Решебник по алгебре 11 класс Никольский Параграф 8. Уравнения-следствия Задание 32

\[\boxed{\mathbf{32.}}\]

\[\textbf{а)}\ \sqrt{2x - 3} + \sqrt{4x + 1} = 4\]

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

\[2x - 3 = 16 - 8\sqrt{4x + 1} + 4x + 1\]

\[8\sqrt{4x + 1} = 2x + 20\ \ |\ :2\]

\[4\sqrt{4x + 1} = x + 10\]

\[16 \cdot (4x + 1) = (x + 10)^{2}\]

\[64x + 16 = x^{2} + 20x + 100\]

\[x^{2} - 44x + 84 = 0\]

\[D_{1} = 484 - 84 = 400\]

\[x_{1} = 22 + 20 = 42;\]

\[x_{2} = 22 - 20 = 2.\]

\[Проверка.\]

\[x = 2:\]

\[\sqrt{4 - 3} + \sqrt{8 + 1} = 4\]

\[1 + 3 = 4\]

\[4 = 4.\]

\[x = 42:\]

\[\sqrt{84 - 3} + \sqrt{168 + 1} = 4\]

\[9 + 13 = 4\]

\[22 \neq 4.\]

\[Ответ:x = 2.\]

\[\textbf{б)}\ \sqrt{2x + 6} = 2 + \sqrt{x + 1}\]

\[2x + 6 = 4 + 4\sqrt{x + 1} + x + 1\]

\[4\sqrt{x + 1} = x + 1\]

\[16(x + 1) = (x + 1)^{2}\]

\[16x + 16 = x^{2} + 2x + 1\]

\[x^{2} - 14x - 15 = 0\]

\[D_{1} = 49 + 15 = 64\]

\[x_{1} = 7 + 8 = 15;\]

\[x_{2} = 7 - 8 = - 1.\]

\[Проверка.\]

\[x = 15:\]

\[\sqrt{30 + 6} = 2 + \sqrt{15 + 1}\]

\[6 = 2 + 4\]

\[6 = 6.\]

\[x = - 1:\]

\[\sqrt{- 2 + 6} = 2 + \sqrt{- 1 + 1}\]

\[\sqrt{4} = 2\]

\[2 = 2.\]

\[Ответ:x = - 1;x = 15.\]

\[\textbf{в)}\ \sqrt{4x + 8} - \sqrt{3x - 2} = 2\]

\[\sqrt{4x + 8} = 2 + \sqrt{3x - 2}\]

\[4x + 8 = 4 + 4\sqrt{3x - 2} + 3x - 2\]

\[4\sqrt{3x - 2} = x + 6\]

\[16(3x - 2) = (x + 6)^{2}\]

\[48x - 32 = x^{2} + 12x + 36\]

\[x^{2} - 36x + 68 = 0\]

\[D_{1} = 324 - 68 = 256\]

\[x_{1} = 18 + 16 = 34;\]

\[x_{2} = 18 - 16 = 2.\]

\[Проверка.\]

\[x = 34:\]

\[\sqrt{136 + 8} - \sqrt{102 - 2} = 2\]

\[12 - 10 = 2\]

\[2 = 2.\]

\[x = 2:\]

\[\sqrt{8 + 8} - \sqrt{6 - 2} = 2\]

\[4 - 2 = 2\]

\[2 = 2.\]

\[Ответ:x = 2;x = 34.\]

\[\textbf{г)}\ \sqrt{3x - 2} + \sqrt{2x + 5} = 5\]

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

\[3x - 2 =\]

\[= 25 - 10\sqrt{2x + 5} + 2x + 5\]

\[10\sqrt{2x + 5} = - x + 32\]

\[100(2x + 5) = (32 - x)^{2}\]

\[200x + 500 = x^{2} - 64x + 1024\]

\[x^{2} - 264x + 524 = 0\]

\[x_{1} + x_{2} = 264;\ \ x_{1} \cdot x_{2} = 524\]

\[x_{1} = 262;\ \ x_{2} = 2.\]

\[Проверка.\]

\[x = 2:\]

\[\sqrt{6 - 2} + \sqrt{4 + 5} = 5\]

\[2 + 3 = 5\]

\[5 = 5.\]

\[x = 262:\]

\[\sqrt{786 - 2} + \sqrt{524 + 5} = 5\]

\[28 + 23 \neq 5.\]

\[Ответ:x = 2.\]