Solve for `x(5+2sqrt(6))^x^(2-3)+(5-2sqrt(6))^x^(2-3)=10` (1985,5M)

Explanation

`(5+2sqrt(6))(5-2sqrt(6))=1`
`:.(5-2sqrt(6))=1/((5+2sqrt(6)))`
`:.(5+2sqrt(6))^(x^(2)-3)+(5-2sqrt(6))^(x^(2)-3)=10`
reduces to `(5+2sqrt(6))^(x^(2)-3)+(1/(5+2sqrt(6))))^(x^(2)-3)=10`
Put `(5+2sqrt(6))^(x^(2)-3)=t`, then `t+1/t=10`
`impliest^(2)-10t+1=0`
or `t=(10+-sqrt((100-4)))/2=(5+-2sqrt(6))`
`implies(5+2sqrt(6))^(x^(2)-3)=(5+-2sqrt(6))=(5+2sqrt(6))^(+-1)`
`:.x^(2)-3=+-1`
`impliesx^(2)-3=1` or `x^(2)-3=-1`
`impliesx^(2)=4` or `x^(2)=2`
Hence `x=+-2, +-sqrt(2)`.