M=\(\frac{3x+3\sqrt{x}-3}{x+\sqrt{x}-2}-\frac{\sqrt{x}+1}{\sqrt{x}+2}+\frac{\sqrt{x}-2}{\sqrt{x}}\left(\frac{1}{1-\sqrt{x}}-1\right)\)
a) Rút gọn M
b) Tính giá trị của M khi x= \(4+2\sqrt{3}\)
c) Tìm x để M =\(\sqrt{x}\)
d) Tìm M để M > \(\frac{1}{2}\)
e) tìm số giá trị nguyên của x để M có giá trị nguyên
ĐKXĐ:...
\(M=\frac{3x+3\sqrt{x}-3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}-\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}+\frac{\sqrt{x}-2}{\sqrt{x}}\left(\frac{\sqrt{x}}{1-\sqrt{x}}\right)\)
\(=\frac{3x+3\sqrt{x}-3-x+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}-\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)
\(=\frac{2x+3\sqrt{x}-2-x+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}=\frac{x+3\sqrt{x}+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)
\(=\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}=\frac{\sqrt{x}+1}{\sqrt{x}-1}\)
\(x=4+2\sqrt{3}\Rightarrow\sqrt{x}=\sqrt{4+2\sqrt{3}}=\sqrt{3}+1\)
\(\Rightarrow P=\frac{\sqrt{3}+1+1}{\sqrt{3}+1-1}=\frac{2+\sqrt{3}}{\sqrt{3}}=\frac{3+2\sqrt{3}}{3}\)
Để \(M=\sqrt{x}\Leftrightarrow\sqrt{x}=\frac{\sqrt{x}+1}{\sqrt{x}-1}\)
\(\Leftrightarrow x-2\sqrt{x}-1=0\Rightarrow\left[{}\begin{matrix}\sqrt{x}=1+\sqrt{2}\\\sqrt{x}=1-\sqrt{2}< 0\left(l\right)\end{matrix}\right.\) \(\Rightarrow x=3+2\sqrt{2}\)
Để \(M>\frac{1}{2}\Rightarrow\frac{\sqrt{x}+1}{\sqrt{x}-1}-\frac{1}{2}>0\Rightarrow\frac{\sqrt{x}+3}{2\left(\sqrt{x}-1\right)}>0\)
\(\Rightarrow\sqrt{x}-1>0\Rightarrow x>1\)
Ta có: \(M=\frac{\sqrt{x}+1}{\sqrt{x}-1}=1+\frac{2}{\sqrt{x}-1}\)
Để M nguyên \(\Rightarrow\sqrt{x}-1=Ư\left(2\right)=\left\{-2;-1;1;2\right\}\)
\(\Rightarrow\sqrt{x}=\left\{2;3\right\}\Rightarrow x=\left\{4;9\right\}\)