Решебник по алгебре 11 класс Никольский Параграф 8. Уравнения-следствия Задание 25

\[\boxed{\mathbf{25.}}\]

\[\textbf{а)}\ \frac{x - 1}{x^{2} + x - 2} = - 3\]

\[ОДЗ:\]

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

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

\[x_{1} \neq - 2;\ \ x_{2} \neq 1.\]

\[\frac{x - 1 + 3 \cdot \left( x^{2} + x - 2 \right)}{x^{2} + x - 2} = 0\]

\[x - 1 + 3x^{2} + 3x - 6 = 0\]

\[3x^{2} + 4x - 7 = 0\]

\[D_{1} = 4 + 21 = 25\]

\[x_{1} = \frac{- 2 + 5}{3} = 1\ (не\ корень);\]

\[x_{2} = \frac{- 2 - 5}{3} = - \frac{7}{3}.\]

\[Ответ:x = - \frac{7}{3}.\]

\[\textbf{б)}\ \frac{3x + 21}{x^{2} + 5x - 14} = 1\]

\[ОДЗ:\]

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

\[x_{1} + x_{2} = - 5;\ \ x_{1} \cdot x_{2} = - 14\]

\[x_{1} \neq - 7;\ \ x_{2} \neq 2.\]

\[\frac{3x + 21 - 1\left( x^{2} + 5x - 14 \right)}{x^{2} + 5x - 14} = 0\]

\[3x + 21 - x^{2} - 5x + 14 = 0\]

\[- x^{2} - 2x + 35 = 0\]

\[x^{2} + 2x - 35 = 0\]

\[x_{1} + x_{2} = - 2;\ \ x_{1} \cdot x_{2} = - 35\]

\[x_{1} = - 7\ (не\ корень);\ \ \ \]

\[x_{2} = 5.\]

\[Ответ:x = 5.\]

\[\textbf{в)}\ \frac{4x - 8}{x^{2} - x - 2} = 3\]

\[ОДЗ:\]

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

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

\[x_{1} \neq 2;x_{2} \neq - 1.\]

\[\frac{4x - 8 - 3 \cdot \left( x^{2} - x - 2 \right)}{x^{2} - x - 2} = 0\]

\[4x - 8 - 3x^{2} + 3x + 6 = 0\]

\[- 3x^{2} + 7x - 2 = 0\]

\[3x^{2} - 7x + 2 = 0\]

\[D = 49 - 24 = 25\]

\[x_{1} = \frac{7 + 5}{6} = 2\ (не\ корень);\]

\[x_{2} = \frac{7 - 5}{6} = \frac{2}{6} = \frac{1}{3}.\]

\[Ответ:x = \frac{1}{3}.\]

\[\textbf{г)}\ \frac{5x + 15}{x^{2} - 9} = - 1\]

\[ОДЗ:\]

\[x^{2} - 9 \neq 0\]

\[x^{2} \neq 9\]

\[x \neq \pm 3.\]

\[\frac{5x + 15 + 1\left( x^{2} - 9 \right)}{x^{2} - 9} = 0\]

\[5x + 15 + x^{2} - 9 = 0\]

\[x^{2} + 5x + 6 = 0\]

\[x_{1} + x_{2} = - 5;\ \ x_{1} \cdot x_{2} = 6\]

\[x_{1} = - 2;\]

\[x_{2} = - 3\ (не\ корень).\]

\[Ответ:x = - 2.\]