Solve the equation `(2+sqrt(3))^(x^(2)-2x+1)+(2-sqrt(3))^(x^(2)-2x-1)=101/(10(2-sqrt(3))`

Answer

`x_(1)=1+sqrt(1+log_(2+sqrt(3))10),x_(22)=1-sqrt(1+log_(2+sqrt(3))10)`

Explanation

`:'(2+sqrt(3))^(x^(2)-2x+1)+(2-sqrt(3))^(x^(2)-2x-1)=101/(10(2-sqrt(3)))`
`implies (2+sqrt(3))^(x^(2)-2x).(2+sqrt(3))(2-sqrt(3))`
`=+(2-sqrt(3))^(x^(2)-2x-1).(2-sqrt(3))=101/10`
`implies(2+sqrt(3))^(x^(2)-2x)+(2-sqrt(3))^(x^(2)-2x)=101/10`
or `(2+sqrt(3))^(x^(2)-2x)+1/((2+sqrt(3))^(x^(2)-2x))=101/10` ..........i
`[ :'2-sqrt(3)=1/(2+sqrt(3))]`
Let `(2+sqrt(3))^(x^(2)-2x)=lamda` then Eq. (i) reduces to
`lamda+1/(lamda)=101/10`
`implies10lamda^(2)-101 lamda+10=0`
or `(lamda-10)(10lamda-1)=0`
`:.lamda=10,1/10`
`=(2+sqrt(3))^(x^(2)-2x)=10,10^(-1)`
`impliesx^(2)-2x=log_(2+sqrt(3))10,-log_(2+sqrt(3))10`
`implies(x-1)^(2)=1+log_(2+sqrt(3))10,1-log_(2+sqrt(3))10`
`:.(x-1)^(2)=1+log_(2+sqrt(3))10`
`:'(x-1)^(2)!=1-log_(2+sqrt(3))10]`
`impliesx=1+-sqrt((1+log_(2+sqrt(3))10))`
`impliesx_(1)=1+sqrt((1+log_(2+sqrt(3)10))`
`=x_(2)=1-sqrt((1+log_(2+sqrt(3))10))`