Question: if x=2+√3 then what is the value of x+1/x.
Starting with 1/x
Multiply top and bottom by 1 = (2-√3)/(2-√3).
1/x = 1/(2+√3) = 1/(2+√3)*(2-√3)/(2-√3) = (2-√3)/(2^2 - (√3)^2) = (2-√3)/(4 - 3) = 2 - √3
i.e. 1/x = 2 - √3
now add on x, to get
x + 1/x = 2 + √3 + 2 - √3 = 4
Answer: x + 1/x = 4