Решебник по алгебре 8 класс Мерзляк ФГОС Задание 812

\[\boxed{\mathbf{812\ (812).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]

\[Пусть\ \text{\ x}\frac{км}{ч} - скорость\ \]

\[течения\ реки,\ тога\ скорость\ \]

\[лодки\ по\ течению\ реки\ \]

\[(x + 12)\ \frac{км}{ч},\ а\ против\ \]

\[течения - \frac{км}{ч}.\]

\[Плот\ прошел\ расстояние\ \]

\[за\ \frac{15}{x}\ \ ч,\ а\ лодка - \frac{15}{x + 12}\ ч.\ \]

\[По\ условию\ задачи\ известно,\ \]

\[что\ лодка\ отплыла\ через\ \]

\[4\ часа.\]

\[Составляем\ уравнение:\]

\[\frac{15}{x} - \frac{15}{x + 12} - 4 = 0;\ \ \ \ \ \ x \neq 0;\ \ \ \ \]

\[x \neq - 12\]

\[- 4x^{2} - 48x + 180 = 0\ \ \ \ |\ :( - 4)\]

\[x^{2} + 12x - 45 = 0\]

\[x_{1} + x_{2} = - 12,\ \ \text{\ x}_{1}x_{2} = - 45\]

\[x_{1} = - 15 \Longrightarrow не\ удовлетворяет\ \]

\[условию.\]

\[x_{2} = 3\ \left( \frac{км}{ч} \right) - скорость\ \]

\[течения\ реки.\]

\[Ответ:3\frac{км}{ч}.\]

\[\boxed{\mathbf{81}\mathbf{2}\mathbf{\text{.\ }}Еуроки\ - \ ДЗ\ без\ мороки}\]

\[1)\ \frac{x^{2} + 3x - 4}{x + 1} = 0;\ \ \ x \neq - 1\]

\[x^{2} + 3x - 4 = 0\]

\[x_{1} + x_{2} = - 3,\ \ x_{1} = - 4\]

\[x_{1} \cdot x_{2} = - 4,\ \ x_{2} = 1\]

\[Ответ:\ x = - 4;x = 1.\]

\[2)\ \frac{x^{2} - 6x - 7}{x - 7} = 0;\ \ \ \ \ \ \ \ \ \ x \neq 7\]

\[x^{2} - 6x - 7 = 0\]

\[x_{1} + x_{2} = 6,\ \ \]

\[x_{1} = 7\ (не\ подходит)\]

\[x_{1} \cdot x_{2} = - 7,\ \ x_{2} = - 1\]

\[Ответ:\ x = - 1.\]

\[3)\ \frac{3x^{2} - x - 2}{1 - x} = 0;\ \ \ \ \ \ x \neq 1\]

\[3x^{2} - x - 2 = 0\]

\[x_{1} + x_{2} = \frac{1}{3},\ \ x_{1} = - \frac{2}{3}\]

\[x_{1} \cdot x_{2} = - \frac{2}{3},\ \ \]

\[x_{2} = 1\ (не\ подходит)\]

\[Ответ:\ x = - \frac{2}{3}.\]

\[4)\ \frac{x^{2} - 8x}{x + 10} = \frac{20}{x + 10}\]

\[\frac{x^{2} - 8x - 20}{x + 10} = 0;\ \ \ \ \ \ x \neq - 10\]

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

\[x_{1} + x_{2} = 8,\ \ x_{1} = 10\]

\[x_{1} \cdot x_{2} = - 20,\ \ x_{2} = - 2\]

\[Ответ:\ x = - 2;x = 10.\]

\[5)\ \frac{x^{2} - 14}{x + 2} = \frac{5x}{x + 2}\]

\[\frac{x^{2} - 14 - 5x}{x + 2} = 0;\ \ \ \ \ \ x \neq - 2\]

\[x^{2} - 5x - 14 = 0\]

\[x_{1} + x_{2} = 5,\ \ x_{1} = 7\]

\[x_{1} \cdot x_{2} = - 14,\ \ \]

\[x_{2} = - 2\ (не\ подходит)\]

\[Ответ:x = 7.\]

\[6)\ \frac{x^{2} + 10x}{x - 8} = \frac{12x + 48}{x - 8}\]

\[\frac{x^{2} + 10x - 12x - 48}{x - 8} = 0;\ \ \ \]

\[x \neq 8\]

\[x^{2} - 2x - 48 = 0\]

\[x_{1} + x_{2} = 2,\ \ \]

\[x_{1} = 8\ (не\ подходит)\]

\[x_{1} \cdot x_{2} = - 48,\ \ x_{2} = - 6\]

\[Ответ:\ x = - 6.\]

\[7)\ \frac{x^{2} + 4x}{x - 5} - \frac{9x + 50}{x - 5} = 0\]

\[\frac{x^{2} + 4x - 9x - 50}{x - 5} = 0;\ \ \ \ \ x \neq 5\]

\[x^{2} - 5x - 50 = 0\]

\[x_{1} + x_{2} = 5,\ \ x_{1} = 10\]

\[x_{1} \cdot x_{2} = - 50,\ \ x_{2} = - 5\]

\[Ответ:\ x = - 5;x = 10.\]

\[8)\ \frac{x^{2} - 6x}{x - 3} + \frac{15 - 2x}{x - 3} = 0\]

\[\frac{x^{2} - 6x + 15 - 2x}{x - 3} = 0;\ \ \ \ x \neq 3\]

\[x^{2} - 8x + 15 = 0\]

\[x_{1} + x_{2} = 8,\ \ x_{1} = 5\]

\[x_{1} \cdot x_{2} = 15,\ \ \]

\[x_{2} = 3\ (не\ подходит)\]

\[Ответ:x = 5.\]