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30 tháng 4 2019

\(P=\left(\frac{x\sqrt{x}-1}{x-\sqrt{x}}-\frac{x\sqrt{x}+1}{x+\sqrt{x}}\right):\left(\frac{2\left(x-2\sqrt{x}+1\right)}{x-1}\right)\)

ĐKXĐ : \(\hept{\begin{cases}x>0\\x\ne1\end{cases}}\)

a, \(P=\left(\frac{\left(x\sqrt{x}-1\right)\left(x+\sqrt{x}\right)-\left(x\sqrt{x}+1\right)\left(x-\sqrt{x}\right)}{\left(x-\sqrt{x}\right)\left(x+\sqrt{x}\right)}\right):\left(\frac{2\left(\sqrt{x}-1\right)^2}{x-1}\right)\)

\(\Leftrightarrow P=\left(\frac{x^2\sqrt{x}+x^2-x-\sqrt{x}-x^2\sqrt{x}+x^2-x+\sqrt{x}}{x\left(x-1\right)}\right):\left(\frac{2\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)

\(\Leftrightarrow P=\frac{2x\left(x-1\right)\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{2\left(\sqrt{x}-1\right)^2x\left(x-1\right)}\)

\(\Leftrightarrow P=\frac{\sqrt{x}+1}{\sqrt{x}-1}\)

b,\(P=\frac{\sqrt{x}-1+2}{\sqrt{x}-1}\)

Để P thuộc Z

\(\Rightarrow2⋮\sqrt{x}-1\)

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

\(\Leftrightarrow\sqrt{x}\in\left(2;0;3;-1\right)\)

\(\Leftrightarrow x=0\)(ko t/m đkxđ)

Vậy ko có x nguyên để P nguyên

26 tháng 12 2020

\(P=\left(\frac{x\sqrt{x}-1}{x-\sqrt{x}}-\frac{x\sqrt{x}+1}{x+\sqrt{x}}\right):\left(\frac{2\left(x-2\sqrt{x}+1\right)}{x-1}\right)\)

\(=\left[\frac{\left(x\sqrt{x}-1\right)\left(x+\sqrt{x}\right)}{\left(x-\sqrt{x}\right)\left(x+\sqrt{x}\right)}-\frac{\left(x\sqrt{x}+1\right)\left(x-\sqrt{x}\right)}{\left(x-\sqrt{x}\right)\left(x+\sqrt{x}\right)}\right]:\left[\frac{2\left(\sqrt{x}-1\right)^2}{x-1}\right]\)

Phương trình tương đương : 

\(=\frac{2x^2-2x}{x^2-x}:\frac{2\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=2:\frac{2\left(\sqrt{x}-1\right)}{\sqrt{x}+1}=\frac{2\left(\sqrt{x}+1\right)}{2\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}+1}{\sqrt{x}-1}\)

13 tháng 8 2019

\(đkxđ\Leftrightarrow\hept{\begin{cases}x\ge0\\x\ne1\end{cases}}\)

\(P=\left(\sqrt{x}-\frac{x+2}{\sqrt{x}+1}\right):\left(\frac{\sqrt{x}}{\sqrt{x}+1}-\frac{\sqrt{x}-4}{1-x}\right).\)

\(=\left(\frac{\sqrt{x}\left(\sqrt{x}+1\right)-x+2}{\sqrt{x}+1}\right):\)\(\left(\frac{\sqrt{x}}{\sqrt{x}+1}+\frac{\sqrt{x}-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)

\(=\left(\frac{x+\sqrt{x}-x+2}{\sqrt{x}+1}\right):\left(\frac{\sqrt{x}\left(\sqrt{x}-1\right)+\sqrt{x}-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)

\(=\left(\frac{\sqrt{x}+2}{\sqrt{x}+1}\right):\left(\frac{x-\sqrt{x}+\sqrt{x}-4}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right)\)

\(=\frac{\left(\sqrt{x}+2\right)\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)\(=\frac{\sqrt{x}-1}{\sqrt{x}+2}\)

Để P âm \(\Rightarrow\frac{\sqrt{x}-1}{\sqrt{x}+2}< 0\)

Mà \(\sqrt{x}+2>0\forall x\Rightarrow\sqrt{x}-1< 0\Rightarrow x< 1\)

Để \(P\in Z\Rightarrow\frac{\sqrt{x}-1}{\sqrt{x}+2}\in Z\)

\(\Rightarrow1-\frac{3}{\sqrt{x}+2}\in Z\Rightarrow\frac{3}{\sqrt{x}+2}\in Z\)

\(\Rightarrow\sqrt{x}+2\inƯ_3\)

Mà \(\sqrt{x}+2\ge2\Rightarrow\sqrt{x}+2=3\Rightarrow x=1\)

Mà để \(P\in Z^-\Rightarrow\hept{\begin{cases}x< 1\\x=1\end{cases}}\)\(\Rightarrow x\in\varnothing\)

Vậy không có giá trị nào của x để P nguyên âm