[Math] Quadratic Equation with imaginary roots.

Question

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.

Answer 1

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 $a$.