Question

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

a) tìm x để biểu thức P có nghĩa b) rút gọn P c) tìm x để P là một số nguyên

Answer 1

\(Đkxđ:x>1\)

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

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

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

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

Để \(P\) nguyên \(\Leftrightarrow\frac{2}{\sqrt{x-1}}\) nguyên

\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x-1}\inƯ\left(2\right)\\\sqrt{x-1}>0\end{matrix}\right.\)

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

\(\Leftrightarrow\left\{{}\begin{matrix}x=2\\x=5\end{matrix}\right.\)

Vậy ..........