Решебник по алгебре 11 класс Никольский Параграф 8. Уравнения-следствия Задание 11

\[\boxed{\mathbf{11.}}\]

\[\textbf{а)}\ \left| x^{2} - 4x + 2 \right| = x^{2} - 6x + 10\]

\[\sqrt{\left( x^{2} - 4x + 2 \right)^{2}} = x^{2} - 6x + 10\]

\[\left( x^{2} - 4x + 2 \right)^{2} = \left( x^{2} - 6x + 10 \right)^{2}\]

\[(2x - 8)\left( 2x^{2} - 10x + 12 \right) = 0\]

\[4 \cdot (x - 4)\left( x^{2} - 5x + 6 \right) = 0\]

\[1)\ x - 4 = 0\]

\[x = 4.\]

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

\[x_{1} + x_{2} = 5;\ \ x_{1} \cdot x_{2} = 6\]

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

\[Проверка:\]

\[|16 - 16 + 2| = 16 - 24 + 10\]

\[|2| = 2\]

\[2 = 2\]

\[x = 4 - корень.\]

\[|4 - 8 + 2| = 4 - 12 + 10\]

\[| - 2| = 2\]

\[2 = 2\]

\[x = 2 - корень.\]

\[|9 - 12 + 2| = 9 - 18 + 10\]

\[| - 1| = 1\]

\[1 = 1\]

\[x = 3 - корень.\]

\[Ответ:x = 2;3;4.\]

\[\textbf{б)}\ \left| x^{2} - 2x - 2 \right| = x^{2} - 4x + 6\]

\[\sqrt{\left( x^{2} - 2x - 2 \right)^{2}} = x^{2} - 4x + 6\]

\[\left( x^{2} - 2x - 2 \right)^{2} = \left( x^{2} - 4x + 6 \right)\]

\[\left( x^{2} - 2x - 2 \right)^{2} - \left( x^{2} - 4x + 6 \right) = 0\]

\[(2x - 8)\left( 2x^{2} - 6x + 4 \right) = 0\]

\[4 \cdot (x - 4)\left( x^{2} - 3x + 2 \right) = 0\]

\[1)\ x - 4 = 0\]

\[x = 4.\]

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

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

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

\[Проверка:\]

\[|16 - 8 - 2| = 16 - 16 + 6\]

\[|6| = 6\]

\[6 = 6\]

\[x = 4 - корень.\]

\[|4 - 4 - 2| = 4 - 8 + 6\]

\[| - 2| = 2\]

\[2 = 2\]

\[x = 2 - корень.\]

\[|1 - 2 - 2| = 1 - 4 + 6\]

\[| - 3| = 3\]

\[3 = 3\]

\[x = 1 - корень.\]

\[Ответ:x = 1;2;4.\]

\[\textbf{в)}\ \left| 2\lg x - 3 \right| = 3\lg x - 2\]

\[\sqrt{\left( 2\lg x - 3 \right)^{2}} = 3\lg x - 2\]

\[\left( 2\lg x - 3 \right)^{2} = \left( 3\lg x - 2 \right)^{2}\]

\[\left( 2\lg x - 3 \right)^{2} - \left( 3\lg x - 2 \right)^{2} = 0\]

\[\left( - \lg x - 1 \right)\left( 5\lg x - 5 \right) = 0\]

\[- 5\left( \lg x + 1 \right)\left( \lg x - 1 \right) = 0\]

\[1)\ \lg x + 1 = 0\]

\[\lg x = - 1\ \]

\[x = 10^{- 1} = \frac{1}{10}.\]

\[2)\ \lg x - 1 = 0\]

\[\lg x = 1\]

\[x = 10.\]

\[Проверка:\]

\[\left| 2\lg\frac{1}{10} - 3 \right| = 3\lg\frac{1}{10} - 2\]

\[\left| 2 \cdot ( - 1) - 3 \right| = 3 \cdot ( - 1) - 2\]

\[| - 5| \neq - 5\]

\[x = \frac{1}{10} - не\ корень.\]

\[\left| 2\lg 10 - 3 \right| = 3\lg 10 - 2\]

\[|2 \cdot 1 - 3| = 3 \cdot 1 - 2\]

\[| - 1| = 1\]

\[1 = 1\]

\[x = 10 - корень.\]

\[Ответ:10.\]

\[\textbf{г)}\ \left| 3\lg x - 4 \right| = 2\lg x - 1\]

\[\sqrt{\left( 3\lg x - 4 \right)^{2}} = 2\lg x - 1\]

\[\left( 3\lg x - 4 \right)^{2} = \left( 2\lg x - 1 \right)^{2}\]

\[\left( 3\lg x - 4 \right)^{2} - \left( 2\lg x - 1 \right)^{2} = 0\]

\[\left( \lg x - 3 \right)\left( 5\lg x - 5 \right) = 0\]

\[5\left( \lg x - 3 \right)\left( \lg x - 1 \right) = 0\]

\[1)\ \lg x = 3\]

\[x = 1000.\]

\[2)\ \lg x = 1\]

\[x = 10.\]

\[Проверка:\]

\[\left| 3\lg 1000 - 4 \right| = 2\lg 1000 - 1\]

\[|3 \cdot 3 - 4| = 2 \cdot 3 - 1\]

\[|5| = 5\]

\[5 = 5\]

\[x = 1000 - корень.\]

\[\left| 3\lg 10 - 4 \right| = 2\lg 10 - 1\]

\[|3 \cdot 1 - 4| = 2 \cdot 1 - 1\]

\[| - 1| = 1\]

\[1 = 1\]

\[x = 10 - корень.\]

\[Ответ:x = 10;x = 1000.\]

\[\textbf{д)}\ \left| 2^{x + 1} - 7 \right| = 5 - 2^{x}\]

\[\sqrt{\left( 2^{x + 1} - 7 \right)^{2}} = 5 - 2^{x}\]

\[\left( 2^{x + 1} - 7 \right)^{2} = \left( 5 - 2^{x} \right)^{2}\]

\[\left( 2^{x + 1} - 7 \right)^{2} - \left( 5 - 2^{x} \right)^{2} = 0\]

\[\left( 3 - 12 \cdot 2^{- x} \right)\left( 1 - 2 \cdot 2^{- x} \right) = 0\]

\[1)\ 3 - 12 \cdot 2^{- x} = 0\ \ |\ :3\]

\[1 - 4 \cdot 2^{- x} = 0\]

\[4 \cdot 2^{- x} = 1\]

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

\[2 - x = 0\]

\[x = 2.\]

\[2)\ 1 - 2 \cdot 2^{- x} = 0\]

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

\[2^{1} \cdot 2^{- x} = 2^{0}\]

\[1 - x = 0\]

\[x = 1.\]

\[Проверка:\]

\[\left| 2^{2 + 1} - 7 \right| = 5 - 2^{2}\]

\[\left| 2^{3} - 7 \right| = 5 - 4\]

\[|8 - 7| = 1\]

\[1 = 1\]

\[x = 2 - корень.\]

\[\left| 2^{1 + 1} - 7 \right| = 5 - 2^{1}\]

\[\left| 2^{2} - 7 \right| = 5 - 2\]

\[|4 - 7| = 3\]

\[| - 3| = 3\]

\[3 = 3\]

\[x = 1 - корень.\]

\[Ответ:x = 1;x = 2.\]

\[\textbf{е)}\ \left| 2^{x + 1} - 7 \right| = 2^{x} + 1\]

\[\sqrt{\left( 2^{x + 1} - 7 \right)^{2}} = 2^{x} + 1\]

\[\left( 2^{x + 1} - 7 \right)^{2} = \left( 2^{x} + 1 \right)^{2}\]

\[\left( 2^{x + 1} - 7 \right)^{2} - \left( 2^{x} + 1 \right)^{2} = 0\]

\[\left( 1 - 8 \cdot 2^{- x} \right)\left( 3 - 6 \cdot 2^{- x} \right) = 0\]

\[1)\ 1 - 8 \cdot 2^{- x} = 0\]

\[2^{3} \cdot 2^{- x} = 1\]

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

\[3 - x = 0\]

\[x = 3.\]

\[2)\ 3 - 6 \cdot 2^{- x} = 0\ \ \ |\ :3\]

\[1 - 2 \cdot 2^{- x} = 0\]

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

\[2^{1} \cdot 2^{- x} = 2^{0}\]

\[1 - x = 0\]

\[x = 1.\]

\[Проверка:\]

\[\left| 2^{3 + 1} - 7 \right| = 2^{3} + 1\]

\[\left| 2^{4} - 7 \right| = 8 + 1\]

\[|16 - 7| = 9\]

\[|9| = 9\]

\[9 = 9\]

\[x = 3 - корень.\]

\[\left| 2^{1 + 1} - 7 \right| = 2^{1} + 1\]

\[\left| 2^{2} - 7 \right| = 2 + 1\]

\[| - 3| = 3\]

\[3 = 3\]

\[x = 1 - корень.\]

\[Ответ:x = 1;x = 3.\]