Question

if `x = 2 +  sqrt3`,find the value of `x^2 + 1/x^2`

Answer 1

We know that `x^3 + 1/x^3 = (x + 1/x)(x^2 - 1 + 1/x^2)`. We have to find the value of `x^3 + 1/x^3`

As x = `2 + sqrt3` therefore

`1/x = 1/(2 + sqrt3)`

We know that rationalization factor for `2 + sqrt3` is `2 - sqrt3`. We will multiply numerator and denominator  of the given expression `1/2 + sqrt3` by `2 - sqrt3` to get

`1/x = 1/(2 + sqrt3) xx (2 - sqrt3)/(2 - sqrt3)`

`= (2 - sqrt3)/((2)^2 - (sqrt3)^2)`

`= (2 - sqrt3)/(4 - 3)`

`= 2 - sqrt3`

Putting the value of x and 1/x we get

`x^3 + 1/x^3= (2 + sqrt3 + 2 - sqrt3)((2 + sqrt3)^2 -  1+ (2 - sqrt3)^2)`

= `4(2^2  + (sqrt3))^2 + 2 xx 2 xx sqrt3 - 1 + 2^2  + (sqrt3)^2 - 2 xx 2 xx sqrt3)`                    

`= 4(4 + 3 + 4sqrt3 - 1 + 4 + 3 - 4sqrt3)`

= 52

Hence the value of the given expression 52.