Question

Cho P= (3x +√9 - 3)/(x + √x - 2) - ( √x - 2)/(√x - 1) + 1/(√x +2) - 1

a) rút gọn P

b) tìm x để P² > P

c) tìm các gt của x thuộc N để P thuộc N

Answer 1

a: \(P=\dfrac{3x+3\sqrt{x}-3}{x+\sqrt{x}-2}-\dfrac{\sqrt{x}-2}{\sqrt{x}-1}+\dfrac{1}{\sqrt{x}+2}-1\)

\(=\dfrac{3x+3\sqrt{x}-3-x+4+\sqrt{x}-1-x-\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{x+3\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}=\dfrac{\sqrt{x}+1}{\sqrt{x}-1}\)

b: Để \(P^2>P\) thì P(P-1)>0

\(\Leftrightarrow\left[{}\begin{matrix}P>1\\P< 0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\dfrac{\sqrt{x}+1-\sqrt{x}+1}{\sqrt{x}-1}>0\\\sqrt{x}-1< 0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x>1\\x< 1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x>=0\\x< >1\end{matrix}\right.\)