Решебник по алгебре 8 класс Мерзляк ФГОС Задание 1144

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\[1)\ \frac{x^{2} - 7x}{x + 1} = \frac{8}{x + 1};\ \ \ \ \ \ \ \ x \neq - 1\]

\[x^{2} - 7x - 8 = 0\]

\[x_{1} + x_{2} = 7,\ \ x_{1}x_{2} = - 8,\ \ \]

\[x_{1} = 8,\ \ x_{2} = - 1\]

\[Ответ:x = 8.\]

\[2)\ \frac{3x^{2} + 4x}{x^{2} - 9} = \frac{3 - 4x}{x^{2} - 9};\ \ \ \ \ x^{2} \neq 9\]

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

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

\[x_{1} + x_{2} = - \frac{8}{3};\ \ x_{1}x_{2} = - \frac{3}{3}\]

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

\[Ответ:x = \frac{1}{3}.\]

\[3)\ \frac{4 - x}{4x - 3} = \frac{2x - 2}{7 - x};\ \ \ \ \ \ \ \ \]

\[x \neq 7;\ \ \ \ \ \ x \neq \frac{3}{4}\]

\[(4 - x)(7 - x) - (2x - 2)(4x - 3) = 0\]

\[28 - 7x - 4x + x^{2} - \left( 8x^{2} - 8x - 6x + 6 \right) = 0\]

\[28 - 11x + x^{2} - 8x^{2} + 8x + 6x - 6 = 0\]

\[- 7x^{2} + 3x + 22 = 0\]

\[D = 9 + 616 = 625\]

\[x_{1} = \frac{- 3 + 25}{- 14} = - \frac{11}{7} = - 1\frac{4}{7}\]

\[x_{2} = \frac{- 3 - 25}{- 14} = 2\]

\[Ответ:x = 2;\ x = - 1\frac{4}{7}.\]

\[4)\ \frac{1}{x + 1} - \frac{1}{x - 6} = \frac{7}{12};\ \ \ \ \ \ \ \ \]

\[x \neq - 1;\ \ \ \ x \neq 6\]

\[12 \cdot (x - 6) - 12 \cdot (x + 1) - 7 \cdot (x + 1)(x - 6) = 0\]

\[12x - 72 - 12x - 12 - 7 \cdot \left( x^{2} + x - 6x - 6 \right) = 0\]

\[- 84 - 7x^{2} - 7x + 42x + 42 = 0\]

\[- 7x^{2} + 35x - 42 = 0\ \ \ \ \ \ \ |\ :( - 7)\]

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

\[x_{1} + x_{2} = 5,\ \ x_{1}x_{2} = 6,\ \ \]

\[x_{1} = 3,\ \ x_{2} = 2\]

\[Ответ:x = 3;x = 2.\]

\[5)\ \frac{63}{x^{2} + 3x} - \frac{2}{x^{2} - 3x} = \frac{7}{x};\ \ \ \ \ \ \]

\[x \neq 0;\ \ x \neq 3;\ \ \ \ x \neq - 3\]

\[63 \cdot (x - 3) - 2 \cdot (x + 3) - 7 \cdot (x + 3)(x - 3) = 0\]

\[63x - 189 - 2x - 6 - 7x^{2} + 63 = 0\]

\[- 7x^{2} + 61x - 132 = 0\]

\[D = 3721 - 3696 = 25\]

\[x_{1} = \frac{- 61 + 5}{- 14} = 4\]

\[x_{2} = \frac{- 61 - 5}{- 14} = \frac{33}{7} = 4\frac{5}{7}\]

\[Ответ:x = 4;x = 4\frac{5}{7}.\]

\[6)\ \frac{2x}{x - 2} + \frac{3}{x + 4} = \frac{4x - 2}{(x + 4)(x - 2)};\ \ \ \ \ \]

\[\ x \neq 2;\ \ \ \ x \neq - 4\]

\[2x(x + 4) + 3 \cdot (x - 2) - 4x + 2 = 0\]

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

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

\[D = 49 + 32 = 81\]

\[x_{1} = \frac{- 7 + 9}{4} = \frac{1}{2} = 0,5\]

\[x_{2} = \frac{- 7 - 9}{4} = - 4\]

\[Ответ:x = 0,5.\]