The Discriminant tells about the nature of the roots of quadratic equation. In quadratic equation formula, we have \( b^2 - 4ac \) under root, this is discriminant of quadratic equations.

There are three cases for discriminant:

if \( b^2 - 4ac = 0 \), then the roots of quadratic equations are real and equal.

if \( b^2 - 4ac > 0 \), then the roots are real but not equal

if \( b^2 - 4ac < 0 \), then the roots are not equal and are imaginary.

# Discriminant Calculator

## What is Discriminant?

Discriminant in math, is a function of coefficients of a polynomial equation. Answers of Discriminant expression can be identified as following:

if \( b^2 – 4ac = 0 \), then we get one real solution.

if \( b^2 – 4ac > 0 \), then we get two real solutions.

if \( b^2 – 4ac < 0 \), then we get two imaginary solutions.

## Discriminant Formula:

Formula for finding discriminant is:

= \( b^2 – 4ac \)

Where:

a is the coefficient of x2.

b is the coefficient of x.

c is a constant term.

Results of this formula, tells us how many solutions our quadratic has.

## Example of discriminant calculation:

### Example 1:

\( x^2 + 6x + 8 = 0 \)

A = 1

B = 6

C = 8

Discriminant:

= \( b^2 – 4ac \)

= \( 6^2 – 4(1)(8) \)

= \( 36 – 32 \)

= \( 4 \)

The number is positive, hence both roots of this equation are real and unequal.

### Example 2:

X2 + 4x + 12 = 0

A = 1

B = 4

C = 12

Discriminant:

= \( b^2 – 4ac \)

= \( 4^2 – 4(1)(4) \)

= \( 16 – 48 \)

= \( -32 \)

The number is negative, hence this equation has no real solution.

## What does it mean if a discriminant is positive?

If discriminant is higher than zero. It represents that the quadratic equation has two real number solutions. Graphical representation can be seen below:

## How to use Discriminant Calculator?

- Enter coefficient values for “a”, “b” and “c” in the above fields.
- Click “Submit”.
- Our discriminant calculator displays the solutions including all the steps and description at the end.

## Cases of Discriminant:

In this section, you will learn about Discriminant, how many forms of Discriminant with examples.

There are 3 cases of Discriminant.

**Case 1: Real and Equal:**

The discriminant is equal to 0 is known as real and equal.

**Example:**

= \( x^2 – 4x + 4 \)

= \( a = 1, b = 4, c = 4 \)

= \( b^2 – 4ac \)

= \( 4^2 – 4(1)(2) \)

= \( 16 – 16 \)

= 0

The root are real and equal.

**Case 2: Real and Unequal**

The discriminant is greater then 0 is known as real and unequal.

**Example:**

= \( 4^2 – 10x + 3 \)

= \( a = 4, b = 10, c = 3 \)

= \( b^2 – 4ac \)

= \( 10^2 – 4(4)(3) \)

= \( 100 – 48 \)

= 52

The root are real and unequal.

**Case 3: Unequal and Imaginary**

The discriminant is less then 0 is known as unequal and imaginary.

**Example:**

= \( 4^2 – 4x + 3 \)

= \( a = 4, b = 4, c = 3 \)

= \( b^2 – 4ac \)

= \( 4^2 – 4(4)(3) \)

= \( 16 – 48 \)

= 52

The root are unequal and imaginary.