SOLUTION: if x=2+root3/2-root3 then find the value of x+1/x

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Question 1113566: if x=2+root3/2-root3 then find the value of x+1/x
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
  x = %282%2Bsqrt%283%29%29%2F%282-sqrt%283%29%29 = rationalize the fraction = %282%2Bsqrt%283%29%29%2F%282-sqrt%283%29%29 . %282%2Bsqrt%283%29%29%2F%282%2Bsqrt%283%29%29  = %282%2Bsqrt%283%29%29%5E2%2F%282%5E2+-+%28sqrt%283%29%29%5E2%29  =  %284+%2B+4%2Asqrt%283%29+%2B+3%29%2F%284-3%29 = 7%2B4%2Asqrt%283%29 .


Then  x + 1%2Fx = 7%2B4%2Asqrt%283%29 + 1%2F%287%2B4%2Asqrt%283%29%29 = rationalize the fraction = 7%2B4%2Asqrt%283%29 + 1%2F%287%2B4%2Asqrt%283%29%29 . %287-4%2Asqrt%283%29%29%2F%287-4%2Asqrt%283%29%29 = 


= 7%2B4%2Asqrt%283%29  + %287-4%2Asqrt%283%29%29%2F%287%5E2+-+%284%2Asqrt%283%29%29%5E2%29 = 7%2B4%2Asqrt%283%29 + %287-4%2Asqrt%283%29%29%2F%2849+-+48%29 = 7%2B4%2Asqrt%283%29 + 7-4%2Asqrt%283%29 = 7 + 7 = 14.


Answer.  14.

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On rationalizing fractions, see the lessons
    - HOW TO rationalize a fraction by making its denominator free of square roots
    - HOW TO rationalize a fraction by making its denominator free of cubic roots
    - Amazing calculations with fractions that contain quadratic irrationalities in denominators
in this site.