ГДЗ по алгебре 8 класс Макарычев Задание 656

\[\boxed{\text{656\ (656).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[\textbf{а)}\ x^{2} - 2x - 5 = 0\]

\[D = 4 + 20 = 24\]

\[x_{1,2} = \frac{2 \pm \sqrt{24}}{2} = \frac{2 \pm 2\sqrt{6}}{2}\]

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

\[Проверка:\]

\[\left( 1 + \sqrt{6} \right)^{2} - 2 \cdot \left( 1 + \sqrt{6} \right) - 5 =\]

\[\left( 1 - \sqrt{6} \right)^{2} - 2 \cdot \left( 1 - \sqrt{6} \right) - 5 =\]

\[\textbf{б)}\ x^{2} + 4x + 1 = 0\]

\[D = 16 + 4 = 20\]

\[x_{1,2} = \frac{- 4 \pm \sqrt{20}}{2} = \frac{- 4 \pm 2\sqrt{5}}{2}\]

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

\[Проверка:\]

\[\left( - 2 + \sqrt{5} \right)^{2} + 4 \cdot \left( - 2 + \sqrt{5} \right) + 1 =\]

\[\left( - 2 - \sqrt{5} \right)^{2} + 4 \cdot \left( - 2 - \sqrt{5} \right) + 1 =\]

\[\textbf{в)}\ 3y^{2} - 4y - 2 = 0\]

\[D = 16 + 24 = 40\]

\[y_{1,2} = \frac{4 \pm \sqrt{40}}{6} = \frac{2 \pm \sqrt{10}}{3}\]

\[Проверка:\]

\[= \frac{4 + 4\sqrt{10}}{3} - \frac{8 + 4\sqrt{10}}{3} - 2 =\]

\[\left( \frac{2 - \sqrt{10}}{3} \right)^{2} \cdot 3 - \left( \frac{2 - \sqrt{10}}{3} \right) \cdot 4 - 2 =\]

\[= \frac{4 - 4\sqrt{10} + 10}{9} \cdot 3 - \frac{8 - 4\sqrt{10}}{3} - 2 =\]

\[= \frac{14 - 4\sqrt{10}}{3} - \frac{8 - 4\sqrt{10}}{3} - \frac{6}{3} =\]

\[\textbf{г)}\ 5y^{2} - 7y + 1 = 0\]

\[D = 49 - 20 = 29\]

\[y_{1,2} = \frac{7 \pm \sqrt{29}}{10}\]

\[Проверка:\]

\[\textbf{д)}\ 2y^{2} + 11y + 10 = 0\]

\[D = 121 - 80 = 41\]

\[y_{1,2} = \frac{- 11 \pm \sqrt{41}}{4}\]

\[Проверка:\]

\[\textbf{е)}\ 4x^{2} - 9x - 2 = 0\]

\[D = 81 + 32 = 113\]

\[x_{1,2} = \frac{9 \pm \sqrt{113}}{8}\]

\[Проверка:\]

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