Question

Cho biểu thức: P = \(\left(\frac{2+\sqrt{x}}{2-\sqrt{x}}-\frac{2-\sqrt{x}}{2+\sqrt{x}}+\frac{4x}{x-4}\right):\frac{\sqrt{x}-3}{2\sqrt{x}-x}\)
a) Rút gọn P
b) Tìm x để P < 0; P > 0

Answer 1

ĐKXĐ: \(x>0;x\ne\left\{4;9\right\}\)

\(P=\left(\frac{-\left(\sqrt{x}+2\right)^2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\frac{\left(\sqrt{x}-2\right)^2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\frac{4x}{x-4}\right).\frac{\sqrt{x}\left(2-\sqrt{x}\right)}{\sqrt{x}-3}\)

\(=\left(\frac{-x-4\sqrt{x}-4+x-4\sqrt{x}+4+4x}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\right).\frac{\sqrt{x}\left(2-\sqrt{x}\right)}{\sqrt{x}-3}\)

\(=\left(\frac{4\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\right).\frac{\sqrt{x}\left(\sqrt{x}-2\right)}{3-\sqrt{x}}\)

\(=\frac{4x\left(\sqrt{x}-2\right)^2}{3-\sqrt{x}}\)

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