Страница 63 номер 163, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение

1) \(\sqrt{4x + 2\sqrt{3x^{2} + 4}} = x + 2\), \(\left(\sqrt{4x + 2\sqrt{3x^{2} + 4}}\right)^{2} = (x + 2)^{2}\),
\(4x + 2\sqrt{3x^{2} + 4} = x^{2} + 4x + 4, \left(2\sqrt{3x^{2} + 4}\right)^{2} = (x^{2} + 4)^{2},\)
\(12x^{2} + 16 = x^{4} + 8x^{2} + 16\), \(x^{4} - 4x^{2} = 0\), \(x^{2}(x^{2} - 4) = 0\), \(x = 0\) или \(x^{2} - 4 = 0\), \(x^{2} = 4\), \(x = \pm 2\).

Проверка: \(\sqrt{0 + 2\sqrt{0 + 4}} = 0 + 2\) – верно;

\(\sqrt{-8 + 2\sqrt{12 + 4}} = -2 + 2\) – верно;
\(\sqrt{8 + 2\sqrt{12 + 4}} = 2 + 2\) – верно.

Ответ: \(x_{1} = 0\), \(x_{2} = 2\), \(x_{3} = -2\).

2) \(3 - x = \sqrt{9 - \sqrt{36x^{2} - 5x^{4}}}\), \((3 - x)^{2} = \left(\sqrt{9 - \sqrt{36x^{2} - 5x^{4}}}\right)^{2}\),
\(9 - 6x + x^{2} = 9 - \sqrt{36x^{2} - 5x^{4}}, (6x - x^{2})^{2} = \left(\sqrt{36x^{2} - 5x^{4}}\right)^{2},\)
\(36x^{2} - 12x^{3} + x^{4} = 36x^{2} - 5x^{4}, 6x^{4} - 12x^{3} = 0, x^{3}(x - 2) = 0,\)
\(x_{1} = 0, x_{2} = 2.\)

Проверка: \(3 - 0 = \sqrt{9 - \sqrt{36 \cdot 0 - 5 \cdot 0}}\) – верно;

\(3 - 2 = \sqrt{9 - \sqrt{36 \cdot 4 - 5 \cdot 16}}\) – верно.

Ответ: \(x_{1} = 0\), \(x_{2} = 2\).

3) \(\sqrt{x^{2} + 3x + 12} - \sqrt{x^{2} + 3x} = 2\),
\(\left(\sqrt{x^{2} + 3x + 12}\right)^{2} = \left(2 + \sqrt{x^{2} + 3x}\right)^{2},\)
\(x^{2} + 3x + 12 = 4 + 4\sqrt{x^{2} + 3x} + x^{2} + 3x, 2^{2} = \left(\sqrt{x^{2} + 3x}\right)^{2},\)
\(4 - x^{2} - 3x = 0, x^{2} + 3x - 4 = 0,\)
\(x_{1{,}2} = \dfrac{-3 \pm \sqrt{9 + 16}}{2}, x_{1} = 1, x_{2} = -4.\)

Проверка: \(\sqrt{1 + 3 + 12} - \sqrt{1 + 3} = 2\) – верно;

\(\sqrt{16 - 12 + 12} - \sqrt{16 - 12} = 2\) – верно.

Ответ: \(x_{1} = 1\), \(x_{2} = -4\).

4) \(\sqrt{x^{2} + 5x + 10} - \sqrt{x^{2} + 5x + 3} = 1\),
\(\left(\sqrt{x^{2} + 5x + 10}\right)^{2} = \left(1 + \sqrt{x^{2} + 5x + 3}\right)^{2},\)
\(x^{2} + 5x + 10 = 1 + 2\sqrt{x^{2} + 5x + 3} + x^{2} + 5x + 3,\)
\(3^{2} = \left(\sqrt{x^{2} + 5x + 3}\right)^{2}, x^{2} + 5x + 3 - 9 = 0, x^{2} + 5x - 6 = 0,\)
\(x_{1{,}2} = \dfrac{-5 \pm \sqrt{25 + 24}}{2}, x_{1} = 1, x_{2} = -6.\)

Проверка: \(\sqrt{1 + 5 + 10} - \sqrt{1 + 5 + 3} = 1\) – верно;

\(\sqrt{36 - 30 + 10} - \sqrt{36 - 30 + 3} = 1\) – верно.

Ответ: \(x_{1} = 1\), \(x_{2} = -6\).