giai phuong trinh  √2x²+11x+19+√2x²+5x+7=3(x+2)

`\sqrt{2x^2 + 11x + 19} + \sqrt{2x^2 + 5x + 7} = 3(x + 2)`

ĐKXĐ: `x > - 2`

Đặt: `\sqrt{2x^2 + 11x + 19} = a`

`\sqrt{2x^2 + 5x + 7} = b`

`⇒ a^2 - b^2 = (2x^2 + 11x + 19) - (2x^2 + 5x + 7) = 6x + 12 = 6(x + 2)`

Ta có hệ PT sau:

$\begin{cases} a + b = 3(x + 2)\ (1)\\a^2 - b^2 = 6(x + 2) = 2.3.(x - 2)\ (2)\\ \end{cases}$

`(2) ⇔ a^2 - b^2 = 2.(a + b)`

`⇔ (a + b)(a - b) = 2(a + b)`

`⇔ a - b = 2`

`⇔ a = b + 2`

Thay vào PT `(1)`, ta có:

`(b + 2) + b = 3(x + 2)`

`⇔ 2b = 3x + 4`

`⇒ 2\sqrt{2x^2 + 5x + 7} = 3x + 4`

`⇔ (2\sqrt{2x^2 + 5x + 7})^2 = (3x + 4)^2` $\text{(ĐK: x >}$ `-4/3)`

`⇔ 4(2x^2 + 5x + 7) = 9x^2 + 24x + 16`

`⇔ x^2 + 4x - 12 = 0`

`⇔ x^2 - 2x + 6x - 12 = 0`

`⇔ x(x - 2) + 6(x - 2) = 0`

`⇔ (x - 2)(x + 6) = 0`

`⇔` $\left[\begin{matrix} x = 2\ \text{(TM)}\\ x = -6\ \text{(Loại)}\end{matrix}\right.$

Vậy PT có nghiệm duy nhất là: `x = 2`