Discriminant Calculator

Discriminant Calculator    

Use the discriminant calculator to find the nature of the roots of a quadratic equation. It will provide the steps you need to find the discriminant of second-degree polynomial equations. In addition to that, users will be able to download, copy, or print the results.

How to use this tool?

Use the following instructions to operate the discriminant calculator with steps.

1. Enter the values of coefficients a,b, and c from the standard polynomial equation.
2. Click “Calculate” to find the result.
3. Click on the “Show steps” for the detailed answer.
4. Reset for the second use.

What is Discriminant? 

The discriminant is the function of a polynomial equation that determines the nature of the roots without actually calculating their value. This value is then used in the quadratic formula to find the x-intercepts for the polynomial. 

Discriminant Formula

The discriminant of a quadratic equation is represented by the symbol Δ (delta). The formula for the discriminant is as follows:

Δ = b² - 4ac

where a, b, and c are the coefficients of the quadratic equation, ax² + bx + c = 0.

This formula is present under the square root of the quadratic equation formula as

x = (-b ± √(b² - 4ac)) / 2a

Nature of the roots:

As mentioned before, the roots are intersecting points of a parabola on the x-axis. The discriminant alone will be able to tell their nature. 

When delta > 0:

If the discriminant is greater than 1, the equation will have 2 real roots because in that case, the quadratic equation will have a positive number under the square root. And the square of a positive number is always a real number. This means its graph will be intersecting the x-axis at two real points.

When delta < 0: 

When the Discriminant results in less than 1, the equation will have two imaginary roots. It is because, in this case, a quadratic equation will have a negative number under the square root and the square root of a negative number is always an imaginary complex number. This means the graph will be intersecting the x-axis at two imaginary points.

When delta = 0:

The quadratic equations which give the discriminant result as zero, have only one real root. You can understand it by putting zero for the discriminant value in the quadratic formula i.e.

x = (-b ± √(b² - 4ac)) / 2a

x = (-b ± √(0)) / 2a

The whole square root part will become zero leaving only x = -b / 2a. It will give one value because the two values in the previous cases were a result of the square root. 

How to find the discriminant?

To find the discriminant manually, the polynomial has to be in the standard form. The coefficient value is found by comparing the variables. The value of variable a corresponds to the value of x², b to x¹, and c to x⁰.

Plug in the extracted values in the discriminant formula mentioned above. Solve to know the nature of the roots.   

Let's take the quadratic equation, 2x² + 5x - 3 = 0.

Using the formula for the discriminant, we can calculate Δ as follows:

Δ = b² - 4ac

Δ = (5)² - 4(2)(-3)

Δ = 25 + 24

Δ = 49

Since Δ > 0, the roots are real and distinct.

FAQ

What is b² - 4ac called?

In the quadratic equation ax2 + bx + c = 0, the expression b2-4ac is known as Discriminant.