Question

if x is equal to 8+3√7 then find the value of x^2 + 1 /x^2

Answer 1

x = 8 + 3√7         ....( i )

∴ 1 / x = 1 / ( 8 + 3√7 )

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Multiply and divide by 8 - 3√7

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⇒ 1 / x = ( 8 - 3√7 ) / { ( 8 + 3√7 ) ( 8 - 3√7 )

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Identity : ( a + b ) ( a - b ) = a^2 - b^2

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⇒ 1 / x = ( 8 - 3√7 ) / { ( 8 )^2 - ( 3√7 )^2 }

⇒ 1 / x =  ( 8 - 3√7 ) / ( 64 - 63 )

⇒ 1 / x = ( 8 - 3√7 )  / 1

⇒ 1 / x = 8 - 3√7            ....( ii )

Adding ( 1 ) & ( 2 ),

⇒ x + 1 / x = 8 + 3√7 + 8 - 3√7

⇒ x + 1 / x = 8 + 8

⇒ x + 1 / x = 16

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Square on both sides,

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⇒ ( x + 1 / x )^2 = 16^2

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( a + b )^2 = a^2 + b^2 + 2ab

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⇒ x^2 + 1 / x^2 + 2( x * 1 / x ) = 256

⇒ x^2 + 1 / x^2 + 2( 1 ) = 256

⇒ x^2 + 1 / x^2 + 2 = 256

⇒ x^2 + 1 / x^2 = 256 - 2

⇒ x^2 + 1 / x^2 =  254

Therefore the value of x^2 + 1 / x^2 satisfying the given equations is 254.

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