a.
\(x=4\Rightarrow A=\dfrac{\sqrt{4}-5}{\sqrt{4}+3}=-\dfrac{3}{5}\)
b.
\(B=\dfrac{2\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}-\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}-\dfrac{2\sqrt{x}+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{2x+4\sqrt{x}-\left(x-1\right)-\left(2\sqrt{x}+4\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{x+2\sqrt{x}-3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}=\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{\sqrt{x}+3}{\sqrt{x}+2}\)
c.
\(P=AB=\dfrac{\sqrt{x}-5}{\sqrt{x}+3}.\dfrac{\sqrt{x}+3}{\sqrt{x}+2}=\dfrac{\sqrt{x}-5}{\sqrt{x}+2}=\dfrac{\sqrt{x}+2-7}{\sqrt{x}+2}=1-\dfrac{7}{\sqrt{x}+2}\)
P nguyên khi \(\dfrac{7}{\sqrt{x}+2}\) nguyên \(\Rightarrow\sqrt{x}+2=Ư\left(7\right)\)
Mà \(\sqrt{x}+2\ge2;\forall x\ge0\)
\(\Rightarrow\sqrt{x}+2=7\)
\(\Rightarrow\sqrt{x}=5\Rightarrow x=25\)
