4:
a: ĐKXĐ: \(\left\{{}\begin{matrix}x>=0\\x\notin\left\{1;9\right\}\end{matrix}\right.\)
\(A=\dfrac{2}{\sqrt{x}-3}+\dfrac{2\sqrt{x}}{x-4\sqrt{x}+3}+\dfrac{\sqrt{x}}{\sqrt{x}-1}\)
\(=\dfrac{2}{\sqrt{x}-3}+\dfrac{2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-3\right)}+\dfrac{\sqrt{x}}{\sqrt{x}-1}\)
\(=\dfrac{2\sqrt{x}-2+2\sqrt{x}+x-3\sqrt{x}}{\left(\sqrt{x}-1\right)\cdot\left(\sqrt{x}-3\right)}\)
\(=\dfrac{x+\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-3\right)}=\dfrac{\sqrt{x}+2}{\sqrt{x}-3}\)
b: A=căn 3
=>\(\dfrac{\sqrt{x}+2}{\sqrt{x}-3}=\sqrt{3}\)
=>\(\sqrt{x}+2=\sqrt{3}\cdot\sqrt{x}-3\sqrt{3}\)
=>\(\sqrt{x}\left(1-\sqrt{3}\right)=-3\sqrt{3}-2\)
=>\(\sqrt{x}=\dfrac{3\sqrt{3}+2}{\sqrt{3}-1}=\dfrac{11+5\sqrt{3}}{2}\)
=>\(x=\dfrac{98+55\sqrt{3}}{2}\)
c: Để A nguyên thì \(\sqrt{x}-3+5⋮\sqrt{x}-3\)
=>\(\sqrt{x}-3\in\left\{1;-1;5;-5\right\}\)
=>\(\sqrt{x}\in\left\{4;2;8;-2\right\}\)
=>\(\sqrt{x}\in\left\{4;2;8\right\}\)
=>\(x\in\left\{16;4;64\right\}\)
