sqrt{3}}{3}\sqrt{\frac{x - 4}{x}} \right)^{2}\]

\[7x + 84 -\]

\[- 2\sqrt{(7x + 84)(7x - 12)} +\]

\[+ 7x - 12 = \frac{64}{3}\left( \frac{x - 4}{x} \right)\]

\[14x + 72 - \frac{64(x - 4)}{3x} =\]

\[= 2\sqrt{49x^{2} - 84x + 588x - 1008}\ \]

\[\ |\ :2\]

\[\left( 21x^{2} + 76x + 128 \right)^{2} =\]

\[= \left( 3x\sqrt{49x^{2} + 504x - 1008} \right)^{2}\ \]

\[441x^{4} + 3192x^{3} + 5376x^{2} +\]

\[+ 5776x^{2} + 19\ 456x + 16384 =\]

\[= 9x^{2}\left( 49x^{2} + 504x - 1008 \right)\]

\[21x^{3} - 316x^{2} - 304x -\]

\[- 256 = 0\]

\[P(16) = 0.\]

\[(x - 16)\left( 21x^{2} + 20x + 16 \right) = 0\]

\[21x^{2} + 20x + 16 = 0\]

\[D_{1} = 100 - 336 < 0\]

\[нет\ корней.\]

\[y = - 3 + \frac{3}{4} \cdot 16 = - 3 + 12 = 9.\]

\[Ответ:(16;9).\]