Question

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

a) Tìm ĐKXĐ

b) Rút gọn K

c) Tính a để K có giá trị âm

Answer 1

a) + b) ĐKXĐ : \(x>0,x\ne1\)

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

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

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

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

c) Để K âm thì : \(\frac{x-1}{\sqrt{x}}< 0\Leftrightarrow x-1< 0\Leftrightarrow x< 1\)

Kết hợp với ĐKXĐ \(\Leftrightarrow x>11\)