Question

[Đĩ Nguyễn Con]

Cho biểu thức \(A=\frac{4}{2\sqrt{x}-x}\) và \(B=\frac{\sqrt{x}-4}{x-2\sqrt{x}}+\frac{3}{\sqrt{x}-2}\)

1, Rút gọn P = B : A

2, Tìm x để M ≥ 0 với \(M=P.\frac{1-\sqrt{x}}{\sqrt{x}-3}\)

Answer 1

[Nguyễn Huy Tú]

a, Ta có : \(B=\frac{\sqrt{x}-4}{x-2\sqrt{x}}+\frac{3}{\sqrt{x}-2}=\frac{\sqrt{x}-4+3\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-2\right)}=\frac{4\sqrt{x}-4}{\sqrt{x}\left(\sqrt{x}-2\right)}\)

\(P=\frac{B}{A}\Rightarrow P=\frac{4\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}-2\right)}.\frac{-\sqrt{x}\left(\sqrt{x}-2\right)}{4}=1-\sqrt{x}\)

b, Ta có : \(M=P.\frac{1-\sqrt{x}}{\sqrt{x}-3}\Rightarrow M=\frac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}-3}\ge0\)

\(\Rightarrow\sqrt{x}-3\ge0\Leftrightarrow\sqrt{x}\ge3\Leftrightarrow x\ge9\)vì \(\left(\sqrt{x}-1\right)^2\ge0\)