Cho ​\(C=\frac{2 \sqrt{x}-9}{x-5 \sqrt{x}+6}-\frac{\sqrt{x}+3}{\sqrt{x}-2}-\frac{2 \sqrt{x}+1}{3-\sqrt{x}} \text { với } x \geq 0 ; x \neq 4 ; x \neq 9\). Tìm x để C...

* Hướng dẫn giải

 \(\begin{array}{l} x - 5\sqrt x + 6 = x - 2\sqrt x - 3\sqrt x + 6 = \sqrt x (\sqrt x - 2) - 3(\sqrt x - 2) = (\sqrt x - 3)(\sqrt x - 2)\\ \Rightarrow C = \frac{{2\sqrt x - 9}}{{x - 5\sqrt x + 6}} - \frac{{\sqrt x + 3}}{{\sqrt x - 2}} - \frac{{2\sqrt x + 1}}{{3 - \sqrt x }} = \frac{{2\sqrt x - 9}}{{(\sqrt x - 2)(\sqrt x - 3)}} - \frac{{\sqrt x + 3}}{{\sqrt x - 2}} + \frac{{2\sqrt x + 1}}{{\sqrt x - 3}}\\ = \frac{{2\sqrt x - 9 - (\sqrt x + 3)(\sqrt x - 3) + (2\sqrt x + 1)(\sqrt x - 2)}}{{(\sqrt x - 2)(\sqrt x - 3)}} = \frac{{2\sqrt x - 9 - x + 9 + 2x - 3\sqrt x - 2}}{{(\sqrt x - 2)(\sqrt x - 3)}}\\ = \frac{{x - \sqrt x - 2}}{{(\sqrt x - 2)(\sqrt x - 3)}} = \frac{{x - 2\sqrt x + \sqrt x - 2}}{{(\sqrt x - 2)(\sqrt x - 3)}} = \frac{{\sqrt x (\sqrt x - 2) + (\sqrt x - 2)}}{{(\sqrt x - 2)(\sqrt x - 3)}}\\ = \frac{{(\sqrt x + 1)(\sqrt x - 2)}}{{(\sqrt x - 2)(\sqrt x - 3)}} = \frac{{\sqrt x + 1}}{{\sqrt x - 3}}\\ C < 1 \Leftrightarrow \frac{{\sqrt x + 1}}{{\sqrt x - 3}} < 1 \Leftrightarrow \frac{{\sqrt x + 1}}{{\sqrt x - 3}} - \frac{{\sqrt x - 3}}{{\sqrt x - 3}} < 0 \Leftrightarrow \frac{4}{{\sqrt x - 3}} < 0\\ \Leftrightarrow \sqrt x - 3 < 0 \Leftrightarrow \sqrt x < 3 \Rightarrow x < 9 \end{array}\)

Kết hợp điều kiện đề bài suy ra \( 0 \le x < 9;x \ne 4\)