Standard Quadratic Form & Discriminant
A quadratic equation in variable $x$ is an algebraic equation of second degree:
$$ax^2 + bx + c = 0 \quad (a \neq 0;\, a, b, c \in \mathbb{R})$$
The Quadratic Formula (Shreedharacharya's Rule):
The roots ($\alpha, \beta$) are given by:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
The expression under radical $D = b^2 - 4ac$ is termed the Discriminant, which dictates the nature of the roots:
| Discriminant Value | Nature of Roots | Graphic Parabola Interpretation |
|---|---|---|
| $D > 0$ | Two distinct real roots | Intersects X-axis at two distinct points. |
| $D = 0$ | Two equal real roots ($x = -\frac{b}{2a}$) | Vertex touches X-axis at single point. |
| $D < 0$ | No real roots (imaginary conjugate roots) | Parabola lies completely above or below X-axis. |
