Đáp án+Giải thích các bước giải:
`P=(x-2sqrtx)/(xsqrtx-1)+(sqrtx+1)/(xsqrtx+x+sqrtx)+(1+2x-sqrtx)/(x^2-sqrtx)(x>0,x ne 1)`
Ta có:`x^2-sqrtx=sqrtx(xsqrtx-1)`
`=sqrtx(sqrtx-1)(x+sqrtx+1)`
`=>P=(x-2sqrtx)/((sqrtx-1)(x+sqrtx+1))+(sqrtx+1)/(sqrtx(x+sqrtx+1))+(2x-2sqrtx+1)/(sqrtx(sqrtx-1)(x+sqrtx+1))`
`P=(sqrtx(x-2sqrtx))/(sqrtx(sqrtx-1)(x+sqrtx+1))+((sqrtx+1)(sqrtx-1))/(sqrtx(sqrtx-1)(x+sqrtx+1))+(2x-2sqrtx+1)/(sqrtx(sqrtx-1)(x+sqrtx+1))`
`P=(xsqrtx-2x+x-1+2x-2sqrtx+1)/(sqrtx(sqrtx-1)(x+sqrtx+1))`
`P=(xsqrtx+x-2sqrtx)/(sqrtx(sqrtx-1)(x+sqrtx+1))`
`P=(sqrtx(x+sqrtx-2))/(sqrtx(sqrtx-1)(x+sqrtx+1))`
`P=(x+sqrtx-2)/((sqrtx-1)(x+sqrtx+1))`
`P=((sqrtx-1)(sqrtx+2))/((sqrtx-1)(x+sqrtx+1))`
`P=(sqrtx+2)/(x+sqrtx+1)`
Vì `x>0=>{(sqrtx+2>2>0),(x+sqrtx+1>1>0):}`
`=>P>0`
Xét `P-2`
`=(sqrtx+2-2x-2sqrtx-2)/(x+sqrtx+1)`
`=(-2x-sqrtx)/(x+sqrtx+1)<0`
`=>P-2<0`
`<=>P<2`
`=>0<P<2`
`=>P=1` do P nguyên
`<=>(sqrtx+2)/(x+sqrtx+1)=1`
`<=>sqrtx+2=x+sqrtx+1`
`<=>x=1(Loại)`
Vậy không có giá trị nào của x để`P` là một số nguyên.
