Hướng dẫn trả lời:
c) `P = {sqrt{x} + 1}/{sqrt{x} - 2}` (ĐK: `x ≥ 0; x ne 4`)
Ta có: `x = {3 - sqrt{5}}/{2}`
`= {2*(3 - sqrt{5})}/{4}`
`= {6 - 2sqrt{5}}/{4}`
`= {5 - 2sqrt{5} + 1}/{4}`
`= {(sqrt{5})^2 - 2*sqrt{5}*1 + 1^2}/{2^2}`
`= {(sqrt{5} - 1)^2}/{2^2}`
`= ({sqrt{5} - 1}/{2})^2` (TMĐK)
`→ sqrt{x} = sqrt{({sqrt{5} - 1}/{2})^2}`
`= |sqrt{5} - 1|/{2}`
`= {sqrt{5} - 1}/{2}` (Vì `sqrt{5} > 1`)
Với `sqrt{x} = {sqrt{5} - 1}/{2}`, ta có:
`P = {{sqrt{5} - 1}/{2} + 1}/{{sqrt{5} - 1}/{2} - 2}`
`= {{sqrt{5} - 1}/{2} + 2/2}/{{sqrt{5} - 1}/{2} - 4/2}`
`= {{sqrt{5} - 1 + 2}/{2}}/{{sqrt{5} - 1 - 4}/{2}}`
`= {{sqrt{5} + 1}/{2}}/{{sqrt{5} - 5}/{2}}`
`= {sqrt{5} + 1}/{2}*{2}/{sqrt{5} - 5}`
`= {sqrt{5} + 1}/{sqrt{5} - 5}`
`= {(sqrt{5} + 1)*(sqrt{5} + 5)}/{(sqrt{5} + 5)*(sqrt{5} - 5)}`
`= {sqrt{5}*(sqrt{5} + 5) + 1*(sqrt{5} + 5)}/{(sqrt{5})^2 - 5^2}`
`= {5 + 5sqrt{5} + sqrt{5} + 5}/{5 - 25}`
`= {(5 + 5) + (5sqrt{5} + sqrt{5})}/{5 - 25}`
`= {10 + 6sqrt{5}}/{- 20}`
`= - {2*(5 + 3sqrt{5})}/{20}`
`= - {5 + 3sqrt{5}}/{10}`
