if x=2+√3 ,find the value of x²+ 1/x²
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Answered by
1088
if x= 2+sqrt3
then
1/x=2-sqrt3 (by rationaliztion)
then (x+1/x)^2 = X^2+(1/X)^2 +2*x*(1/X)
that is (2+sqrt3 +2-sqrt3)^2=x^2 +(1/x)^2+2
4^2 -2=x^2+(1/x)^2
i.e, 14 is the answer
then
1/x=2-sqrt3 (by rationaliztion)
then (x+1/x)^2 = X^2+(1/X)^2 +2*x*(1/X)
that is (2+sqrt3 +2-sqrt3)^2=x^2 +(1/x)^2+2
4^2 -2=x^2+(1/x)^2
i.e, 14 is the answer
Answered by
1126
x²+1/x² = (2+√3)²+1/(2+√3)² = (4+3+4√3)+1/(4+3+4√3) = (7+4√3)+1/(7+4√3) = (7+4√3)²+1/(7+4√3) = (49+48+56√3+1)/(7+4√3) = [(98+56√3)/(7+4√3)]×[(7-4√3)/(7-4√3)] = [(98+56√3)(7-4√3)]/7²-(4√3)² = (686+392√3-392√3-672)/(49-48) = 14/1 = 14
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