Quadratic Equation with imaginary roots. I know that if the discriminant of a quadratic equation is less than $0$, the roots are imaginary.
But why is this quadratic expression (with imaginary roots) always positive for all values of $x$?
Can you explain me the logic? My text book has directly stated that fact. 
Thanks.

Recall the geometric interpretation for the quadratic equation
$$ax^2+bx+c=0$$
which is the solution of the system


*

*$y=ax^2+bx+c$

*$y=0$
which represents the intersection of a parabola with the $x$ axis and we can have three cases


*

*$2$ real solutions that is the parabola intersects the $x$ axis ($\Delta >0$)

*$1$ real solution that is the parabola is tangent to $x$ axis ($\Delta =0$)

*$2$ complex solutions that is the parabola does not intersect the $x$ axis ($\Delta <0$)



and in the latter case the expression is positive or negative depending upon the sign of the coefficient $a$.
