Đáp án:
Giải thích các bước giải:
`P=\frac{\sqrt{a}+1}{\sqrt{a}-2}+\frac{2\sqrt{a}}{\sqrt{a}+2}+\frac{2+5\sqrt{a}}{4-a}`
ĐK: `a \ge 0, a \ne 4`
a) `P=\frac{(\sqrt{a}+1)(\sqrt{a}+2)}{(\sqrt{a}-2)(\sqrt{a}+2)}+\frac{2\sqrt{a}(\sqrt{a}-2)}{(\sqrt{a}-2)(\sqrt{a}+2)}-\frac{2+5\sqrt{a}}{(\sqrt{a}-2)(\sqrt{a}+2)}`
`P=\frac{a+3\sqrt{a}+2+2a-4\sqrt{a}-2-5\sqrt{a}}{(\sqrt{a}-2)(\sqrt{a}+2)}`
`P=\frac{3a-6\sqrt{a}}{(\sqrt{a}-2)(\sqrt{a}+2)}`
`P=\frac{3\sqrt{a}(\sqrt{a}-2)}{(\sqrt{a}-2)(\sqrt{a}+2)}`
`P=\frac{3\sqrt{a}}{\sqrt{a}+2}`
b) `x=3-2\sqrt{2}`
`x=2+1-2\sqrt{2}`
`x=(\sqrt{2}-1)^2`
`⇒ \sqrt{x}=\sqrt{(\sqrt{2}-1)^2}=|\sqrt{2}-1|=\sqrt{2}-1`
Thay vào P ta có:
`P=\frac{3(\sqrt{2}-1)}{\sqrt{2}-1+2}=\frac{3\sqrt{2}-3}{\sqrt{2}+1}=9-6\sqrt{2}`
Vậy với `x=3-2\sqrt{2}` thì `P=9-6\sqrt{2}`
c) `P=2`
`⇔ \frac{3\sqrt{a}}{\sqrt{a}+2}=2`
`⇔ 3\sqrt{a}=2(\sqrt{a}+2)`
`⇔ 3\sqrt{a}-2\sqrt{a}=4`
`⇔ \sqrt{a}=4`
`⇔ a=16\ (TM)`
Vậy khi `a=16` thì `P=2`
