a. Rút gọn biểu thức \(A\).

\(A = \left(\right. \frac{\sqrt{x} + 2}{x - 5 \sqrt{x} + 6} - \frac{\sqrt{x} + 3}{2 - \sqrt{x}} - \frac{\sqrt{x} + 2}{\sqrt{x} - 3} \left.\right) : \left(\right. 2 - \frac{\sqrt{x}}{\sqrt{x} + 1} \left.\right)\) 

\(A = \left(\right. \frac{\sqrt{x} + 2}{\left(\right. \sqrt{x} - 2 \left.\right) \left(\right. \sqrt{x} - 3 \left.\right)} - \frac{\sqrt{x} + 3}{2 - \sqrt{x}} - \frac{\sqrt{x} + 2}{\sqrt{x} - 3} \left.\right) : \left(\right. 2 - \frac{\sqrt{x}}{\sqrt{x} + 1} \left.\right)\) 

\(A = \left(\right. \frac{\sqrt{x} + 2}{\left(\right. \sqrt{x} - 2 \left.\right) \left(\right. \sqrt{x} - 3 \left.\right)} + \frac{\left(\right. \sqrt{x} + 3 \left.\right) \left(\right. \sqrt{x} - 3 \left.\right)}{\left(\right. \sqrt{x} - 2 \left.\right) \left(\right. \sqrt{x} - 3 \left.\right)} - \frac{\left(\right. \sqrt{x} + 2 \left.\right) \left(\right. \sqrt{x} - 2 \left.\right)}{\left(\right. \sqrt{x} - 3 \left.\right) \left(\right. \sqrt{x} - 2 \left.\right)} \left.\right) : \left(\right. 2 - \frac{\sqrt{x}}{\sqrt{x} + 1} \left.\right)\) 

\(A = \frac{\sqrt{x} + 2 + x - 9 - x + 4}{\left(\right. \sqrt{x} - 2 \left.\right) \left(\right. \sqrt{x} - 3 \left.\right)} : \left(\right. 2 - \frac{\sqrt{x}}{\sqrt{x} + 1} \left.\right)\) 

\(A = \frac{\sqrt{x} - 3}{\left(\right. \sqrt{x} - 2 \left.\right) \left(\right. \sqrt{x} - 3 \left.\right)} : \frac{\sqrt{x} + 2}{\sqrt{x} + 1}\) 

\(A = \frac{1}{\sqrt{x} - 2} . \frac{\sqrt{x} + 1}{\sqrt{x} + 2}\) 

\(A = \frac{\sqrt{x} + 1}{x - 4}\).

b. Tìm các giá trị của x để \(\frac{1}{A} \leq - \frac{5}{2}\).

(ĐK: \(x \geq 0 , x \neq 4 , x \neq 9\))

Để \(\frac{1}{A} \leq - \frac{5}{2}\)thì

\(\frac{x - 4}{\sqrt{x} + 1} \leq - \frac{5}{2}\)

\(2 x - 8 \leq - 5 \sqrt{x} - 5\)

\(2 x + 5 \sqrt{x} - 3 \leq 0\)

\(- 3 \leq \sqrt{x} \leq \frac{1}{2}\)

\(\&\text{nbsp}; 0 \leq \sqrt{x} \leq \frac{1}{2}\)

\(\&\text{nbsp}; 0 \leq x \leq \frac{1}{4}\).

Kết hợp với điều kiện ta được \(0 \leq x \leq \frac{1}{4}\)thì \(\frac{1}{A} \leq - \frac{5}{2}\).

​A=2−3​​.(6​+2​)

\(​​ A = \sqrt{2} . \left(\right. \sqrt{2 - \sqrt{3}} \left.\right) + \sqrt{6} . \left(\right. \sqrt{2 - \sqrt{3}} \left.\right)\)

\(A = \sqrt{4 - 2 \sqrt{3}} + \sqrt{12 - 6 \sqrt{3}}\)

\(A = \sqrt{1 + 3 - 2 \sqrt{1.3}} + \sqrt{12 - 2.3 \sqrt{3}}\)

\(A = \sqrt{1^{2} - 2 \sqrt{1.3} + \left(\right. \sqrt{3} \left.\right)^{2}} + \sqrt{3^{2} - 2.3 \sqrt{3} + \left(\right. \sqrt{3} \left.\right)^{2}}\)

\(A = \sqrt{\left(\right. 1 - \sqrt{3} \left.\right)^{2}} + \sqrt{\left(\right. 3 - \sqrt{3} \left.\right)^{2}}\)

\(A = \sqrt{3} - 1 + 3 - \sqrt{3} = 2\).