CAT 2025 Lesson : Quadratic Equations - Discriminant & Graphical Representation

3.1 Discriminant & Nature of Roots

The roots of an equation are the solution or the values of x where the quadratic equation holds good. Every quadratic equation has two roots.

For a general quadratic equation written as $ax^{2} + bx + c = 0$,

$x = \dfrac{-b + \sqrt{b^{2} - 4ac}}{2a}, \dfrac{-b - \sqrt{b^{2} - 4ac}}{2a}$

Note that the two roots are generally read together as $\bm{x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}}$ .

The expression inside the square root is commonly called the discriminant, i.e. $\bm{D = b^{2} - 4ac}$.

∴ The roots can be written as $x = \dfrac{-b \pm \sqrt{D}}{2a}$

As the discriminant is inside a square root, the nature of roots can be defined as follows.

Discriminant	Nature of Roots
$D \gt 0$	Two real and distinct roots.
$D = 0$	Two real and equal roots
$D \lt 0$	Two imaginary roots

3.2 Graphical Representation

The following graphical representation of quadratic expressions will help you better understand the roots of the equations. The curves are for the expression $f(x) = ax^{2} + bx + c$.

When the curve cuts the $x$-axis, then at these points $f(x) = 0$. The $x$-values at these points are the roots of the quadratic expression.

$1$) All quadratic expressions are either U-shaped (when a > 0) or inverted U-shaped (when a < 0)
$2$) In a U-shaped curve, we can find the minimum value that the quadratic expression can attain. And, in an inverted U-shaped curve, we can find the maximum value that the quadratic expression can attain. This is covered later in this lesson.
$3$) When D $\bm{\gt}$ 0, the $2$ distinct real roots are points at which the curve cuts the $x$-axis.
$4$) When D = 0, the real and equal root is the point at which the curve touches the x-axis.
$5$) When D $\bm{\lt}$ 0, the curve does not cut the x-axis and, therefore, the roots are imaginary.

$\bm{f(x) = ax^{2} + bx + c}$	$\bm{a \gt 0}$	$\bm{a \lt 0}$
D $\gt$ 0 (Real & Distinct roots)
D $=$ 0 (Real & Equal roots)
D $\lt$ 0 (Imaginary roots)