Решебник по алгебре 11 класс Никольский Параграф 9. Равносильность уравнений и неравенств системам Задание 38

\[\boxed{\mathbf{38.}}\]

\[\textbf{а)}\arcsin{(x^{2} - 80,5}) =\]

\[= \arcsin(x - 8,5)\]

\[\left\{ \begin{matrix} x^{2} - 80,5 = x - 8,5 \\ - 1 \leq x - 8,5 \leq 1\ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x^{2} - x - 72 = 0 \\ 7,5 \leq x \leq 9,5\ \ \ \ \ \\ \end{matrix} \right.\ \]

\[x^{2} - x - 72 = 0\]

\[x_{1} + x_{2} = 1;\ \ x_{1} \cdot x_{2} = - 72;\]

\[x_{1} = 9;\]

\[x_{2} = - 8\ (не\ подходит).\]

\[Ответ:x = 9.\]

\[\textbf{б)}\arccos\left( x^{2} - 9 \right) =\]

\[= \arccos(7x + 21)\]

\[\left\{ \begin{matrix} x^{2} - 9 = 7x + 21 \\ - 1 \leq 7x + 21 \leq 1 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x^{2} - 7x - 30 = 0 \\ - 22 \leq 7x \leq - 20 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x^{2} - 7x - 30 = 0 \\ - 3\frac{1}{7} \leq x \leq - 2\frac{6}{7} \\ \end{matrix} \right.\ \]

\[x^{2} - 7x - 30 = 0\]

\[x_{1} + x_{2} = 7;\ \ \ x_{1} \cdot x_{2} = - 30\]

\[x_{1} = - 3;\]

\[x_{2} = 10\ (не\ подходит).\]

\[Ответ:x = - 3.\]

\[\textbf{в)}\ arctg\ \left( x^{2} - 1 \right) =\]

\[= arctg\ (5x - 5)\]

\[x = R;\]

\[x^{2} - 1 = 5x - 5\]

\[x^{2} - 5x + 4 = 0\]

\[x_{1} + x_{2} = 5;\ \ x_{1} \cdot x_{2} = 4\]

\[x_{1} = 1;\ \ x_{2} = 4.\]

\[Ответ:x = 1;x = 4.\]

\[\textbf{г)}\ arcctg\ \left( x^{2} - 1 \right) =\]

\[= arcctg\ (6x - 6)\]

\[x = R;\]

\[x^{2} - 1 = 6x - 6\]

\[x^{2} - 6x + 5 = 0\]

\[D_{1} = 9 - 5 = 4\]

\[x_{1} = 3 + 2 = 5;\]

\[x_{2} = 3 - 2 = 1.\]

\[Ответ:x = 1;x = 5.\]