Question

Here's an interesting question.Best of luck.

Answer 1

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Given,

x = 1 / ( 2 - √3 )

By multiplying numerator and denominator by ( 2 + √3 )

x = 1 ( 2 + √3 ) / ( 2 - √3 ) ( 2  + √3 )

x = ( 2 + √3 ) / { ( 2 ) ^ 2 - ( √3 )^2 }

x = ( 2 + √3 ) / ( 4 - 3 )

x = ( 2 + √3 ).

Now,

x³ = ( 2 + √3 )³

x³ = 2³ + ( √3 )³ + 3 * 2 * √3 ( 2 + √3 )

x³ = 8 + 3√3 + 6√3 ( 2 + √3 )

x³ = 8 + 3√3 + 12√3 + 18

x³ = 26 + 15√3.

And,

x^2 = ( 2 + √3 ) ^ 2

x^2 = 2^2 + ( √3 ) ^2 + 2 * 2 *√3

x^2 = 4 + 3 + 4√3

x^2 = 7 + 4√3.

Now,

 = x³ - 2 x^2 - 7 x + 5 

 = 26 + 15√3 - 2 ( 7 + 4√3 ) - 7 ( 2 + √3 ) + 5

= 31 + 15√3 - 14 - 8√3 - 14 - 7√3

= 31 - 28 + 15√3 - 15√3

= 3

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