Discriminant Calculator

Discover the nature of quadratic equation roots effortlessly with our discriminant calculator. it reveals real, repeated, or complex solutions.

Last Updated: Jan 12, 2025

Welcome to the CalculatorValue, whether you're a student or someone looking for a discriminant calculator and want to grasp the concept behind it. In this guide, I have explained everything about quadratic equations and discriminants, including the following: What is the discriminant? What are the formulas and real-world examples? And a step-by-step guide on how to use a discriminant calculator?

What is a Discriminant?

In quadratic equations, the discriminant is a mathematical expression that determines the nature of the roots of the equation. The discriminant gives us valuable insights about the solution without solving the equations. It helps us determine whether the equations have one real root, two distinct real roots or two complex (also called imaginary) roots).

Quadratic equations are polynomial equations of the form:

$a x^{2} + b x + c = 0$ Where is, $a$ , $b$ , and $c$ are the coefficients with $a \neq 0$ .

What is the Discriminant Formula? — A Guide To Understanding It

The discriminant (sign is $D$ ) is calculated with the help of the coefficients of the quadratic equation. So, the formula of the discriminant is the following:

$D = b^{2} - 4 a c$

Where is,

$a$ is the coefficient of $x^{2}$
$b$ is the coefficient of $x$
While the $c$ is the constant term

What are the Types of Discriminants

The types of discriminants depend on the value of the discriminant, while the equation has different types of roots:

Two Distinct Real Roots ( $D & g t; 0$ )
- When the discriminant is positive, the equation has two distinct real solutions.
- Example: $x^{2} - 5 x + 6 = 0$
$D = (- 5) 2 - 4 (1) (6)$ $D = 25 - 24$ $D = 1 & g t; 0$
- Roots: $x = 2$ and $x = 3$ (two distinct real)
One Real Root ( $D = 0$ )
- A zero discriminant tells that both roots are real and equal (a repeated or double root).
- Example: $x^{2} - 4 x + 4 = 0$
$D = (- 4)^{2} - 4 (1) (4)$ $D = 16 - 16$ $D = 0$
- Root: $x = 2$ (double root)
Two Complex Roots ( $D & l t; 0$ )
- A negative discriminant determines the equation has two complex (imaginary) roots.
- Example: $x^{2} + x + 1 = 0$
$D = (1)^{2} - 4 (1) (1)$ $D = 1 - 4$ $D = - 3 & l t; 0$
- Roots: $x = \frac{- 1 \pm \sqrt{- 3}}{2}$ (complex roots)

How to Use a Discriminant Calculator

Our discriminant calculator is designed to ease the calculation of the discriminant of a quadratic equation. Instead of manually performing the calculation, you can simply calculate by inputting the values. It provides a complete step-by-step procedure to get the result, which helps you understand the formulas and methods involved.

Steps to use the calculator:

Enter value of $a$ : ________

Enter value of $b$ : ________

Enter value of $c$ : ________

You have to input the Coefficients values of $a$ , $b$ , and $c$ from the quadratic equation $a x^{2} + b x + c = 0$ . After inputting the values, you will get the result along with a box displaying the steps involved in the calculation to obtain it, included formula.

Real-World Example: Manual Discriminant Calculation

The discriminant calculator is super easy and provides the complete procedure, including formulas, steps involved, etc., but below, I will go through a real-world example of calculating the discriminant, how to apply the formula, etc.

Step 1: Identify the Coefficients

Determine the values of the $a$ , $b$ , and $c$ from the quadratic equation $a x^{2} + b x + c = 0$ .

Let's say a quadratic equation of $3 x^{2} + 6 x + 2 = 0$

Where is,

$a$ = 3
$b$ = 6
$c$ = 2

Step 2: Apply the Discriminant Formula

Use that formula $D = b^{2} - 4 a c$

Now Perform calculation:

$D = (6)^{2} - 4 (3) (2)$ $D = 36 - 24$ $D = 12$

Step 3: Interpret the Discriminant

Now let's determine which type of discriminant we got from result as we know:

If $D & g t; 0$ : Two distinct real roots.
If $D = 0$ : One real root (double root).
If $D & l t; 0$ : Two complex roots.

In our example, we got the $D$ = 12 which is greater than 0 $D & g t; 0$ , determining that the equation has two distinct real roots.

Where are the Discriminants used?

While the discriminants, the discriminant is a math equation that is specifically used to find out the nature of the root of the quadratic equations. However, its uses are broadened beyond mathematics, so below are the many real-world fields in which the discriminants are being used:

Mathematics and Education
Physics
Engineering
Finance and Economics
Computer Science
Astronomy
Biology
Environmental Science
Architecture
Chemistry

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is a mathematical equation representing a second-degree polynomial in a single variable, which is denoted as $x$ . While the common form of the quadratic equation is: $a x^{2} + b x + c = 0$

Where is:

$a$ , $b$ , and $c$ are the constants,
While $a$ can't be equal to 0 ( $a \neq 0$ ), Because if the $a$ become equal to 0 ( $a = 0$ ), the equation becomes linear rather than quadratic.

Can the calculator handle complex roots?

Yes, the calculator can identify if a quadratic equation has complex roots based on the discriminant. However, it does not provide the complex roots themselves.

What input values are required for the calculator?

The calculator requires three inputs:

$(a)$ : The coefficient of $x^{2}$ (must be non-zero)
$(b)$ : The coefficient of $x$
$(c)$ : The constant term

Ensure that the input values for $(a)$ , $(b)$ , and $(c)$ are valid numbers. For example, enter the coefficients as numerical values without any additional characters or spaces.