ГДЗ по алгебре 9 класс Макарычев Задание 42

\[\boxed{\text{42\ (42).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[\textbf{а)}\ y = \frac{x - \sqrt{x + 6}}{x + 5}\]

\[\left\{ \begin{matrix} x + 6 \geq 0 \\ x + 5 \neq 0 \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x \geq - 6 \\ x \neq - 5 \\ \end{matrix} \right.\ \]

\[Область\ определения:\ \ \ \]

\[x \in \lbrack - 6;\ - 5) \cup ( - 5;\ + \infty);\]

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

\[x = \sqrt{x + 6},\ \ x > 0\]

\[x^{2} = x + 6\]

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

\[D = 1 + 24 = 25 = 5^{2}\]

\[x_{1.2} = \frac{1 \pm 5}{2} = 3;\ - 2.\]

\[Так\ как\ x > 0;\ \ то\ x =\]

\[= - 2 \Longrightarrow не\ подходит.\]

\[Ноль\ функции:\ \ \ x = 3.\]

\[\textbf{б)}\ y = \frac{4x^{2} + 25x}{2x - \sqrt{10 - 6x}}\]

\[\left\{ \begin{matrix} 10 - 6x \geq 0\ \ \ \ \ \ \ \ \ \ \ \ \\ 2x - \sqrt{10 - 6x} \neq 0 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} 6x \leq 10\ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2x \neq \sqrt{10 - 6x} \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x \leq \frac{10}{6}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 4x^{2} \neq 10 - 6x \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x \leq \frac{5}{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 2x^{2} + 3x - 5 = 0 \\ \end{matrix} \right.\ \]

\[2x^{2} + 3x - 5 = 0\]

\[D = 9 + 4 \cdot 2 \cdot 5 = 49 = 7^{2}\]

\[x_{1,2} = \frac{- 3 \pm 7}{4} = 1;\ - 2,5.\]

\[Но\ при\ x = - 2,5:\ \ \ \ \]

\[2x = - 5,а\ \]

\[\sqrt{10 - 6x} = 5.\]

\[Область\ определения:\ \ \ \]

\[x \in ( - \infty;1) \cup \left( 1;\frac{5}{3} \right\rbrack.\]

\[Нули\ функции:\]

\[4x^{2} + 25x = 0\]

\[x(4x + 25) = 0\]

\[x_{1} = 0\ \ \ и\ \ \ \ 4x_{2} = - 25,\ \ \]

\[\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }x_{2} = - 6\frac{1}{4}\]

\[y = 0\ \ \ при\ \ x_{1} = 0\ \ \ и\ \ x_{2} =\]

\[= - 6\frac{1}{4}\text{\ .\ \ }\]