Решебник по алгебре 11 класс Никольский Параграф 8. Уравнения-следствия Задание 24

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\[\textbf{а)}\ tg\frac{\text{πx}}{2} + x^{2} - 7x = tg\frac{\text{πx}}{2} - 6\]

\[x^{2} - 7x + 6 = 0\]

\[x_{1} + x_{2} = 7;\ \ x_{1} \cdot x_{2} = 6\]

\[x_{1} = 1;\ \ x_{2} = 6.\]

\[x = 1:\]

\[\text{tg}\frac{\pi}{2} + 1 - 7 = tg\frac{\pi}{2} - 6\]

\[\text{tg}\frac{\pi}{2} - 6 = tg\frac{\pi}{2} - 6\]

\[tg\ не\ определен\ в\ точке\ \frac{\pi}{2};\]

\[x = 1 - не\ корень.\]

\[x = 6:\]

\[\text{tg}\frac{\pi \cdot 6}{2} + 6^{2} - 7 \cdot 6 =\]

\[= tg\frac{\pi \cdot 6}{2} - 6\]

\[0 + 36 - 45 = 0 - 6\]

\[- 6 = - 6\]

\[x = 6 - корень.\]

\[Ответ:x = 6.\]

\[\textbf{б)}\ ctg\frac{\text{πx}}{3} + x^{2} - 2x =\]

\[= ctg\frac{\text{πx}}{3} + 24\]

\[x^{2} - 2x - 24 = 0\]

\[D_{1} = 1 + 24 = 25\]

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

\[x_{2} = 1 - 5 = - 4.\]

\[x = 6:\]

\[\text{ctg}\frac{\pi \cdot 6}{3} + 6^{2} - 2 \cdot 6 =\]

\[= ctg\frac{\pi \cdot 6}{3} + 24\]

\[ctg\ \ не\ определен\ в\ точке\ 2\pi;\]

\[x = 6 - не\ корень.\]

\[x = - 4:\]

\[\text{ctg}\frac{\pi \cdot ( - 4)}{3} + ( - 4)^{2} + 2 \cdot 4 =\]

\[= ctg\frac{\pi \cdot ( - 4)}{3} + 24\]

\[\text{ctg}\frac{- 4\pi}{3} + 24 = ctg\frac{- 4\pi}{3} + 24\]

\[x = - 4 - корень.\]

\[Ответ:x = - 4.\]

\[\textbf{в)}\ x^{2} + 13 + \log_{2}\left( x^{3} - 9 \right) =\]

\[= 8x + \log_{2}\left( 2x^{3} - 18 \right)\text{\ \ }\]

\[x^{2} + 13 + \log_{2}\left( x^{3} - 9 \right) =\]

\[= 8x + \log_{2}{2\left( x^{3} - 9 \right)}\]

\[x^{2} + 13 + \log_{2}\left( x^{3} - 9 \right) =\]

\[= 8x + \log_{2}2 + \log_{2}\left( x^{3} - 9 \right)\]

\[x^{2} + 13 + \log_{2}\left( x^{3} - 9 \right) =\]

\[= 8x + 1 + \log_{2}\left( x^{3} - 9 \right)\]

\[x^{2} - 8x + 12 = 0\]

\[D_{1} = 16 - 12 = 4\]

\[x_{1} = 4 - 2 = 2;\]

\[x_{2} = 4 + 2 = 6.\]

\[Проверка:\]

\[2^{2} + 13 + \log_{2}\left( 2^{3} - 9 \right) =\]

\[= 8 \cdot 2 + \log_{2}\left( 2 \cdot 2^{3} - 18 \right)\]

\[17 + \log_{2}( - 1) = 16 + \log_{2}( - 2)\]

\[a < 0;\]

\[нет\ решения;\]

\[x = 2 - не\ корень.\]

\[6^{2} + 13 + \log_{2}\left( 6^{3} - 9 \right) =\]

\[= 8 \cdot 6 + 1 + \log_{2}\left( 6^{3} - 9 \right)\]

\[49 + \log_{2}\left( 6^{3} - 9 \right) =\]

\[= 49 + \log_{2}\left( 6^{3} - 9 \right)\]

\[x = 6 - корень.\]

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

\[= 23 + \log_{3}\left( 3x^{3} + 12 \right)\]

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

\[= 23 + \log_{3}{3\left( x^{3} + 4 \right)}\]

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

\[= 23 + \log_{3}3 + \log_{3}\left( x^{3} + 4 \right)\]

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

\[= 23 + 1 + \log_{3}\left( x^{3} + 4 \right)\]

\[x^{2} + 2x - 24 = 0\]

\[D_{1} = 1 + 24 = 25\]

\[x_{1} = - 1 + 5 = 4;\]

\[x_{2} = - 1 - 5 = - 6.\]

\[Проверка:\]

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

\[= 23 + 1 + \log_{3}\left( 4^{3} + 4 \right)\]

\[24 + \log_{3}\left( 4^{3} + 4 \right) =\]

\[= 24 + \log_{3}\left( 4^{3} + 4 \right)\]

\[x = 4 - корень\ уравнения.\]

\[( - 6)^{2} - 2 \cdot 6 + \log_{3}\left( ( - 6)^{3} + 4 \right) =\]

\[= 23 + 1 + \log_{3}\left( ( - 6)^{3} + 4 \right)\]

\[a < 0;\]

\[нет\ решений;\]

\[x = - 6 - не\ корень.\]

\[Ответ:x = 4.\]