Страница 62 номер 155, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение

1) \(\sqrt{x} - x = -12\), \(\left(\sqrt{x}\right)^{2} = (x - 12)^{2}\), \(x - x^{2} + 24x - 144 = 0\),
\(x^{2} - 25x + 144 = 0, x_{1{,}2} = \dfrac{25 \pm \sqrt{625 - 576}}{2}, x_{1} = 16, x_{2} = 9.\)

Проверка: \(\sqrt{16} - 16 = -12\) – верно; \(\sqrt{9} - 9 = -12\) – неверно.

Ответ: \(x = 16\).

2) \(x + \sqrt{x} = 2(x - 1)\), \(\left(\sqrt{x}\right)^{2} = (x - 2)^{2}\), \(x - x^{2} + 4x - 4 = 0\),
\(x^{2} - 5x + 4 = 0, x_{1{,}2} = \dfrac{5 \pm \sqrt{25 - 16}}{2}, x_{1} = 4, x_{2} = 1.\)

Проверка: \(4 + \sqrt{4} = 2(4 - 1)\) – верно; \(1 + \sqrt{1} = 2(1 - 1)\) – неверно.

Ответ: \(x = 4\).

3) \(\sqrt{x - 1} = x - 3\), \(\left(\sqrt{x - 1}\right)^{2} = (x - 3)^{2}\), \(x - 1 - x^{2} + 6x - 9 = 0\),
\(x^{2} - 7x + 10 = 0, x_{1{,}2} = \dfrac{7 \pm \sqrt{49 - 40}}{2}, x_{1} = 5, x_{2} = 2.\)

Проверка: \(\sqrt{5 - 1} = 5 - 3\) – верно; \(\sqrt{2 - 1} = 2 - 3\) – неверно.

Ответ: \(x = 5\).

4) \(\sqrt{6 + x - x^{2}} = 1 - x\), \(\left(\sqrt{6 + x - x^{2}}\right)^{2} = (1 - x)^{2}\),
\(6 + x - x^{2} - 1 + 2x - x^{2} = 0,\)
\(2x^{2} - 3x - 5 = 0, x_{1{,}2} = \dfrac{3 \pm \sqrt{9 + 40}}{4},\)
\(x_{1} = 2{,}5, x_{2} = -1.\)

Проверка: \(\sqrt{6 + 2{,}5 - 6{,}25} = 1 - 2{,}5\), \(1{,}5 = -1{,}5\) – неверно; \(\sqrt{6 - 1 - 1} = 1 + 1\) – верно.

Ответ: \(x = -1\).