Question

Rút gọn biểu thức:

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

Answer 1

Với `x >= 0` có:

`P=x/[\sqrt{x}+1]+\sqrt{x}(1/[\sqrt{x}+1]+1/[\sqrt{x}+2])+2/[\sqrt{x}+2]`

`P=x/[\sqrt{x}+1]+\sqrt{x}/[\sqrt{x}+1]+\sqrt{x}/[\sqrt{x}+2]+2/[\sqrt{x}+2]`

`P=[\sqrt{x}(\sqrt{x}+1)]/[\sqrt{x}+1]+[\sqrt{x}+2]/[\sqrt{x}+2]`

`P=\sqrt{x}+1`

Answer 2

\(ĐK:x\ge0\\ P=\dfrac{x}{\sqrt{x}+1}+\sqrt{x}\cdot\dfrac{\sqrt{x}+2+\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)}+\dfrac{2}{\sqrt{x}+2}\\ P=\dfrac{x}{\sqrt{x}+1}+\dfrac{2x+3\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)}+\dfrac{2}{\sqrt{x}+2}\\ P=\dfrac{x\sqrt{x}+2x+2x+3\sqrt{x}+2\sqrt{x}+2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)}\\ P=\dfrac{x\sqrt{x}+4x+5\sqrt{x}+2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)}\\ P=\dfrac{x\sqrt{x}+2x+2x+4\sqrt{x}+\sqrt{x}+2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)}\\ P=\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}+1\right)^2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)}=\sqrt{x}+1\)