\(a.P=\dfrac{x\sqrt{x}-47}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+1\right)}-\dfrac{4\sqrt{x}+12}{\sqrt{x}+1}+\dfrac{\sqrt{x}+2}{\sqrt{x}-3}=\dfrac{x\sqrt{x}-47-4\left(x-9\right)+\left(\sqrt{x}+2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+1\right)}=\dfrac{x\sqrt{x}-3x+3\sqrt{x}-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+1\right)}=\dfrac{\left(\sqrt{x}-3\right)\left(x+3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+1\right)}=\dfrac{x+3}{\sqrt{x}+1}\left(x\ne9;x\ge0\right)\)
\(b.P=\dfrac{x+3}{\sqrt{x}+1}=\dfrac{x-1+4}{\sqrt{x}+1}=\sqrt{x}-1+\dfrac{4}{\sqrt{x}+1}=\sqrt{x}+1+\dfrac{4}{\sqrt{x}+1}-2\)
Áp dụng BĐT Cauchy cho các số dương , ta có :
\(\sqrt{x}+1+\dfrac{4}{\sqrt{x}+1}\ge2\sqrt{\left(\sqrt{x}+1\right).\dfrac{4}{\sqrt{x}+1}}=2\sqrt{4}=4\)
\(\Leftrightarrow\sqrt{x}+1+\dfrac{4}{\sqrt{x}+1}-2\ge4-2=2\)
\(\Rightarrow P_{Min}=2."="\Leftrightarrow x=1\left(TM\right)\)