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МатематикаРешебник по алгебре 11 класс Никольский Параграф 9. Равносильность уравнений и неравенств системам Задание 13
Задание 13
\[\boxed{\mathbf{13.}}\]
\[\textbf{а)}\lg\left( x^{2} - 17 \right) = \lg(11x - 45)\]
\[\left\{ \begin{matrix} x^{2} - 17 = 11x - 45 \\ 11x - 45 > 0\ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x^{2} - 11x + 28 = 0 \\ 11x > 45\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x^{2} - 11x + 28 = 0 \\ x > \frac{45}{11} > 4\frac{1}{11}\text{\ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]
\[x^{2} - 11x + 28 = 0\]
\[x_{1} + x_{2} = 11;\ \ x_{1} \cdot x_{2} = 28\]
\[x_{1} = 4 < 4\frac{1}{11} - не\ корень;\]
\[x_{2} = 7 > 4\frac{1}{11} - корень.\]
\[Ответ:x = 7.\]
\[\textbf{б)}\lg\left( x^{2} - 7x + 14 \right) =\]
\[= \lg(3x - 16)\]
\[\left\{ \begin{matrix} x^{2} - 7x + 14 = 3x - 16 \\ 3x - 16 > 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x^{2} - 10x + 30 = 0 \\ 3x > 16\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x^{2} - 10x + 30 = 0 \\ x > \frac{16}{3} > 5\frac{1}{3}\text{\ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]
\[x^{2} - 10x + 30 = 0\]
\[D_{1} = 25 - 30 = - 5 < 0\]
\[нет\ корней.\]
\[Ответ:нет\ решений.\]
\[\textbf{в)}\lg\left( 25 - x^{2} \right) = \lg(2x - 10)\]
\[\left\{ \begin{matrix} 25 - x^{2} = 2x - 10 \\ 2x - 10 > 0\ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x^{2} + 2x - 35 = 0 \\ x > 5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[x^{2} + 2x - 35 = 0\]
\[D_{1} = 1 + 35 = 36\]
\[x_{1} = - 1 + 6 = 5;\]
\[x_{2} = - 1 - 6 = - 7 < 5.\]
\[Ответ:нет\ решений.\]
\[\textbf{г)}\lg\left( x^{2} - 5x - 24 \right) = \lg(8 - x)\]
\[\left\{ \begin{matrix} x^{2} - 5x - 24 = 8 - x \\ 8 - x > 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x^{2} - 4x - 32 = 0 \\ x < 8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[x^{2} - 4x - 32 = 0\]
\[D_{1} = 4 + 32 = 36\]
\[x_{1} = 2 + 6 = 8;\]
\[x_{2} = 2 - 6 = - 4 < 8.\]
\[Ответ:x = - 4.\]
