a) \(P=\dfrac{3m+\sqrt{9m}-3}{m+\sqrt{m}-2}-\dfrac{\sqrt{m}-2}{\sqrt{m}-1}+\dfrac{1}{\sqrt{m}+2}-1\)
\(=\dfrac{3m+3\sqrt{m}-3}{\left(\sqrt{m}-1\right)\left(\sqrt{m}+2\right)}-\dfrac{m-4}{\left(\sqrt{m}-1\right)\left(\sqrt{m}+2\right)}+\dfrac{\sqrt{m}-1}{\left(\sqrt{m}-1\right)\left(\sqrt{m}+2\right)}-\dfrac{m+\sqrt{m}-2}{\left(\sqrt{m}-1\right)\left(\sqrt{m}+2\right)}\)
\(=\dfrac{3m+3\sqrt{m}-3-m+4+\sqrt{m}-1-m-\sqrt{m}+2}{\left(\sqrt{m}-1\right)\left(\sqrt{m}+2\right)}\)
\(=\dfrac{m+3\sqrt{m}+2}{\left(\sqrt{m}-1\right)\left(\sqrt{m}+2\right)}=\dfrac{\left(\sqrt{m}+1\right)\left(\sqrt{m}+2\right)}{\left(\sqrt{m}-1\right)\left(\sqrt{m}+2\right)}=\dfrac{\sqrt{m}+1}{\sqrt{m}-1}\)
b) Đk: \(\left\{{}\begin{matrix}m\ge0\\m\ne1\end{matrix}\right.\)
\(\left|P\right|=2\Leftrightarrow\left|\dfrac{\sqrt{m}+1}{\sqrt{m}-1}\right|=2\Leftrightarrow\left[{}\begin{matrix}\dfrac{\sqrt{m}+1}{\sqrt{m}-1}=-2\\\dfrac{\sqrt{m}+1}{\sqrt{m}-1}=2\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{m}+1=-2\sqrt{m}+2\\\sqrt{m}+1=2\sqrt{m}-2\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}3\sqrt{m}=1\\\sqrt{m}=3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}m=\dfrac{1}{9}\left(N\right)\\m=9\left(N\right)\end{matrix}\right.\)
c) \(P=\dfrac{\sqrt{m}+1}{\sqrt{m}-1}=\dfrac{\sqrt{m}-1+2}{\sqrt{m}-1}=1+\dfrac{2}{\sqrt{m}-1}\)
\(P\in N\Rightarrow\dfrac{2}{\sqrt{m}-1}\in Z\)
\(\Rightarrow\left[{}\begin{matrix}\sqrt{m}-1=-2\\\sqrt{m}-1=-1\\\sqrt{m}-1=1\\\sqrt{m}-1=2\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\sqrt{m}=-1\left(VN\right)\\\sqrt{m}=0\left(1\right)\\\sqrt{m}=2\left(VN,m\ne N\right)\\\sqrt{m}=3\left(2\right)\end{matrix}\right.\)
(1) \(\Leftrightarrow m=0\left(loại,P\notin N\right)\)
(2) \(\Leftrightarrow m=9\left(N\right)\)
Kl: a) \(P=\dfrac{\sqrt{m}+1}{\sqrt{m}-1}\)
b) m=1/9 , m = 9
c) m =9
