Introduction:

For a quadratic equation \( a{x}^2+bx+c=0 \):

Discriminant \( {b}^2-4ac \)	Nature of roots	Characteristics of curve
> 0	2 real and distinct roots	The curve cuts the x-axis (\(y=0\)) at 2 different points
Perfect square	2 real and rational roots
Not a perfect square	2 real and irrational roots
= 0	2 real and equal roots	The curve touches the x-axis at \(x=-\frac { b }{ 2a }  \).  The x-axis is a tangent to the curve.
< 0	No real roots	The curve does not cut or touch the x-axis. It lies entirely above (a > 0) or below (a < 0) the x-axis.

From the first 2 points, we conclude that

                                      \({ b }^{ 2 }-4ac\ge 0\Longleftrightarrow \) the roots are real