Đáp án:
`a.`Rút gọn:
`M=\frac{sqrtx}{sqrtx-2}-\frac{4sqrtx-4}{sqrtx(sqrtx-2)}` (`x>0, x\ne4`)
`= \frac{x-(4sqrtx-4)}{sqrtx(sqrtx-2)}`
`= \frac{x-4sqrtx+4}{sqrtx(sqrtx-2)}`
`= \frac{(sqrtx-2)^2}{sqrtx(sqrtx-2)}`
`= \frac{sqrtx-2}{sqrtx}`
`= \frac{sqrtx-2}{sqrtx}xx\frac{sqrtx}{sqrtx}`
`= \frac{(sqrtx-2}sqrtx}{sqrtx xx sqrtx}`
`= \frac{x-2sqrtx}{x}`
`b.`Tính giá trị:
Thay `3+2sqrt2` (tm) vào biểu thức `M`, ta được:
`M=\frac{3+2sqrt2-2sqrt{3+2sqrt2}}{3+2sqrt2}`
`= \frac{3+2sqrt2-2sqrt{(1+sqrt2)^2}}{3+2sqrt2}`
`= \frac{3+2sqrt2-2(1+sqrt2)}{3+2sqrt2}`
`= \frac{3+2sqrt2-2-2sqrt2}{3+2sqrt2}`
`= \frac{1}{3+2sqrt2}`
`= \frac{3-2sqrt2}{9-4xx2}`
`= 3-2sqrt2`
`c.`Tìm giá trị của `x`:
Để `M>0` thì `\frac{x-2sqrt2}{x}>0`
`<=> \frac{sqrtx(sqrtx-2)}{x}>0`
`<=>`\(\left[ \begin{array}{l}\begin{cases}\sqrt{x}(\sqrt{x}-2)>0\\x>0\end{cases}\\\begin{cases}\sqrt{x}(\sqrt{x}-2)<0\\x<0\end{cases}\end{array} \right.\)
`<=>`\(\left[ \begin{array}{l}\begin{cases}x>4\\x>0\end{cases}\\\begin{cases}x<0\\x<4\end{cases}\end{array} \right.\)
`<=>`\(\left[ \begin{array}{l}x>4\\x<0\end{array} \right.\)
`<=> x>4`
