Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 687

\[\boxed{\mathbf{687}.}\]

\[1)\ {0,3}^{x^{3} - x^{2} + x - 1} = 1;\]

\[{0,3}^{x^{3} - x^{2} + x - 1} = {0,3}^{0};\]

\[x^{3} - x^{2} + x - 1 = 0;\]

\[x^{2}(x - 1) + (x - 1) = 0;\]

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

\[x - 1 = 0;\]

\[x = 1;\]

\[Ответ:\ \ x = 1.\]

\[2)\ \left( 2\frac{1}{3} \right)^{- x^{2} - 2x + 3} = 1;\]

\[\left( 2\frac{1}{3} \right)^{- x^{2} - 2x + 3} = \left( 2\frac{1}{3} \right)^{0};\]

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

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

\[D = 2^{2} + 4 \bullet 3 = 4 + 12 = 16\]

\[x_{1} = \frac{- 2 - 4}{2} = - 3\ \ и\ \]

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

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

\[3)\ {5,1}^{\frac{1}{2}(x - 3)} = 5,1\sqrt{5,1};\]

\[{5,1}^{\frac{x - 3}{2}} = {5,1}^{1} \bullet {5,1}^{\frac{1}{2}};\]

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

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

\[x - 3 = 2 + 1;\]

\[x - 3 = 3;\]

\[x = 6;\]

\[Ответ:\ \ x = 6.\]

\[4)\ 100^{x^{2} - 1} = 10^{1 - 5x};\]

\[10^{2\left( x^{2} - 1 \right)} = 10^{1 - 5x};\]

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

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

\[2x^{2} + 5x - 3 = 0;\]

\[D = 5^{2} + 4 \bullet 2 \bullet 3 =\]

\[= 25 + 24 = 49\]

\[x_{1} = \frac{- 5 - 7}{2 \bullet 2} = - \frac{12}{4} = - 3;\]

\[x_{2} = \frac{- 5 + 7}{2 \bullet 2} = \frac{2}{4} = 0,5;\]

\[Ответ:\ \ x_{1} = - 3;\ \ x_{2} = 0,5.\]