Đáp án:

a/ $A=\dfrac{\sqrt{x}+2}{\sqrt{x}-3}$

b/ $x=\dfrac{98+55\sqrt{3}}{2}$

Giải thích các bước giải:

a/ $A=\dfrac{2}{\sqrt{x}-3}+\dfrac{2\sqrt{x}}{x-4\sqrt{x}+3}+\dfrac{\sqrt{x}}{\sqrt{x}-1}$

$\text{ĐKXĐ: $x \geq 0$; $x \neq 1$ và $x \neq 9$}$

$=\dfrac{2(\sqrt{x}-1)+2\sqrt{x}+\sqrt{x}(\sqrt{x}-3)}{(\sqrt{x}-3)(\sqrt{x}-1)}$

$=\dfrac{2\sqrt{x}-2+2\sqrt{x}+x-3\sqrt{x}}{(\sqrt{x}-3)(\sqrt{x}-1)}$

$=\dfrac{x+\sqrt{x}-2}{(\sqrt{x}-3)(\sqrt{x}-1)}$

$=\dfrac{(\sqrt{x}-1)(\sqrt{x}+2)}{(\sqrt{x}-3)(\sqrt{x}-1)}$

$=\dfrac{\sqrt{x}+2}{\sqrt{x}-3}$

b/ $\text{Để $A=\sqrt{3}$ thì: $\dfrac{\sqrt{x}+2}{\sqrt{x}-3}=\sqrt{3}$}$

$⇔ \sqrt{x}+2=\sqrt{3x}-3\sqrt{3}$

$⇔ \sqrt{3x}-\sqrt{x}=2+3\sqrt{3}$

$⇔ \sqrt{x}(\sqrt{3}-1)=2+3\sqrt{3}$

$⇔ \sqrt{x}=\dfrac{2+3\sqrt{3}}{\sqrt{3}-1}$

$⇔ x=\dfrac{4+9.3+12\sqrt{3}}{3+1-2\sqrt{3}}$

$⇔ x=\dfrac{31+12\sqrt{3}}{4-2\sqrt{3}}$

$⇔ x=\dfrac{(31+12\sqrt{3})(4+2\sqrt{3})}{(4-2\sqrt{3})(4+2\sqrt{3})}$

$⇔ x=\dfrac{124+48\sqrt{3}+62\sqrt{3}+72}{16-4.3}$

$⇔ x=\dfrac{98+55\sqrt{3}}{2}$

$\text{Vậy $x=\dfrac{98+55\sqrt{3}}{2}$ thì $A=\sqrt{3}$}$

Đáp án:

a) `(\sqrt{x}+2)/(\sqrt{x}-3)`

b) `x=(98+55\sqrt{3})/2`

Giải thích các bước giải:

a)

`x>=0;x\ne1;9`

Xét `x-4\sqrt{x}+3`

`=x-\sqrt{x}-3\sqrt{x}+3`

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

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

Ta có:

`A=2/(\sqrt{x}-3)+(2\sqrt{x})/(x-4\sqrt{x}+3)+\sqrt{x}/(\sqrt{x}-1)`

`=2/(\sqrt{x}-3)+(2\sqrt{x})/((\sqrt{x}-3)(\sqrt{x}-1))+\sqrt{x}/(\sqrt{x}-1)`

`=(2(\sqrt{x}-1)+2\sqrt{x}+\sqrt{x}(\sqrt{x}-3))/((\sqrt{x}-3)(\sqrt{x}-1))`

`=(2\sqrt{x}-2+2\sqrt{x}+x-3\sqrt{x})/((\sqrt{x}-3)(\sqrt{x}-1))`

`=(x+\sqrt{x}-2)/((\sqrt{x}-3)(\sqrt{x}-1))`

`=(x-\sqrt{x}+2\sqrt{x}-2)/((\sqrt{x}-3)(\sqrt{x}-1))`

`=(x(\sqrt{x}-1)+2(\sqrt{x}-1))/((\sqrt{x}-3)(\sqrt{x}-1))`

`=((\sqrt{x}-1)(\sqrt{x}+2))/((\sqrt{x}-3)(\sqrt{x}-1))`

`=(\sqrt{x}+2)/(\sqrt{x}-3)`

b) 

`A=\sqrt{3}`

`->(\sqrt{x}+2)/(\sqrt{x}-3)=\sqrt{3}`

`->\sqrt{3}(\sqrt{x}-3)=\sqrt{x}+2`

`->3(\sqrt{x}-3)^2=(\sqrt{x}+2)^2`

`->3(x-6\sqrt{x}+9)=x+4\sqrt{x}+4`

`->3x-18\sqrt{x}+27=x+4\sqrt{x}+4`

`->2x-22\sqrt{x}+23=0`

`->2(x-11\sqrt{x}+121/4)-75/2=0`

`->2(\sqrt{x}-11/2)^2=75/2`

`->(\sqrt{x}-11/2)^2=75/4`

`->`\(\left[ \begin{array}{l}\sqrt{x}-\dfrac{11}{2}=\dfrac{5\sqrt{3}}{2}\\\sqrt{x}-\dfrac{11}{2}=\dfrac{-5\sqrt{3}}{2}\end{array} \right.\) 

`->` \(\left[ \begin{array}{l}\sqrt{x}=\dfrac{5\sqrt{3}+11}{2}\\\sqrt{x}=\dfrac{-5\sqrt{3}+11}{2}\end{array} \right.\) 

`->` \(\left[ \begin{array}{l}x=\dfrac{98+55\sqrt{3}}{2}\\x=\dfrac{98-55\sqrt{3}}{2}\end{array} \right.\) 

Do phép bình phương chưa tương đương nên thử lại, nghiệm duy nhất là `(98+55\sqrt{3})/2`