Решебник по алгебре 11 класс Никольский Параграф 8. Уравнения-следствия Задание 38

\[\boxed{\mathbf{38.}}\]

\[\textbf{а)}\lg{2x} - \lg(x + 4) = \lg{0,4}\]

\[\lg{2x} = \lg{0,4} + \lg{(x + 4)}\]

\[\lg{0,4(x + 4)} = \lg 2\]

\[x > 0.\]

\[0,4(x + 4) = 2\]

\[0,4x + 1,6 = 2\]

\[0,4x = 0,4\]

\[x = 1.\]

\[Ответ:x = 1.\]

\[\textbf{б)}\lg{(x - 4)} + \lg{(x - 6)} = \lg 8\]

\[\lg\left( (x - 4)(x - 6) \right) = 8\]

\[x - 6 > 0\]

\[x > 6.\]

\[(x - 4)(x - 6) = 8\]

\[x^{2} - 4x - 6x + 24 - 8 = 0\]

\[x^{2} - 10x + 16 = 0\]

\[D_{1} = 25 - 16 = 9\]

\[x_{1} = 5 + 3 = 8;\]

\[x_{2} = 5 - 3 = 3 < 6 - не\ корень.\]

\[Ответ:x = 8.\]

\[\textbf{в)}\lg(x + 5) + \lg(x - 4) =\]

\[= \lg{(x + 16)}\]

\[\lg\left( (x + 5)(x - 4) \right) = \lg(x + 16)\]

\[x - 4 > 0\]

\[x > 4.\]

\[(x + 5)(x - 4) = x + 16\]

\[x^{2} + 5x - 4x - 20 - x - 16 = 0\]

\[x^{2} - 36 = 0\]

\[x^{2} = 36\]

\[x_{1} = - 6 < 4\ (не\ корень).\]

\[x_{2} = 6.\]

\[Ответ:x = 6.\]

\[\textbf{г)}\lg(x - 3) + \lg(x + 4) =\]

\[= \lg(7x - 20)\]

\[\lg\left( (x - 3)(x + 4) \right) =\]

\[= \lg(7x - 20)\]

\[x - 3 > 0\]

\[x > 3.\]

\[(x - 3)(x + 4) = 7x - 20\]

\[x^{2} - 3x + 4x - 12 - 7x + 20 = 0\]

\[x^{2} - 6x + 8 = 0\]

\[D_{1} = 9 - 8 = 1\]

\[x_{1} = 3 + 1 = 4;\]

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

\[= 2 < 3\ (не\ корень).\]

\[Ответ:x = 4.\]