# Discriminant Calculator

## Calculate the discriminant of a quadratic equation step by step

The calculator will find the discriminant of the given quadratic equation, with steps shown.

Related calculator: Quadratic Equation Calculator

During your algebra exploration, you'll inevitably encounter quadratic equations. Our Discriminant Calculator is an efficient and potent tool to assist you in effortlessly calculating the discriminant. By computing the discriminant, you gain insights into the character of the roots of the quadratic equation.

## How to Use the Discriminant Calculator?

### Input

Input your quadratic equation in the designated field. Make sure you enter it correctly to get accurate results.

### Calculation

Once you've entered the coefficients, click on the "Calculate" button. The calculator will calculate the discriminant.

### Result

After the calculation, the Discriminant Calculator will display the discriminant value instantly on the screen.

## What Is a Discriminant?

In algebra, the discriminant plays a crucial role in determining the nature of the roots of a quadratic equation. A quadratic equation is a second-order polynomial equation in a single variable $$$x$$$, with a non-zero coefficient for $$$x^2$$$. The general form of a quadratic equation is:

$$ax^2+bx+c=0,$$where $$$a$$$ is the coefficient of $$$x^2$$$ $$$\left(a\ne0\right)$$$, $$$b$$$ is the coefficient of $$$x$$$, and $$$c$$$ is the constant term.

The discriminant $$$D$$$ of this equation is given by the formula:

$$D=b^2-4ac$$Let's look at an example. Consider the quadratic equation $$$2x^2-6x+3=0$$$. Here, $$$a=2$$$, $$$b=-6$$$, and $$$c=3$$$.

Substituting these values into the discriminant formula gives:

$$D=(-6)^2 - 4\cdot2\cdot3=36-24=12$$Since $$$D\gt0$$$, this equation has two distinct real roots.

## What Does a Positive and Negative Discriminant Represent?

In the context of a quadratic equation, the discriminant, represented by the formula $$$D=b^2-4ac$$$, carries crucial information about the nature of the roots (solutions) of the equation.

**Positive Discriminant $$$\left(D\gt0\right)$$$**

The quadratic equation has two distinct real roots when the discriminant is positive. This means the parabola represented by the equation crosses the x-axis at two distinct points. For example, in the equation $$$x^2-5x+6=0$$$, the discriminant is $$$1$$$ (a positive number), so there are two real and distinct solutions, namely, $$$x=2$$$ and $$$x=3$$$.

**Zero Discriminant $$$\left(D=0\right)$$$**

When the discriminant is zero, the quadratic equation has exactly one real root or two real roots that are the same (also known as repeated roots). In other words, the parabola touches the x-axis at exactly one point. For example, in the equation $$$x^2-6x+9=0$$$, the discriminant is $$$0$$$, so there is one real solution (or two identical real solutions), namely, $$$x=3$$$.

**Negative Discriminant $$$\left(D\lt0\right)$$$**

When the discriminant is negative, the quadratic equation has no real roots. Instead, it has two complex roots (solutions). This means the parabola does not intersect the x-axis at all. For example, in the equation $$$x^2+4=0$$$, the discriminant is $$$-16$$$ (a negative number), so there are two complex solutions, namely, $$$x=\pm2i$$$.

## Why Choose Our Discriminant Calculator?

### Accuracy

The calculator ensures that all calculations are error-free, delivering accurate results every time.

### Ease of Use

With a straightforward and user-friendly interface, our Discriminant Calculator is easy to use. Just input your quadratic equation, and the tool will do the rest.

### Step-by-Step Solutions

Not only does this calculator give you the discriminant value, but it also provides a step-by-step solution. This feature makes it an excellent learning tool for those trying to understand the process behind the calculation.

### Speed

Get instantaneous results! Our calculator handles any quadratic equation quickly, saving you time and effort.

### FAQ

#### Why do we need to find the discriminant?

We find the discriminant to determine the nature of the roots of a quadratic equation. The discriminant can tell us whether the roots are real or complex, and if real, whether they are identical or distinct.

#### What is the formula of a discriminant?

The formula for the discriminant is $$$b^2-4ac$$$, where $$$a$$$, $$$b$$$, and $$$c$$$ are the coefficients of the quadratic equation $$$ax^2+bx+c=0$$$.

#### What is the symbol of a discriminant?

The symbol for the discriminant is $$$D$$$ or $$$\Delta$$$ in many mathematical contexts.

#### What is a discriminant?

The discriminant is a component of the quadratic formula that provides information about the nature of the roots of a quadratic equation. Specifically, it's part of the formula under the square root sign, calculated as $$$b^2-4ac$$$.

#### Can the discriminant be negative?

Yes, the discriminant can be negative. A negative discriminant indicates that the quadratic equation has two distinct complex roots.

#### What does a discriminant of zero mean?

A discriminant of zero means that the quadratic equation has exactly one real root, often referred to as a repeated or double root.