Giải phương trình sau

Để giải phương trình này, ta bắt đầu bằng cách đặt một biến tạm thời để giảm bớt căn bậc hai trong phương trình.

Đặt \(y = \sqrt{x + 1}\), ta có \(y^2 = x + 1\).

Thay thế \(y\) vào phương trình ban đầu, ta được:

\(\sqrt{x + 3} + 2x\sqrt{x + 1} = 2x + \sqrt{x^2 + 4x + 3}\)

\(\sqrt{x + 3} + 2xy = 2x + \sqrt{x^2 + 4x + 3}\)

\(2xy - \sqrt{x^2 + 4x + 3} = 2x - \sqrt{x + 3}\)

Bình phương cả hai vế của phương trình, ta có:

\((2xy - \sqrt{x^2 + 4x + 3})^2 = (2x - \sqrt{x + 3})^2\)

\(4x^2y^2 - 4xy\sqrt{x^2 + 4x + 3} + x^2 + 4x + 3 = 4x^2 - 4x\sqrt{x + 3} + x + 3\)

\(4x^2y^2 - 4xy\sqrt{x^2 + 4x + 3} + x^2 + 4x + 3 = 4x^2 - 4x\sqrt{x + 3} + x + 3\)

\(4x^2y^2 - 4xy\sqrt{x^2 + 4x + 3} = 4x^2 - 4x\sqrt{x + 3}\)

\(4xy(xy - \sqrt{x^2 + 4x + 3}) = 4x(x - \sqrt{x + 3})\)

\(xy - \sqrt{x^2 + 4x + 3} = x - \sqrt{x + 3}\)

\(xy - x = \sqrt{x^2 + 4x + 3} - \sqrt{x + 3}\)

\(x(y - 1) = \sqrt{x^2 + 4x + 3} - \sqrt{x + 3}\)

\(x(y - 1) = \frac{(x^2 + 4x + 3) - (x + 3)}{\sqrt{x^2 + 4x + 3} + \sqrt{x + 3}}\)

\(x(y - 1) = \frac{x^2 + 3x}{\sqrt{x^2 + 4x + 3} + \sqrt{x + 3}}\)

\(x(y - 1) = \frac{x(x + 3)}{\sqrt{x^2 + 4x + 3} + \sqrt{x + 3}}\)

\(x(y - 1) = \frac{x(x + 3)}{\sqrt{(x + 1)(x + 3)} + \sqrt{x + 3}}\)

\(x(y - 1) = \frac{x(x + 3)}{(\sqrt{x + 1} + 1)\sqrt{x + 3}}\)

\(x(y - 1) = \frac{x(x + 3)}{\sqrt{x + 1}\sqrt{x + 3} + \sqrt{x + 3}}\)

\(x(y - 1) = \frac{x(x + 3)}{\sqrt{x + 3}(\sqrt{x + 1} + 1)}\)

\(x(y - 1) = \frac{x(x + 3)}{(\sqrt{x + 3} + 1)(\sqrt{x + 1} + 1)}\)

\(x(y - 1) = \frac{x(x + 3)}{(\sqrt{x + 3} + 1)^2}\)

\(x(y - 1) = \frac{x(x + 3)}{x + 3 + 2\sqrt{x + 3} + 1}\)

\(x(y - 1) = \frac{x(x + 3)}{x + 4 + 2\sqrt{x + 3}}\)

\(x(y - 1) = \frac{x(x + 3)}{(x + 4) + 2\sqrt{x + 3}}\)

\(x(y - 1) = \frac{x(x + 3)}{(x + 4) + 2\sqrt{x + 3}} \cdot \frac{(x + 4) - 2\sqrt{x + 3}}{(x + 4) - 2\sqrt{x + 3}}\)

\(x(y - 1) = \frac{x(x + 3)((x + 4) - 2\sqrt{x + 3})}{(x + 4)^2 - (2\sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{x^2 + 8x + 16 - 4(x + 3)}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{x^2 + 8x + 16 - 4x - 12}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{x^2 + 4x + 4 - 2\sqrt{x + 3}}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})(x + 2 - \sqrt{x + 3})}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y - 1) = \frac{x(x + 3)(x + 4 - 2\sqrt{x + 3})}{(x + 2 - \sqrt{x + 3})^2}\)

\(x(y