Question

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

Answer 1

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 \(\Rightarrow\sqrt{x}-1=Ư\left(2\right)=\left\{-2;-1;1;2\right\}\)

\(\Rightarrow\sqrt{x}=\left\{2;3\right\}\Rightarrow x=\left\{4;9\right\}\)