If x=2+√3, then (x+1/x) equals to:
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x = 2 + root 3
x + 1/x = (x^2 +1) / x
= [(2 + root 3)^2 +1 ]/ (2 + root 3)
= (4 + 3 + 4 root 3 + 1)/(2 + root 3)
= (8 + 4 root 3)/(2 + root 3)
= 4(2 + root 3)/(2 + root 3)
= 4
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