Đáp án+Giải thích các bước giải:
a,
`P=\frac{\sqrt{x}+1}{\sqrt{x}}+\frac{2}{1-\sqrt{x}}+\frac{3\sqrt{x}+1}{x-\sqrt{x}}(x>0;x\ne1)`
`P=\frac{\sqrt{x}+1}{\sqrt{x}}-\frac{2}{\sqrt{x}-1}+\frac{3\sqrt{x}+1}{x-\sqrt{x}}`
`P=\frac{(\sqrt{x}+1)(\sqrt{x}-1)-2\sqrt{x}+3\sqrt{x}+1}{\sqrt{x}(\sqrt{x}-1)}`
`P=\frac{x-1+\sqrt{x}+1}{\sqrt{x}(\sqrt{x}-1)}`
`P=\frac{x+\sqrt{x}}{\sqrt{x}(\sqrt{x}-1)}`
`P=\frac{\sqrt{x}(\sqrt{x}+1)}{\sqrt{x}(\sqrt{x}-1)}`
`P=\frac{\sqrt{x}+1}{\sqrt{x}-1}(đpcm)`
b,
`x=4+2\sqrt{3}(tm)`
`⇒\sqrt{x}=\sqrt{4+2\sqrt{3}}`
`⇔\sqrt{x}=\sqrt{3+2.\sqrt{3}.1+1}`
`⇔\sqrt{x}=\sqrt{(\sqrt{3}+1)^2}`
`⇔\sqrt{x}=|\sqrt{3}+1|`
`⇔\sqrt{x}=\sqrt{3}+1`
Thay `\sqrt{x}=\sqrt{3}+1` vào `P` ta có:
`P=\frac{\sqrt{3}+1+1}{\sqrt{3}+1-1}`
`P=\frac{\sqrt{3}+2}{\sqrt{3}}`
`P=\frac{\sqrt{3}(\sqrt{3}+2)}{3}`
`P=\frac{3+2\sqrt{3}}{3}`
Vậy tại `x=4+2\sqrt{3}` thì `P=\frac{3+2\sqrt{3}}{3}`
