\(\dfrac{2\sqrt{x}}{\sqrt{x}+3}+\dfrac{x+6\sqrt{x}+9}{9-x}-\dfrac{\sqrt{x}}{\sqrt{x}+3}\left(dkxd:x\ge0,x\ne9\right)\)

\(=\dfrac{2\sqrt{x}}{\sqrt{x}+3}-\dfrac{\left(\sqrt{x}+3\right)^2}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}-\dfrac{\sqrt{x}}{\sqrt{x}+3}\)

\(=\dfrac{2\sqrt{x}\left(\sqrt{x}-3\right)-\left(x+6\sqrt{x}+9\right)-\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)

\(=\dfrac{2x-6\sqrt{x}-x-6\sqrt{x}-9-x+3\sqrt{x}}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)

\(=\dfrac{-9\sqrt{x}-9}{x-9}\) với \(x\ge0,x\ne9\)

\(VT=\left(\dfrac{\sqrt{x}}{\sqrt{x}+3}+\dfrac{3}{\sqrt{x}-3}\right).\dfrac{\sqrt{x}+3}{x+9}\\ =\left(\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}+\dfrac{3\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\right).\dfrac{\sqrt{x}+3}{x+9}\\ =\dfrac{x-3\sqrt{x}+3\sqrt{x}+9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}.\dfrac{\sqrt{x}+3}{x+9}\\ =\dfrac{x+9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}.\dfrac{\sqrt{x}+3}{x+9}\\ =\dfrac{1}{\sqrt{x}-3}=VP\)

\(VT=\dfrac{x-3\sqrt{x}+3\sqrt{x}+9}{x-9}\cdot\dfrac{\sqrt{x}+3}{x+9}\)

\(=\dfrac{x+9}{x+9}\cdot\dfrac{1}{\sqrt{x}-3}=\dfrac{1}{\sqrt{x}-3}=VP\)

a) \(\dfrac{\sqrt{x}}{\sqrt{x}+3}+\dfrac{2\sqrt{x}}{\sqrt{x}-3}-\dfrac{3x+9}{x-9}\left(x\ge0;x\ne0\right)\)

\(=\dfrac{\sqrt{x}.\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right).\left(\sqrt{x-3}\right)}+\dfrac{2\sqrt{x}.\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}-\dfrac{3x+9}{\left(\sqrt{x}-3\right).\left(\sqrt{x+3}\right)}\)

\(=\dfrac{x-3\sqrt{x}+2x+6\sqrt{x}-3x-9}{\left(\sqrt{x}+3\right).\left(\sqrt{x-3}\right)}\)

\(=\dfrac{3\sqrt{x}-9}{\left(\sqrt{x}+3\right).\left(\sqrt{x}-3\right)}\)

\(=\dfrac{3.\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right).\left(\sqrt{x}-3\right)}\)

\(=\dfrac{3}{\sqrt{x}+3}\)

b) \(\dfrac{3}{\sqrt{x}-1}-\dfrac{\sqrt{x}+5}{x-1}\left(x\ge0;x\ne1\right)\)

\(=\dfrac{3.\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right).\left(\sqrt{x}+1\right)}-\dfrac{\sqrt{x}+5}{\left(\sqrt{x}-1\right).\left(\sqrt{x}+1\right)}\)

\(=\dfrac{3\sqrt{x}+3-\sqrt{x}-5}{\left(\sqrt{x}-1\right).\left(\sqrt{x}+1\right)}\)

\(=\dfrac{2\sqrt{x}-2}{\left(\sqrt{x}-1\right).\left(\sqrt{x}+1\right)}\)

\(=\dfrac{2.\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right).\left(\sqrt{x}+1\right)}\)

\(=\dfrac{2}{\sqrt{x}+1}\)

c) \(\left(\dfrac{15-\sqrt{x}}{x-25}+\dfrac{2}{\sqrt{x}+5}\right):\dfrac{\sqrt{x}+1}{\sqrt{x}-5}\left(x\ge0;x\ne1\right)\)

\(=\left(\dfrac{15-\sqrt{x}}{\left(\sqrt{x}-5\right).\left(\sqrt{x}+5\right)}+\dfrac{2.\left(\sqrt{x}-5\right)}{\left(\sqrt{x}-5\right).\left(\sqrt{x}+5\right)}\right):\dfrac{\sqrt{x}+1}{\sqrt{x}-5}\)

\(=\dfrac{15-\sqrt{x}+2\sqrt{x}-10}{\left(\sqrt{x}-5\right).\left(\sqrt{x}+5\right)}:\dfrac{\sqrt{x}+1}{\sqrt{x}-5}\)

\(=\dfrac{\sqrt{x}+5}{\left(\sqrt{x}-5\right).\left(\sqrt{x}+5\right)}.\dfrac{\sqrt{x}-5}{\sqrt{x}+1}\)

\(\dfrac{1}{\sqrt{x}+1}\)

Sửa đề: \(\dfrac{\sqrt{x}}{\sqrt{x}+3}+\dfrac{2\sqrt{x}}{\sqrt{x}-3}-\dfrac{3x+9}{x-9}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)+2\sqrt{x}\left(\sqrt{x}+3\right)-3x-9}{x-9}\)

\(=\dfrac{x+3\sqrt{x}+2x+6\sqrt{x}-3x-9}{x-9}\)

\(=\dfrac{9\sqrt{x}-9}{x-9}\)

Với x ≥ 0; x ≠ 9 ta có:

\(A=\dfrac{\sqrt{x}}{\sqrt{x}+3}+\dfrac{2\sqrt{x}}{\sqrt{x}-3}-\dfrac{3x+9}{x-9}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x-3}\right)+2\sqrt{x}\left(\sqrt{x}+3\right)-3x-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)

\(=\dfrac{x-3\sqrt{x}+2x+6\sqrt{x}-3x-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)

\(=\dfrac{3\sqrt{x}-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)

\(=\dfrac{3\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)

\(=\dfrac{3}{\sqrt{x}+3}\)

Vậy \(A=\dfrac{3}{\sqrt{x}+3}\).

Ta có: \(A=\dfrac{\sqrt{x}}{\sqrt{x}-3}+\dfrac{2\sqrt{x}-1}{\sqrt{x}-3}-\dfrac{2x-\sqrt{x}-3}{x-9}\)

\(=\dfrac{x-3\sqrt{x}+2x-6\sqrt{x}-\sqrt{x}+3-2x+\sqrt{x}+3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)

\(=\dfrac{x-9\sqrt{x}+6}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)

\(=\left(\dfrac{x+9}{x-9}-\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)}{x-9}\right).\dfrac{x-3\sqrt{x}}{\sqrt{x}}\\ =\left(\dfrac{x+9-x+3\sqrt{x}}{x-9}\right).\dfrac{x-3\sqrt{x}}{\sqrt{x}}\\ =\dfrac{3\left(3+\sqrt{x}\right)}{\left(\sqrt{x}-3\right)\left(3+\sqrt{x}\right)}.\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)}{\sqrt{x}}\\ =3\)

\(=\dfrac{3\sqrt{x}-x+2x}{9-x}:\dfrac{\sqrt{x}-1-2\sqrt{x}+6}{\sqrt{x}\left(\sqrt{x}-3\right)}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}+3\right)}{-\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\cdot\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)}{-\sqrt{x}+5}\)

\(=\dfrac{x}{\sqrt{x}-5}\)

\(\left(\dfrac{\sqrt{x}}{\sqrt{x}-3}-\dfrac{2\sqrt{x}+6}{x-9}\right):\dfrac{x-2\sqrt{x}}{\sqrt{x}-3}\left(x>3;x\ne9\right)\)

\(=\left(\dfrac{\sqrt{x}}{\sqrt{x}-3}-\dfrac{2\sqrt{x}+6}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right):\dfrac{\sqrt{x}-3}{x-2\sqrt{x}}\)

\(=\left(\dfrac{\sqrt{x}\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-3\right)\left( \sqrt{x}+3\right)}-\dfrac{2\sqrt{x}+6}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right)\cdot\dfrac{x-2\sqrt{x}}{\sqrt{x}-3}\)

\(=\dfrac{x+3\sqrt{x}-2\sqrt{x}-6}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\cdot\dfrac{\sqrt{x}-3}{x-2\sqrt{x}}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}+3\right)-2\left(\sqrt{x}+3\right)}{\sqrt{x}+3}\cdot\dfrac{1}{x-2\sqrt{x}}\)

\(=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+3\right)}{\sqrt{x}+3}\cdot\dfrac{1}{\sqrt{x}\left(\sqrt{x}-2\right)}\)

\(=\dfrac{1}{\sqrt{x}}\)