Question

[Nguyễn Thị Thanh Tâm]

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

a) Pút gọn P

b) Tìm x để P có giá trị nguyên

Answer 1

[Nguyễn Huy Tú]

a, Với  \(x\ge0;x\ne1\)

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

\(=\left(\frac{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}+\frac{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}\right):\left(\frac{2\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}\pm1\right)}\right)\)

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

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