\[\boxed{\mathbf{Упражнения}\mathbf{\ }\mathbf{стр}\mathbf{.\ 196}}\]
\[\boxed{\mathbf{1.}}\]
\[\frac{3x^{2} - 9x}{2} - \frac{12}{x^{2} - 3x} = 3\]
\[\frac{3 \cdot \left( x^{2} - 3x \right)}{2} - \frac{12}{x^{2} - 3x} = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \]
\[ОДЗ:\ \ x^{2} - 3x \neq 0\]
\[x(x - 3) \neq 0\]
\[x \neq 0;\ \ \ \ \ x \neq 3\]
\[Пусть\ \ y = x^{2} - 3x:\]
\[\frac{3y}{2} - \frac{12}{y} = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ | \cdot 2y\]
\[3y^{2} - 24 = 6y\]
\[3y^{2} - 6y - 24 = 0\]
\[y^{2} - 2y - 8 = 0\]
\[y_{1} + y_{2} = 2;\ \ \ \ \ \ y_{1} \cdot y_{2} = - 8\]
\[y_{1} = 4;\ \ \ \ \ \ \ \ y_{2} = - 2\]
\[1)\ x^{2} - 3x = 4\]
\[x^{2} - 3x - 4 = 0\]
\[x_{1} + x_{2} = 3;\ \ \ \ x_{1} \cdot x_{2} = - 4\]
\[x_{1} = 4;\ \ \ \ x_{2} = - 1.\]
\[2)\ x^{2} - 3x = - 2\]
\[x^{2} - 3x + 2 = 0\]
\[x_{1} + x_{2} = 3;\ \ \ \ \ x_{1} \cdot x_{2} = 2\]
\[x_{1} = 1;\ \ \ \ \ x_{2} = 2.\]
\[Ответ:x = 1;\ x = - 1;x = 2;\]
\[x = 4.\]
\[\boxed{\mathbf{2.}}\]
\[ОДЗ:\ \ \ \ \ x \neq - 1;\ \ \ x \neq - 2;\ \ \ \ \]
\[x \neq 1;\ \ x \neq - 4.\]
\[(x + 1)(x + 2) =\]
\[= x^{2} + x + 2x + 2 =\]
\[= x^{2} + 3x + 2\]
\[(x - 1)(x + 4) =\]
\[= x^{2} - x + 4x - 4 =\]
\[= x^{2} + 3x - 4\]
\[\frac{6}{x^{2} + 3x + 2} + \frac{8}{x^{2} + 3x - 4} = 1\]
\[Пусть\ y = x^{2} + 3x:\]
\[\frac{6}{y + 2} + \frac{8}{y - 4} = 1\ \ \ \ \ \ \ \ \ \ \ y \neq - 2;\ \ \ \ \]
\[y \neq 4\]
\[6 \cdot (y - 4) + 8 \cdot (y + 2) =\]
\[= (y + 2)(y - 4)\]
\[6y - 24 + 8y + 16 =\]
\[= y^{2} + 2y - 4y - 8\]
\[14y - 8 - y^{2} + 2y + 8 = 0\]
\[- y^{2} + 16y = 0\]
\[y^{2} - 16y = 0\]
\[y(y - 16) = 0\]
\[y = 0;\ \ \ \ \ y = 16.\]
\[1)\ x^{2} + 3x = 0\]
\[x(x + 3) = 0\]
\[x = 0;\ \ \ \ x = - 3.\]
\[2)\ x^{2} + 3x = 16\]
\[{x^{2} + 3x - 16 = 0 }{D = 9 + 64 = 73}\]
\[x_{1,2} = \frac{- 3 \pm \sqrt{73}}{2}\]
\[Ответ:x = 0;\ x = - 3;\ \]
\[x = \ \frac{- 3 \pm \sqrt{73}}{2}.\]
\[\boxed{\mathbf{3.}}\]
\[x(x + 3)(x + 5)(x + 8) = 100\]
\[\left( x^{2} + 8x \right)\left( (x + 3)(x + 5) \right) =\]
\[= 100\]
\[\left( x^{2} + 8x \right)\left( x^{2} + 3x + 5x + 15 \right) =\]
\[= 100\]
\[\left( x^{2} + 8x \right)\left( x^{2} + 8x + 15 \right) = 100\]
\[Пусть\ y = x^{2} + 8x:\]
\[y(y + 15) = 100\]
\[y^{2} + 15y - 100 = 0\]
\[y_{1} + y_{2} = - 15;\ \ \ \ \ y_{1} \cdot y_{2} = - 100\]
\[y_{1} = - 20;\ \ \ \ \ y_{2} = 5.\]
\[1)\ x^{2} + 8x + 20 = 0\]
\[D = 64 - 80 < 0\]
\[нет\ корней.\]
\[2)\ x^{2} + 8x - 5 = 0\]
\[D = 64 + 20 = 84\]
\[x_{1,2} = \frac{- 8 \pm \sqrt{84}}{2} = - 4 \pm \sqrt{21}\]
\[Ответ:x = - 4 \pm \sqrt{21}.\]
\[\boxed{\mathbf{4.}}\]
\[(x + 2)(x + 3)(x + 8)(x + 12) =\]
\[= 4x^{2}\]
\[\frac{x^{2} + 14x + 24}{x} \cdot \frac{x^{2} + 11x + 24}{x} =\]
\[= 4\]
\[\left( x + 14 + \frac{24}{x} \right)\left( x + 11 + \frac{24}{x} \right) = 4\]
\[Пусть\ \ \ \ y = x + \frac{24}{x}:\]
\[(y + 14)(y + 11) = 4\]
\[y^{2} + 14y + 11y + 154 - 4 = 0\]
\[y^{2} + 25y + 150 = 0\]
\[y_{1} + y_{2} = - 25;\ \ \ \ \ \ y_{1} \cdot y_{2} = 150\]
\[y_{1} = - 15;\ \ \ \ \ \ y_{2} = - 10.\]
\[1)\ x + \frac{24}{x} = - 15\]
\[x^{2} + 24 = - 15x\]
\[x^{2} + 15x + 24 = 0\]
\[D = 225 - 96 = 129\]
\[x_{1,2} = \frac{- 15 \pm \sqrt{129}\ }{2}\]
\[2)\ x + \frac{24}{x} = - 10\]
\[x^{2} + 24 = - 10x\]
\[x^{2} + 10x + 24 = 0\]
\[x_{1} + x_{2} = - 10;\ \ \ \ \ x_{1} \cdot x_{2} = 24\]
\[x_{1} = - 6;\ \ \ \ \ \ x_{2} = - 4\]
\[Ответ:x = - 6;\ \ \ x = - 4;\ \ \ \ \]
\[x = \frac{- 15 \pm \sqrt{129}}{2}.\]
\[\boxed{\mathbf{5.}}\]
\[7 \cdot \left( x + \frac{1}{x} \right) - 2 \cdot \left( x^{2} + \frac{1}{x^{2}} \right) = 9;\ \ \ \ \ \ \ \ \ \]
\[x \neq 0\]
\[Пусть\ y = x + \frac{1}{x}:\]
\[y^{2} = \left( x + \frac{1}{x} \right)^{2} = x^{2} + 2 + \frac{1}{x^{2}}\]
\[x^{2} + \frac{1}{x^{2}} = y^{2} - 2\]
\[Подставим:\]
\[7y - 2 \cdot \left( y^{2} - 2 \right) = 9\]
\[7y - 2y^{2} + 4 - 9 = 0\]
\[- 2y^{2} + 7y - 5 = 0\]
\[2y^{2} - 7y + 5 = 0\]
\[D = 49 - 40 = 9\]
\[y_{1} = \frac{7 + 3}{4} = \frac{10}{4} = 2,5;\ \ \ \ \ \]
\[y_{2} = \frac{7 - 3}{4} = 1\]
\[1)\ x + \frac{1}{x} = 2,5\ \ \ \ \ \ \ \ | \cdot 2x\]
\[2x^{2} + 2 = 5x\]
\[2x^{2} - 5x + 2 = 0\]
\[D = 25 - 16 = 9\]
\[x_{1} = \frac{5 + 3}{4} = 2;\ \ \ \ \ \ \]
\[x_{2} = \frac{5 - 3}{4} = \frac{2}{4} = 0,5\]
\[2)\ x + \frac{1}{x} = 1\ \ \ \ \ \ \ \ | \cdot x\]
\[x^{2} + 1 = x\]
\[x^{2} - x + 1 = 0\]
\[D = 1 - 4 = - 3 < 0\]
\[корней\ нет.\]
\[Ответ:x = 2;\ \ \ x = 0,5.\]
\[\boxed{\mathbf{6.}}\]
\[2 \cdot \left( x^{2} + x + 1 \right)^{2} - 7 \cdot (x - 1)^{2} =\]
\[= 13 \cdot \left( x^{3} - 1 \right)\]
\[2 \cdot \left( x^{2} + x + 1 \right)^{2} - 7 \cdot (x - 1)^{2} =\]
\[ОДЗ:\ \ \ \ x \neq 1.\]
\[x^{3} - 1 = (x - 1)\left( x^{2} + x + 1 \right)\]
\[\frac{2 \cdot \left( x^{2} + x + 1 \right)}{x - 1} - \frac{7 \cdot (x - 1)}{x^{2} + x + 1} =\]
\[= 13\]
\[Пусть\ \ y = \frac{x^{2} + x + 1}{x - 1}:\]
\[2y - \frac{7}{y} = 13\]
\[2y^{2} - 7 = 13y\]
\[2y^{2} - 13y - 7 = 0\]
\[D = 169 + 56 = 225\]
\[y_{1} = \frac{13 + 15}{4} = 7;\ \ \ \ \ \]
\[y_{2} = \frac{13 - 15}{4} = - \frac{2}{4} = - 0,5\]
\[1)\frac{x^{2} + x + 1}{x - 1} = 7\]
\[x^{2} + x + 1 = 7 \cdot (x - 1)\]
\[x^{2} + x + 1 - 7x + 7 = 0\]
\[x^{2} - 6x + 8 = 0\]
\[x_{1} + x_{2} = 6;\ \ \ x_{1} \cdot x_{2} = 8\]
\[x_{1} = 2;\ \ \ \ \ x_{2} = 4\ \]
\[2)\ \frac{x^{2} + x + 1}{x - 1} = - 0,5\]
\[x^{2} + x + 1 = - 0,5 \cdot (x - 1)\]
\[x^{2} + x + 1 = - 0,5x + 0,5\]
\[x^{2} + 1,5x + 0,5 = 0\ \ \ \ | \cdot 2\]
\[2x^{2} + 3x + 1 = 0\]
\[D = 9 - 8 = 1\]
\[x_{1} = \frac{- 3 + 1}{4} = - 0,5;\ \ \ \]
\[x_{2} = \frac{- 3 - 1}{4} = - 1\]
\[Ответ:x = 2;\ \ x = 4;\ \ x = - 1;\ \ \ \]
\[x = - 0,5.\]
\[\boxed{\mathbf{7.}}\]
\[(x - 6)^{4} + (x - 4)^{4} = 82\]
\[Пусть\ y = x - 5:\]
\[(y - 1)^{4} + (y + 1)^{4} = 82\]
\[2y^{4} + 12y^{2} + 2 = 82\]
\[2y^{4} + 12y^{2} - 80 = 0\ \ \ \ \ \ |\ :2\]
\[y^{4} + 6y^{2} - 40 = 0\]
\[Пусть\ y^{2} = z:\]
\[z^{2} + 6z - 40 = 0\]
\[z_{1} + z_{2} = - 6;\ \ \ \ z_{1} \cdot z_{2} = - 40\]
\[z_{1} = - 10;\ \ \ \ z_{2} = 4.\]
\[1)\ y^{2} = - 10\]
\[нет\ корней.\]
\[2)\ y^{2} = 4\]
\[y = \pm 2.\]
\[x_{1} = 2 + 5 = 7\]
\[x_{2} = - 2 + 5 = 3\]
\[Ответ:x = 7;\ \ x = 3.\]