Introduction:
For a quadratic equation \( a{x}^2+bx+c=0 \):
Discriminant
\( {b}^24ac \)

Nature of roots

Characteristics of curve

> 0

2 real and distinct
roots

The curve cuts the xaxis
(\(y=0\)) at 2 different points

Perfect square

2 real and rational
roots


Not a perfect square


= 0

2 real and equal
roots

The curve touches the xaxis
at \(x=\frac { b }{ 2a } \). The xaxis is a tangent to the curve.

< 0

No real roots

The curve does not cut or touch the xaxis.
It lies entirely above (a > 0) or below
(a < 0) the xaxis.

From the first 2 points, we conclude that
\({ b }^{ 2 }4ac\ge 0\Longleftrightarrow \) the roots are real
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