6. Решите уравнение: 1) √2x + 48 = -x; 2) √x−2+√x = 3) 3) √2x + 7 - √x - 5 = 3.

1) Решим уравнение \(\sqrt{2x + 48}\) = -x.

ОДЗ: 2x + 48 ≥ 0 => x ≥ -24.

Так как \(\sqrt{2x + 48}\) ≥ 0, то -x ≥ 0 => x ≤ 0. Получаем -24 ≤ x ≤ 0.

2x + 48 = x².

x² - 2x - 48 = 0.

D = (-2)² - 4 ⋅ 1 ⋅ (-48) = 4 + 192 = 196.

x₁ = (2 + \(\sqrt{196}\))/(2 ⋅ 1) = (2 + 14)/2 = 16/2 = 8 (не подходит).

x₂ = (2 - \(\sqrt{196}\))/(2 ⋅ 1) = (2 - 14)/2 = -12/2 = -6 (подходит).

2) Решим уравнение \(\sqrt{x - 2}\) + \(\sqrt{x}\) = 3.

ОДЗ: x ≥ 2.

\(\sqrt{x - 2}\) = 3 - \(\sqrt{x}\).

x - 2 = 9 - 6\(\sqrt{x}\) + x.

-11 = -6\(\sqrt{x}\).

\(\sqrt{x}\) = 11/6.

x = (11/6)² = 121/36 = 3 целых 13/36.

3) Решим уравнение \(\sqrt{2x + 7}\) - \(\sqrt{x - 5}\) = 3.

ОДЗ: x ≥ 5.

\(\sqrt{2x + 7}\) = \(\sqrt{x - 5}\) + 3.

2x + 7 = x - 5 + 6\(\sqrt{x - 5}\) + 9.

x + 3 = 6\(\sqrt{x - 5}\).

x² + 6x + 9 = 36(x - 5).

x² + 6x + 9 = 36x - 180.

x² - 30x + 189 = 0.

D = (-30)² - 4 ⋅ 1 ⋅ 189 = 900 - 756 = 144.

x₁ = (30 + \(\sqrt{144}\))/(2 ⋅ 1) = (30 + 12)/2 = 42/2 = 21.

x₂ = (30 - \(\sqrt{144}\))/(2 ⋅ 1) = (30 - 12)/2 = 18/2 = 9.

Ответ: 1) -6; 2) 3 целых 13/36; 3) 21 и 9.