a: \(P=\frac{\sqrt{x}-3}{2-\sqrt{x}}+\frac{\sqrt{x}-2}{3+\sqrt{x}}-\frac{9-x}{x+\sqrt{x}-6}\)
\(=\frac{-\left(\sqrt{x}-3\right)}{\sqrt{x}-2}+\frac{\sqrt{x}-2}{\sqrt{x}+3}+\frac{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}\)
\(=\frac{-\left(\sqrt{x}-3\right)}{\sqrt{x}-2}+\frac{\sqrt{x}-2}{\sqrt{x}+3}+\frac{\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-2\right)}=\frac{\sqrt{x}-2}{\sqrt{x}+3}\)
b: \(P=\frac{7}{12}\)
=>\(\frac{\sqrt{x}-2}{\sqrt{x}+3}=\frac{7}{12}\)
=>\(12\left(\sqrt{x}-2\right)=7\left(\sqrt{x}+3\right)\)
=>\(12\sqrt{x}-24=7\sqrt{x}+21\)
=>\(5\sqrt{x}=45\)
=>\(\sqrt{x}=9\)
=>x=81(nhận)
c: \(P>\frac12\)
=>\(P-\frac12>0\)
=>\(\frac{\sqrt{x}-2}{\sqrt{x}+3}-\frac12>0\)
=>\(\frac{2\sqrt{x}-4-\sqrt{x}-3}{2\left(\sqrt{x}+3\right)}>0\)
=>\(\sqrt{x}-7>0\)
=>\(\sqrt{x}>7\)
=>x>49
d: Để 1/P nguyên thì \(\sqrt{x}+3\) ⋮\(\sqrt{x}-2\)
=>\(\sqrt{x}-2+5\) ⋮\(\sqrt{x}-2\)
=>5⋮\(\sqrt{x}-2\)
=>\(\sqrt{x}-2\in\left\lbrace1;-1;5\right\rbrace\)
=>\(\sqrt{x}\in\left\lbrace3;1;7\right\rbrace\)
=>x∈{1;9;49}
e: Để P nguyên thì \(\sqrt{x}-2\vdots\sqrt{x}+3\)
=>\(\sqrt{x}+3-5\vdots\sqrt{x}+3\)
=>\(-5\vdots\sqrt{x}+3\)
=>\(\sqrt{x}+3=5\)
=>\(\sqrt{x}=2\)
=>x=4(loại)
