ГДЗ по алгебре 9 класс Макарычев Задание 354

\[\boxed{\text{354\ (354).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[\textbf{а)}\ x^{3} - x^{2} - 4 \cdot (x - 1)^{2} = 0\]

\[x^{2}(x - 1) - 4 \cdot\]

\[\cdot (x - 1)(x - 1) = 0\]

\[(x - 1)\left( x^{2} - 4 \cdot (x - 1) \right) = 0\]

\[(x - 1)\left( x^{2} - 4x + 4 \right) = 0\]

\[(x - 1)(x - 2)^{2} = 0\]

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

\[Ответ:x = 1;x = 2.\]

\[\textbf{б)}\ 2y^{3} + 2y^{2} - (y + 1)^{2} = 0\]

\[2y^{2}(y + 1) - (y + 1)(y + 1) = 0\]

\[(y + 1)\left( 2y^{2} - y - 1 \right) = 0\]

\[1)\ y + 1 = 0\ \ \ \]

\[y_{1} = - 1.\]

\[2)\ 2y^{2} - y - 1 = 0\]

\[D = 1 + 4 \cdot 2 = 9\]

\[y_{2,3} = \frac{1 \pm 3}{4} = 1;\ - 0,5.\ \]

\[Ответ:\ y = 0,5;\ \ y = \pm 1.\]

\[\textbf{в)}\ 5x^{3} - 19x^{2} - 38x + 40 = 0\]

\[5x^{3} + 40 - \left( 19x^{2} + 38x \right) = 0\]

\[5 \cdot \left( x^{3} + 8 \right) - 19x(x + 2) = 0\]

\[5 \cdot (x + 2)\left( x^{2} - 2x + 4 \right) -\]

\[- 19x(x + 2) = 0\]

\[(x + 2) \cdot\]

\[\cdot \left( 5 \cdot \left( x^{2} - 2x + 4 \right) - 19x \right) = 0\]

\[(x + 2)\left( 5x^{2} - 29x + 20 \right) = 0\]

\[1)\ x + 2 = 0\ \ \ \ \ \]

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

\[2)\ 5x^{2} - 29x + 20 = 0\]

\[D = 29^{2} - 4 \cdot 5 \cdot 20 = 441\]

\[x_{2,3} = \frac{29 \pm 21}{10} = 5;0,8.\]

\[Ответ:x = - 2;x = 5;x = 0,8.\]

\[\textbf{г)}\ 6x^{3} - 31x^{2} - 31x + 6 = 0\]

\[6 \cdot \left( x^{3} + 1 \right) - 31x(x + 1) = 0\]

\[6 \cdot (x + 1)\left( x^{2} - x + 1 \right) -\]

\[- 31x(x + 1) = 0\]

\[(x + 1) \cdot\]

\[\cdot \left( 6 \cdot \left( x^{2} - x + 1 \right) - 31x \right) = 0\]

\[(x + 1)\left( 6x^{2} - 6x + 6 - 31x \right) = 0\]

\[(x + 1)\left( 6x^{2} - 37x + 6 \right) = 0\]

\[1)\ x + 1 = 0\ \]

\[x_{1} = - 1.\]

\[2)\ 6x^{2} - 37x + 6 = 0\]

\[D = 37^{2} - 4 \cdot 6 \cdot 6 =\]

\[= 1369 - 144 = 1225\]

\[x_{2,3} = \frac{37 \pm 35}{12} = 6;\frac{1}{6}.\]

\[Ответ:x = - 1;\ \ x = \frac{1}{6};\ \ x = 6.\]