Решебник по алгебре 8 класс Мерзляк ФГОС Задание 782

 

\[\boxed{\mathbf{782\ (782).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]

\[1)\ x - 6\sqrt{x} + 8 = 0\]

\[\left\{ \begin{matrix} \sqrt{x} = t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ t^{2} - 6t + 8 = 0 \\ \end{matrix} \right.\ \text{\ \ }\]

\[t_{1} + t_{2} = 6,\ \ t_{1} \cdot t_{2} = 8,\ \]

\[\ t_{1} = 4,\ \ t_{2} = 2\]

\[Ответ:x = 16;x = 4.\]

\[2)\ x - 5\sqrt{x} - 50 = 0\]

\[\left\{ \begin{matrix} \sqrt{x} = t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ t^{2} - 5t - 50 = 0 \\ \end{matrix} \right.\ \]

\[t_{1} + t_{2} = 5,\ \ t_{1} \cdot t_{2} = - 50,\ \ \]

\[t_{1} = 10,\ \ t_{2} = - 5\]

\[\left\{ \begin{matrix} \sqrt{x} = t \\ t = 10 \\ t = - 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} \sqrt{x} = 10 \\ \sqrt{x} = - 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[x = 100\]

\[Ответ:x = 100.\]

\[3)\ 2x - 3\sqrt{x} + 1 = 0\]

\[\left\{ \begin{matrix} \sqrt{x} = t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2t^{2} - 3t + 1 = 0 \\ \end{matrix} \right.\ \text{\ \ }\]

\[D = 9 - 8 = 1\]

\[\text{\ \ }t_{1,2} = \frac{3 \pm 1}{4}\]

\[Ответ:x = 1;x = \frac{1}{4}.\]

\[\boxed{\mathbf{7}\mathbf{8}\mathbf{2}\mathbf{\text{.\ }}Еуроки\ - \ ДЗ\ без\ мороки}\]

\[1)\ \frac{4a - 16}{a^{2} - 16} = \frac{4 \cdot (a - 4)}{(a - 4)(a + 4)} =\]

\[= \frac{4}{a + 4}\]

\[2)\ \frac{12b^{3} - 8b^{2}}{2 - 3b} = \frac{4b^{2}(3b - 2)}{(2 - 3b)} =\]

\[= - 4b²\]

\[3)\ \frac{c^{2} + 10c + 25}{5c + 25} = \frac{(c + 5)^{2}}{5 \cdot (c + 5)} =\]

\[= \frac{c + 5}{5}\]

\[4)\ \frac{4 - m^{2}}{m^{2} - 4m + 4} =\]

\[= \frac{(2 - m)(2 + m)}{(m - 2)^{2}} = \frac{- 2 - m}{m - 2} =\]

\[= \frac{2 + m}{2 - m}\]

\[5)\ \frac{n^{3} - n^{5}}{n^{3} - n} = \frac{n^{3}\left( 1 - n^{2} \right)}{n\left( n^{2} - 1 \right)} = - n²\]

\[6)\ \frac{2 - 2x^{2}}{4x^{2} - 8x + 4} = \frac{2 \cdot \left( 1 - x^{2} \right)}{(2x - 2)^{2}} =\]

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

\[= \frac{1 + x}{2 - 2x}\]