To find the equation (quadratic), we use 1-√6 and 1+√6 as the zeroes:
(x-1+√6)(x-1-√6)=0=x²-2x+1-6=x²-2x-5. So x²-2x-5=0 will include the solution 1-√6. This zero has a negative value. The other zero is 1+√6, a positive value.
If we want an expression that gives us the unique solution 1-√6, then we need to rationalise:
1/(a+b√6), where a and b are rational constants.
Multiply top and bottom by a-b√6: (a-b√6)/(a²-6b²), which must be equivalent to 1-√6.
Therefore, a/(a²-6b²)=1 and b/(a²-6b²)=1, so a=b.
a=a²-6b²=-5a², so since a can’t be zero, a=-⅕. So -5/(1+√6)≡1-√6.