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प्रश्न
If x = 5 - 2√6, find `x^2 + 1/x^2`
उत्तर
Given `x = 5 - 2sqrt6`
We need to find `x^2 + 1/x^2`
Since x = `5 - 2sqrt6` , we have
`1/x = 1/[ 5 - 2sqrt6]`
⇒ `1/x = 1/[ 5 - 2sqrt6 ] xx [ 5 + 2sqrt6 ]/[ 5 + 2sqrt6 ]`
⇒ `1/x = ( 5 - 2sqrt6 )/[( 5 - 2sqrt6 )( 5 + 2sqrt6 )]`
⇒ `1/x = ( 5 + 2sqrt6)/ [ 5^2 - (2sqrt6)^2 ]`
⇒ `1/x = ( 5 + 2sqrt6)/ [ 25 - 24 ]`
⇒ `1/x = ( 5 + 2sqrt6)/1`
⇒ `1/x = ( 5 + 2sqrt6)` .....(1)
Thus,`( x - 1/x ) = ( 5 - cancel(2sqrt6) ) - ( 5 + cancel(2sqrt6))` = 10
`x^2 + 1/x^2 = (x + 1/x)^2 - 2`
`x^2 + 1/x^2= (10)^2 - 2`
`x^2 + 1/x^2= 100 - 2`
`x^2 + 1/x^2 = 98`
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