Đáp án+Giải thích các bước giải:
a,
`P=\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}+1}{x+\sqrt{x}+1}-\frac{1}{\sqrt{x}-1}(x≥0;x\ne1)`
`P=\frac{x+2+(\sqrt{x}+1)(\sqrt{x}-1)-(x+\sqrt{x}+1)}{(\sqrt{x}-1)(x+\sqrt{x}+1)}`
`P=\frac{x+2+x-1-x-\sqrt{x}-1}{(\sqrt{x}-1)(x+\sqrt{x}+1)}`
`P=\frac{x-\sqrt{x}}{(\sqrt{x}-1)(x+\sqrt{x}+1)}`
`P=\frac{\sqrt{x}(\sqrt{x}-1)}{(\sqrt{x}-1)(x+\sqrt{x}+1)}`
`P=\frac{\sqrt{x}}{x+\sqrt{x}+1}`
b,
`x=6-2\sqrt{5}`
`⇔\sqrt{x}=\sqrt{6-2\sqrt{5}}`
`⇔\sqrt{x}=\sqrt{5-2.\sqrt{5}.1+1}`
`⇔\sqrt{x}=\sqrt{(\sqrt{5}-1)^2}`
`⇔\sqrt{x}=\sqrt{5}-1`
Thay giá trị `x` và `\sqrt{x}` vào P` ta có:
`P=\frac{\sqrt{5}-1}{6-2\sqrt{5}+\sqrt{5}-1+1}`
`P=\frac{\sqrt{5}-1}{6-\sqrt{5}}`
`P=\frac{-1+5\sqrt{5}}{31}`
Vậy tại `x=6-2\sqrt{5}` thì `P=\frac{-1+5\sqrt{5}}{31}`
c,
`P<\frac{1}{3}`
`⇔\frac{\sqrt{x}}{x+\sqrt{x}+1}<\frac{1}{3}`
`⇔\frac{\sqrt{x}}{x+\sqrt{x}+1}-\frac{1}{3}<0`
`⇔\frac{3\sqrt{x}-x-\sqrt{x}-1}{3(x+\sqrt{x}+1)}<0`
`⇔\frac{-x+2\sqrt{x}-1}{3(x+\sqrt{x}+1)}<0`
`x+\sqrt{x}+1≥1`
`⇔3(x+\sqrt{x}+1)≥3>0`
`⇒-x+2\sqrt{x}-1<0`
`⇒-(\sqrt{x}-1)^2<0`
`⇔(\sqrt{x}-1)^2>0`
`\sqrt{x}-1\ne0`
`⇔\sqrt{x}\ne1`
`⇔x\ne1`
Vậy `P<\frac{1}{3}` khi `x≥0;x\ne1`
