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

Example 2

Which of the following equations have real and distinct roots?

(1)

x^{2} - 2x + 7 = 0

(2)

x^{2} - 2x - 7 = 0

(3)

4x^{2} - 12x + 9 = 0

(4)

2x^{2} + 5x + 2 = 0

Solution

The following discriminants help us identify the nature of roots for each of the equations.

Equation	Discriminant	Nature of Roots
$1$	$(-2)^{2} - 4 \times 1 \times 7 = -24$	Imaginary roots
$2$	$(-2)^{2} - 4 \times 1 \times -7 = 32$	Real and Distinct roots
$3$	$(-12)^{2} - 4 \times 4 \times 9 = 0$	Real and Equal roots
$4$	$(5)^{2} - 4 \times 2 \times 2 = 9$	Real and Distinct roots

Answer:

(2)

and

(4)

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.