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МатематикаРешебник по алгебре 11 класс Никольский Параграф 9. Равносильность уравнений и неравенств системам Задание 22
Задание 22
\[\boxed{\mathbf{22.}}\]
\[\textbf{а)}\ \frac{2x^{2} + x - 15}{\sqrt{4x^{2} - 2x + 25}} = 0\]
\[\left\{ \begin{matrix} 2x^{2} + x - 15 = 0\ \ \\ 4x^{2} - 2x + 25 > 0 \\ \end{matrix} \right.\ \]
\[4x^{2} - 2x + 25 > 0\]
\[D_{1} = 1 - 100 = - 99 < 0\]
\[нет\ корней.\]
\[\left\{ \begin{matrix} 2x^{2} + x - 15 = 0 \\ x \in R\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[2x^{2} + x - 15 = 0\]
\[D = 1 + 120 = 121\]
\[x_{1} = \frac{- 1 + 11}{4} = \frac{10}{4} = 2,5;\]
\[x_{2} = \frac{- 1 - 11}{4} = - 3.\]
\[Ответ:x = - 3;x = 2,5.\]
\[\textbf{б)}\ \frac{3x^{2} - 10x - 8}{\sqrt{9x^{2} + 12x + 4}} = 0\]
\[\left\{ \begin{matrix} 3x^{2} - 10x - 8 = 0 \\ 9x^{2} + 12x + 4 > 0 \\ \end{matrix} \right.\ \]
\[9x^{2} + 12x + 4 > 0\]
\[(3x + 2)^{2} > 0\]
\[x \neq - \frac{2}{3}.\]
\[\left\{ \begin{matrix} 3x^{2} - 10x - 8 = 0 \\ x \neq - \frac{2}{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]
\[3x^{2} - 10x - 8 = 0\]
\[D_{1} = 25 + 24 = 49\]
\[x_{1} = \frac{5 + 7}{3} = 4;\]
\[x_{2} = \frac{5 - 7}{3} = - \frac{2}{3}\ (не\ подходит).\]
\[Ответ:x = 4.\]
\[\textbf{в)}\ \frac{2x^{2} + 9x - 18}{\sqrt{4x^{2} - 12x + 9}} = 0\]
\[\left\{ \begin{matrix} 2x^{2} + 9x - 18 = 0 \\ 4x^{2} - 12x + 9 > 0 \\ \end{matrix} \right.\ \]
\[4x^{2} - 12x + 9 > 0\]
\[(2x - 3)^{2} > 0\]
\[x \neq \frac{3}{2} \neq 1,5.\]
\[\left\{ \begin{matrix} 2x^{2} + 9x - 18 = 0 \\ x \neq 1,5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[2x^{2} + 9x - 18 = 0\]
\[D = 81 + 144 = 225\]
\[x_{1} = \frac{- 9 + 15}{4} = \frac{6}{4} =\]
\[= 1,5\ (не\ подходит);\]
\[x_{2} = \frac{- 9 - 15}{4} = - 6.\]
\[Ответ:x = - 6.\]
\[\textbf{г)}\ \frac{3x^{2} - 19x + 20}{\sqrt{9x^{2} - 24x + 16}} = 0\]
\[\left\{ \begin{matrix} 3x^{2} - 19x + 20 = 0 \\ 9x^{2} - 24x + 16 > 0 \\ \end{matrix} \right.\ \]
\[9x^{2} - 24x + 16 > 0\]
\[(3x - 4)^{2} > 0\]
\[x \neq \frac{4}{3}.\]
\[\left\{ \begin{matrix} 3x^{2} - 19x + 20 = 0 \\ x \neq \frac{4}{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]
\[3x^{2} - 19x + 20 = 0\]
\[D = 361 - 240 = 121\]
\[x_{1} = \frac{19 + 11}{6} = 5;\]
\[x_{2} = \frac{19 - 11}{6} = \frac{8}{6} =\]
\[= \frac{4}{3}\ (не\ подходит).\]
\[Ответ:x = \frac{4}{3}.\]
