Quadratic Equations with Complex Roots

Question

I don't understand what is the geometric ( or intuitive ) meaning of complex roots of a quadratic polynomial :

$ax^{2}+bx+c=0$

Can you help me ?

Answer 1

The system

• $y=ax^2+bx+c$

• $y=0$

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 ($\Delta <0$)

Answer 2

It means that $ax^2+bx+c$ gives $0$ when $x$ is a particular complex number. since, $$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
The discriminant $b^2-4ac$ can be less than $0$ equal to $0$ or greater than $0$
if the first case is there it means that we have a square root of negative number which is a complex number.
I think you can understand it if you have basic knowledge about complex numbers.

Answer 3

The geometric interpretation of complex roots is that the parabola does not intersect the x-axis.

It stays above the x-axis if $a$ is positive and it stays below the x-axis if $a$ is negative.