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If x=sqrt(2)+1, then find the values of ...

If `x=sqrt(2)+1`, then find the values of the following :
`(i)(1)/(x)" "(ii)x+(1)/(x)" "(iii)x-(1)/(x)" "(iv)x^(2)+(1)/(x^(2))`

Text Solution

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The correct Answer is:
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(i) `(1)/(x)=(1)/(sqrt(2)+1)=(1)/(sqrt(2)+1)xx(sqrt(2)-1)/(sqrt(2)-1)=(sqrt(2)-1)/((sqrt(2))^(2)-(1)^(2))=(sqrt(2)-1)/(2-1)=sqrt(2)-1`
(ii) `x+(1)/(x)=(sqrt(2)+1)+(sqrt(2)-1)=2sqrt(2)`
(iii) `x-(1)/(x)=(sqrt(2)+1)-(sqrt(2)-1)=2`
(iv) `x+(1)/(x)=2sqrt(2)`
`rArr" "(x+(1)/(x))^(2)=(2sqrt(2))^(2)" "rArr" "x^(2)+(1)/(x^(2))+2=8`
`rArr" "x^(2)+(1)/(x^(2))=6`
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