given [tex] - 3 {x}^{2} + x - 4[/tex] finding the determinant will tell us how many solutions the determinant is [tex] {b}^{2} - 4ac[/tex] so we have a=-3 b=1 c=-4
so the determinant is [tex] {1}^{2} - 4( - 3)( - 4) = 1 - 48 = - 47[/tex] now we know if d=0 there is 1 real solution if d<0 there are complex solutions and if d>0 there are 2 real solutions. since d=-47<0 there are no real solutions only complex ones
