Question

Cho biểu thức Q=\(\dfrac{5\sqrt{x}-11}{x+2\sqrt{x}-3}+\dfrac{3\sqrt{x}-2}{1-\sqrt{x}}-\dfrac{2\sqrt{x}-3}{\sqrt{x}+3}\)

a.Rút gọn Q

b.Chứng minh rằng Q\(\le\dfrac{2}{3}\)

Answer 1

a: \(A=\dfrac{5\sqrt{x}-11-\left(3x+9\sqrt{x}-2\sqrt{x}-6\right)-\left(2x-5\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{5\sqrt{x}-11-3x-7\sqrt{x}+6-2x+5\sqrt{x}-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{-5x+3\sqrt{x}-8}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

b: Q-2/3

\(=\dfrac{-15x+9\sqrt{x}-24-2\left(x+2\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{-17x+5\sqrt{x}-18}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}< 0\)

=>Q<2/3