Find the sum of the squares of all the real solution of the equation `2log_((2+sqrt3)) (sqrt(x^2+1)+x)+log_((2-sqrt3)) (sqrt(x^2+1)-x)=3`

Answer

` x= sqrt3`

Explanation

`2 log_(2+sqrt3)(sqrt(x^(2)+1+x))+ log_(2-sqrt3)(sqrt(x^(2)+1-x)) = 3`
` rArr 2 log_(2+sqrt3)(sqrt(x^(2)+1+x))+log_((2+sqrt3)^(-1))(sqrt(x^(2) + 1)-x)=3`
` rArr 2 log_(2+sqrt3)(sqrt(x^(2)+1+x))-log_(2+sqrt3)(sqrt(x^(2) + 1)-x)=3`
` rArr log_(2+sqrt3).((sqrt(x^(2)+1)+x)^(2))/((sqrt(x^(2)+1)+x)) = 3`
` rArr log_(2+sqrt3)(sqrt(x^(2)+1)+x)^(3)=3`
` rArr (sqrt(x^(2)+1)+x)^(3) = (2+sqrt3)^(3)`
` rArr (sqrt(x^(2)+1)+x)^(3) = (2+sqrt3)^(3)`
` rArr sqrt(x^(2)+1)+x = 2 + sqrt3`
` rArr x = sqrt3`