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

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

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

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

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

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

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

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

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

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

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

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

\(B=\left(\dfrac{3\sqrt{x}+6}{x-4}+\dfrac{\sqrt{x}}{\sqrt{x}-2}\right):\dfrac{x-9}{\sqrt{x}-3}\left(x\ge0;x\ne4;x\ne9\right)\)

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

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

\(=\dfrac{x+5\sqrt{x}+6}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\cdot\dfrac{1}{\sqrt{x}+3}\)

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

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

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

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

#\(Toru\)

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

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

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

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

c) \(\dfrac{x-4\sqrt{x}+4}{x-2\sqrt{x}}\left(đk:x>0\right)=\dfrac{\left(\sqrt{x}-2\right)^2}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}-2}{\sqrt{x}}\)

d) \(\dfrac{x-9}{x+6\sqrt{x}+9}\left(đk:x\ge0\right)=\dfrac{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)^2}=\dfrac{\sqrt{x}-3}{\sqrt{x}+3}\)

e) \(\dfrac{x-10\sqrt{x}+25}{25-x}\left(đk:x\ge0,x\ne25\right)=\dfrac{\left(\sqrt{x}-5\right)^2}{\left(\sqrt{x}-5\right)\left(\sqrt{x}+5\right)}=\dfrac{\sqrt{x}-5}{\sqrt{x}+5}\)

c: \(\dfrac{x-4\sqrt{x}+4}{x-2\sqrt{x}}=\dfrac{\sqrt{x}-2}{\sqrt{x}}\)

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

e: \(\dfrac{x-10\sqrt{x}+25}{25-x}=\dfrac{5-\sqrt{x}}{\sqrt{x}+5}\)

`A=(2\sqrtx-9)(x-5sqrtx+6)-(sqrtx+3)/(sqrtx-2)-(2sqrtx+1)(3-sqrtx)(x>=0,x ne 4, x ne 9)`

`=(2\sqrtx-9)(x-5sqrtx+6)-(sqrtx+3)/(sqrtx-2)+(2sqrtx+1)(sqrtx-3)`

`=(2sqrtx-9-x+9+2x-3sqrtx-2)/(x-5sqrtx+6)`
`=(x-sqrtx-2)/(x-5sqrtx+6)`
`=((\sqrtx+1)(sqrtx-2))/((sqrtx-2)(sqrtx-3))`
`=(sqrtx+1)/(sqrtx-3)`

`A=(2\sqrtx-9)/(x-5sqrtx+6)-(sqrtx+3)/(sqrtx-2)-(2sqrtx+1)/(3-sqrtx)(x>=0,x ne 4, x ne 9)`

`=(2\sqrtx-9)/(x-5sqrtx+6)-(sqrtx+3)/(sqrtx-2)+(2sqrtx+1)/(sqrtx-3)`

`=(2sqrtx-9-x+9+2x-3sqrtx-2)/(x-5sqrtx+6)`
`=(x-sqrtx-2)/(x-5sqrtx+6)`
`=((\sqrtx+1)(sqrtx-2))/((sqrtx-2)(sqrtx-3))`
`=(sqrtx+1)/(sqrtx-3)`

a: \(\frac{x+3+2\cdot\sqrt{x^2-9}}{2x-6+\sqrt{x^2-9}}\)

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

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

b: \(T=\frac{x^2+5x+6+x\cdot\sqrt{9-x^2}}{3x-x^2+\left(x+2\right)\cdot\sqrt{9-x^2}}\)

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

\(=\frac{\sqrt{x+3}\left\lbrack\left(x+2\right)\cdot\sqrt{x+3}+x\cdot\sqrt{3-x}\right\rbrack}{\sqrt{3-x}\left\lbrack x\cdot\sqrt{3-x}+\left(x+2\right)\cdot\sqrt{x+3}\right\rbrack}=\frac{\sqrt{3+x}}{\sqrt{3-x}}\)

Ta có: \(\frac{x+\sqrt{x}-6}{x-9}+\frac{x-7\sqrt{x}+19}{x+\sqrt{x}-12}-\frac{x-5\sqrt{x}}{x+4\sqrt{x}}\)

\(=\frac{x+3\sqrt{x}-2\sqrt{x}-6}{x-9}+\frac{x-7\sqrt{x}+19}{x+4\sqrt{x}-3\sqrt{x}-12}-\frac{\sqrt{x}\left(\sqrt{x}-5\right)}{\sqrt{x}\left(\sqrt{x}+4\right)}\)

\(=\frac{\sqrt{x}\left(\sqrt{x}+3\right)-2\left(\sqrt{x}+3\right)}{x-9}+\frac{x-7\sqrt{x}+19}{\sqrt{x}\left(\sqrt{x}+4\right)-3\left(\sqrt{x}+4\right)}-\frac{\sqrt{x}-5}{\sqrt{x}+4}\)

\(=\frac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}+\frac{x-7\sqrt{x}+19}{\left(\sqrt{x}+4\right)\left(\sqrt{x}-3\right)}-\frac{\left(\sqrt{x}-5\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+4\right)\left(\sqrt{x}-3\right)}\)

\(=\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+4\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+4\right)}+\frac{x-7\sqrt{x}+19}{\left(\sqrt{x}+4\right)\left(\sqrt{x}-3\right)}-\frac{x-8\sqrt{x}+15}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+4\right)}\)

\(=\frac{x+2\sqrt{x}-8+x-7\sqrt{x}+19-x+8\sqrt{x}-15}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+4\right)}\)

\(=\frac{x+3\sqrt{x}-4}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+4\right)}\)

\(=\frac{x+4\sqrt{x}-\sqrt{x}-4}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+4\right)}\)

\(=\frac{\sqrt{x}\left(\sqrt{x}+4\right)-\left(\sqrt{x}+4\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+4\right)}\)

\(=\frac{\left(\sqrt{x}+4\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+4\right)\left(\sqrt{x}-3\right)}\)

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

a) \(\dfrac{3-\sqrt{x}}{x-9}=\dfrac{-\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}=-\dfrac{1}{\sqrt{x+3}}\)(\(x\ge0,x\ne9\))

b) \(\dfrac{x-5\sqrt{x}+6}{\sqrt{x}-3}=\dfrac{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}{\sqrt{x}-3}=\sqrt{x}-2\left(x\ge0,x\ne9\right)\)

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

b) \(\dfrac{x-5\sqrt{x}+6}{\sqrt{x}-3}=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}{\sqrt{x}-3}=\sqrt{x}-2\)

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

mà \(x< 3\Rightarrow3-x>0\Rightarrow6-2x-\left|3-x\right|=6-2x-3+x=3-x\)

\(C=\left(\dfrac{\sqrt{x}+1}{x-4}-\dfrac{\sqrt{x}-1}{x+4\sqrt{x}+4}\right).\dfrac{x\sqrt{x}+2x-4\sqrt{x}-8}{\sqrt{x}-2}\)

\(=\left[\dfrac{\sqrt{x}+1}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}-\dfrac{\sqrt{x}-1}{\left(\sqrt{x}+2\right)^2}\right].\dfrac{x\left(\sqrt{x}+2\right)-4\left(\sqrt{x}+2\right)}{\sqrt{x}-2}\)

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

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

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

\(C=\left[\dfrac{\sqrt{x}+1}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}-\dfrac{\sqrt{x}-1}{\left(\sqrt{x}+2\right)^2}\right].\dfrac{\sqrt{x}\left(x-4\right)+2\left(x-4\right)}{\sqrt{x}-2}\) (\(x\ge0,x\ne4,x\ne9\))
\(C=\left[\dfrac{\sqrt{x}+1-\sqrt{x}+1}{\left(\sqrt{x}-2\right)\left(\sqrt{x+2}\right)^2}\right].\dfrac{\left(\sqrt{x}+2\right)\left(x-4\right)}{\sqrt{x}-2}\)
\(C=\dfrac{2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)^2}.\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}+2\right)\left(\sqrt{x-2}\right)}{\sqrt{x}-2}\)
\(C=\dfrac{2}{\left(\sqrt{x}-2\right)\left(\sqrt{x+2}\right)^2}.\left(\sqrt{x}+2\right)^2\)
\(C=\dfrac{2}{\sqrt{x}-2}\)