Question

cho A=\(\dfrac{2x+2}{\sqrt{x}}+\dfrac{2\sqrt{x}-1}{x-\sqrt{x}}-\dfrac{x\sqrt{x}+1}{x+\sqrt{x}}\)

a)Rút gọn

b)CMR a>5

c)tìm x để \(\dfrac{8}{A}\) nguyên

Answer 1

ĐKXĐ : \(x>0\) và \(x\ne1\) .

Câu a : \(A=\dfrac{2x+2}{\sqrt{x}}+\dfrac{2\sqrt{x}-1}{x-\sqrt{x}}-\dfrac{x\sqrt{x}+1}{x+\sqrt{x}}\)

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

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

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

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