a.
\(P=\dfrac{2}{\sqrt{x}+2}-\dfrac{1}{\sqrt{x}-2}+\dfrac{2\sqrt{x}}{x-4}\)
\(=\dfrac{2\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}-\dfrac{\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\dfrac{2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{2\left(\sqrt{x}-2\right)-\left(\sqrt{x}+2\right)+2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{3\sqrt{x}-6}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{3\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{3}{\sqrt{x}+2}\)
\(Q=\sqrt{12}.\sqrt{3}-3\sqrt{6}.\sqrt{3}+\sqrt{3}.\sqrt{3}+3\sqrt{18}\)
\(=\sqrt{36}-3\sqrt{18}+3+3\sqrt{18}\)
\(=6+3=9\)
b.
\(\dfrac{P}{Q}=\dfrac{1}{9}\Leftrightarrow\dfrac{3}{9\left(\sqrt{x}+2\right)}=\dfrac{1}{9}\)
\(\Leftrightarrow\dfrac{3}{\sqrt{x}+2}=1\)
\(\Rightarrow\sqrt{x}+2=3\)
\(\Leftrightarrow\sqrt{x}=1\)
\(\Rightarrow x=1\)
