Discriminant 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:

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:

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:

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.

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