If `x=2+sqrt(3)` , find the value of `x^2+1/(x^2)`

Explanation

`:'x=3+sqrt3`
`:.(1)/(x)=(1)/(2+sqrt3)=(1)/(2+sqrt3)xx(2-sqrt3)/(2-sqrt3)=(2-sqrt3)/(2^(2)(sqrt3)^(2))=(2-sqrt3)/(4-3)=2-sqrt3`
Now, `x^(2)+(1)/(x^(2))=(x+(1)/(x))^(2)-2`
`impliesx^(2)+(1)/(x^(2))=(2+sqrt3+2-sqrt3)^(2)-2=4^(2)=16-2=14`
optional method:
`:'x=2+sqrt3`
`:.(1)/(x)=(1)/(2+sqrt3)=(1)/(2+sqrt3)xx(2-sqrt3)/(2-sqrt3)=(2-sqrt3)/(2^(2)-(sqrt3)^(2))=(2-sqrt3)/(2^(2)-(sqrt3)^(2))=(2-sqrt3)/(4-3)=2-sqrt3`
Therefore: , `x^(2)+(1)/(x^(2))=2^(2)+(2+sqrt3)^(2)+(2-sqrt3)^(2)`
`x^(2)+(1)/(x^(2))=2^(2)+(sqrt3)^(2)+2xx2sqrt3+2^(2)+(sqrt3)^(2)-2xx2sqrt3`
`impliesx^(2)+(1)/(x^(2))=4+3+4sqrt3+4+3-4sqrt3=14`