Решебник по алгебре 11 класс Никольский Параграф 8. Уравнения-следствия Задание 28

\[\boxed{\mathbf{28.}}\]

\[\textbf{а)}\ 10^{\lg\left( x^{2} + 5x - 1 \right)} = 3x + 2\]

\[x^{2} + 5x - 1 = 3x + 2\]

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

\[D_{1} = 1 + 3 = 4\]

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

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

\[Проверка.\]

\[x = - 3:\]

\[10^{\lg( - 7)} - не\ существует.\]

\[x = 1:\]

\[10^{\lg 5} = 3 + 2\]

\[5 = 5.\]

\[Ответ:x = 1.\]

\[\textbf{б)}\ 10^{\lg\left( x^{2} - 3x + 1 \right)} = x - 2\]

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

\[x^{2} - 4x + 3 = 0\]

\[D_{1} = 4 - 3 = 1\]

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

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

\[Проверка.\]

\[x = 3:\]

\[10^{\lg 1} = 3 - 2\]

\[1 = 1.\]

\[x = 1:\]

\[10^{\lg( - 1)} - не\ существует.\]

\[Ответ:x = 3.\]

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

\[2x^{2} + 5x - 1 = x^{2} - 7\]

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

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

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

\[Проверка:\]

\[x = - 3:\]

\[2^{\log_{2}2} = 9 - 7\]

\[2 = 2.\]

\[x = - 2:\]

\[2^{\log_{2}( - 3)} - не\ существует.\]

\[Ответ:x = - 3.\]

\[\textbf{г)}\ 5^{\log_{5}\left( 3x^{2} + 4x - 1 \right)} = 2x^{2} - 4\]

\[3x^{2} + 4x - 1 = 2x^{2} - 4\]

\[x^{2} + 4x + 3 = 0\]

\[D_{1} = 4 - 3 = 1\]

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

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

\[Проверка.\]

\[x = - 1:\]

\[5^{\log_{5}( - 2)} - не\ существует.\]

\[x = - 3:\]

\[5^{\log_{5}14} = 18 - 4\]

\[4 = 4.\]

\[Ответ:x = - 3.\]