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МатематикаРешебник по алгебре 11 класс Никольский Параграф 9. Равносильность уравнений и неравенств системам Задание 9
Задание 9
\[\boxed{\mathbf{9.}}\]
\[\textbf{а)}\ \sqrt{2x + 1} = x - 1\]
\[\left\{ \begin{matrix} 2x + 1 = (x - 1)^{2} \\ x - 1 \geq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ }\]
\[\left\{ \begin{matrix} 2x + 1 = x^{2} - 2x + 1 \\ x \geq 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x^{2} - 4x = 0 \\ x \geq 1\ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[x^{2} - 4x = 0\]
\[x(x - 4) = 0\]
\[x = 0\ (не\ подходит);\]
\[x = 4.\]
\[Ответ:x = 4.\]
\[\textbf{б)}\ \sqrt{2x - 1} = x - 2\]
\[\left\{ \begin{matrix} 2x - 1 = (x - 2)^{2} \\ x - 2 \geq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} 2x - 1 = x^{2} - 4x + 4 \\ x \geq 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x^{2} - 6x + 5 = 0 \\ x \geq 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[x^{2} - 6x + 5 = 0\]
\[D_{1} = 9 - 5 = 4\]
\[x_{1} = 3 + 2 = 5;\]
\[x_{2} = 3 - 2 = 1\ (не\ подходит).\]
\[Ответ:x = 5.\]
\[\textbf{в)}\ \sqrt{147 - 2x} = x - 2\]
\[\left\{ \begin{matrix} 147 - 2x = (x - 2)^{2} \\ x - 2 \geq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} 147 - 2x = x^{2} - 4x + 4 \\ x \geq 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x^{2} - 2x - 143 = 0 \\ x \geq 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[x^{2} - 2x - 143 = 0\]
\[D_{1} = 1 + 143 = 144\]
\[x_{1} = 1 + 12 = 13;\]
\[x_{2} = 1 - 12 =\]
\[= - 11\ (не\ подходит).\]
\[Ответ:x = 13.\]
\[\textbf{г)}\ \sqrt{- 8x + 108} = x - 3\]
\[\left\{ \begin{matrix} - 8x + 108 = (x - 3)^{2} \\ x - 3 \geq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} - 8x + 108 = x^{2} - 6x + 9 \\ x \geq 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x^{2} + 2x - 99 = 0 \\ x \geq 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[x^{2} + 2x - 99 = 0\]
\[D_{1} = 1 + 99 = 100\]
\[x_{1} = - 1 + 10 = 9;\]
\[x_{2} = - 1 - 10 =\]
\[= - 11\ (не\ подходит).\]
\[Ответ:x = 9.\]
