The discriminant of quadratic equation ax^2+bx+c=0 is Delta=b^2-4ac and roots are given by (-b+-sqrtDelta)/(2a)
If Delta is positive and a square of a rational number (and so are a and b), roots are rational .
If Delta is positive but not a square of a rational number, roots are irrational and real .
If Delta=0, we have only one root given by (-b)/(2a).
If Delta is negative but a, b and c are real numbers, roots are two complex conjugate. But if Delta is negative but a, b are not real numbers, roots are complex .
Hence, one needs to first convert the equation -4r^2-4r=6 in this form. This can be easily done by shifting terms on left hand side to RHS and this become 0=4r^2+4r+6 or 4r^2+4r+6=0
As we have a=4, b=4 and c=6,
the determinant is 4^2-4xx4xx6=16-96=-80
As the determinant is negative and a, b and c are real numbers, we have complex conjugate roots for the given equation.
These are (-4+-sqrt(-80))/(2xx4)=-1/2+-isqrt5/2
i.e. -1/2-isqrt5/2 and -1/2+isqrt5/2
