Question

rút gọn biểu thức với x ≥0 và x ≠ 1

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

sau khi rút gọn, hãy tìm GTLN của biểu thức A

Answer 1

\(=\left(\dfrac{\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\dfrac{\sqrt{x}+2}{\left(\sqrt{x}-1\right)^2}\right).\dfrac{\left[\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\right]^2}{2}\)
\(=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}+2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)^2}.\dfrac{\left(\sqrt{x}-1\right)^2\left(\sqrt{x}+1\right)^2}{2}\)
\(=\left(x-\sqrt{x}\right)-2\sqrt{x}+2-x-\sqrt{x}-2\sqrt{x}.\dfrac{\left(\sqrt{x}+1\right)}{2}\)
\(=-6\sqrt{x}.\dfrac{\left(\sqrt{x}+1\right)}{2}=-3x-3\sqrt{x}\)