Discriminant: Definition & Explanation

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

The discriminant helps you to solve quadratic equations. In this lesson, you will learn how to identify the discriminant and how to use it to help you find the number of solutions in a quadratic equation.


The discriminant is easy to find when you look at the quadratic formula. The quadratic formula is the equation you use to find the solutions to quadratic equations. It looks like this:

This is the quadratic formula. Look for the square root symbol - the contents inside are the discriminant.
Quadratic formula

The discriminant is the part inside the square root. Take a moment to look for the square root and find what is inside the square root. Once you see it, you will have found the discriminant. If we isolate that part, we get the formula for finding the discriminant, which is this:

This is the formula for finding the discriminant.

Using the Discriminant

The discriminant tells you how many possible solutions a particular quadratic equation has. Before we can use the quadratic equation, though, we first have to change it to standard form. Standard form is when all the variables and constants are on one side of the equation, and the other side is a zero. It looks like this:

This is a quadratic equation in standard form.
standard form of quadratic equation

Once you have your quadratic equation in standard form, you can label your numbers with the appropriate letters and plug the values into your formula for finding the discriminant. The result of your discriminant tells you how many solutions your quadratic has.


Let's look at an example:

Example 1
positive discriminant example

In our example, our quadratic equation gives us 1 for our letter a, 5 for letter b, and 4 for letter c. We take these values and plug them into their appropriate places in the discriminant formula, and we'll find that our discriminant equals 9, a positive number. This tells us that our quadratic equation has two possible real solutions. Real solutions are solutions that can be calculated using the quadratic formula. When you graph this quadratic equation, you will see that the curve crosses the x-axis in two places, exactly where your solutions are.

While the discriminant tells us the number of possible solutions, it does not tell us what those solutions are. But, it does give us a clue as to how many solutions we need to look for.

Remember that, when there are no numbers in front of variables, it is assumed that there is a 1 in front. We don't write the 1 because it is a mathematical convention and because it looks neater, especially when you have a lot of letters to work with.

Let's look at another example:

Example 2
negative discriminant example

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