Question

 

Cho biểu thức : `C = (3x+\sqrt(9x)-3)/(x+\sqrt(x)-2)-(\sqrt(x)+1)/(\sqrt(x)+2)-(\sqrt(x)-2)/(\sqrt(x)-1)`

`a.` Rút gọn biểu thức A

`b.` Tính giá trị của `A` khi `x = 3 + 2\sqrt(2)`

 

Answer 1

`a)`

`C = (3x + \sqrt{9x} - 3)/(x + \sqrt{x} - 2) - (\sqrt{x}+1)/(\sqrt{x} + 2) - (\sqrt{x} - 2)/(\sqrt{x} - 1)`

`(x >= 0 ; x` $\neq$ `1)`

`C = (3x + 3\sqrt{x} - 3)/((\sqrt{x} - 1)(\sqrt{x} + 2)) - ((\sqrt{x} + 1)(\sqrt{x} - 1))/((\sqrt{x}- 1)(\sqrt{x} + 2)) - ((\sqrt{x} - 2)(\sqrt{x} + 2))/((\sqrt{x} - 1)(\sqrt{x} + 2))`

`C = (3x + 3\sqrt{x} - 3 - x + 1 - x + 4)/((\sqrt{x} - 1)(\sqrt{x} + 2))`

`C = (3x + 3\sqrt{x} - 2x + 2)/((\sqrt{x} - 1)(\sqrt{x} + 2))`

`C = (x + 3\sqrt{x} + 2)/((\sqrt{x} - 1)(\sqrt{x} + 2))`

`C = ((\sqrt{x} + 1)(\sqrt{x} + 2))/((\sqrt{x} - 1)(\sqrt{x} + 2))`

`C = (\sqrt{x} + 1)/(\sqrt{x} - 1)`

`b)`

`x = 3 + 2\sqrt{2} = 2 + 2\sqrt{2} + 1 = (\sqrt{2} + 1)^2`

`=> \sqrt{x} = \sqrt{(\sqrt{2} + 1)^2} = \sqrt{2} + 1`

`C = (\sqrt{2} + 1 + 1)/(\sqrt{2} - 1 + 1)`

`C = (\sqrt{2} + 2)/(\sqrt{2})`

`C = 1 + \sqrt{2}`