What is the Discriminant?
We can use the Quadratic Formula to solve quadratic equations if we want the exact answers. The discriminant is just part of the Quadratic Formula and can tell us quite a bit about a quadratic equation. The discriminant is the number we end up with under the square root in the Quadratic Formula: the b squared minus 4ac part.
How Do You Calculate the Discriminant?
If you've used the Quadratic Formula before, finding the discriminant should feel like a shortcut. You don't need to use the whole formula, you just find the value of b squared minus 4ac. To do this, first make sure the equation is set equal to 0 and identify a, b, and c. a is the coefficient of the x-squared term, b is the coefficient of the x term and c is the constant. Plug the values into the expression b squared minus 4ac and simplify to find the discriminant.
Here's an example:
Why Is the Discriminant Important?
Why do we need to know the discriminant anyway? If we already have the quadratic formula to find the exact answers to a quadratic equation, why would we look at just part of the formula?
The discriminant gives us important information about the quadratic equation. First, it can tell us how many solutions the quadratic equation has. It can also tell us how many times the graph crosses the x-axis and if the solutions are real or complex.
But the discriminant is just a number! How does it give us all that information? Remember, the discriminant is the same as the number you get under the square root in the quadratic formula. The plus or minus in the quadratic formula is the part that often gives you two different solutions.
Open the next tabe to see at what we can tell about the equation based on different values of the discriminant.
A Positive Discriminant
If the discriminant is positive, this means that you have a positive number under the square root in the quadratic formula. This means you will end up with 2 real solutions. You can always take the square root of a positive number. It might not come out to a whole number, but it's going to be a real number. The plus or minus in the quadratic formula means you'll have to add this number to get one answer and you'll subtract it to find the second answer.
If there are 2 real solutions, this also means that the graph will have 2 x-intercepts. Remember, the solutions to a quadratic equation are often called roots or zeros. The roots/zeros/solutions are the the values for x that make the equation equal to 0. On a graph, this will be where the parabola crosses the x-axis. Anytime the discriminant is positive, the graph will cross the x-axis twice.
The discriminant won't tell you the actual answers. It doesn't tell you exactly where the graph crosses the x-axis, but it can tell you how many solutions and how many times it crosses. If you want to know more specifics, you have to use the entire Quadratic Formula to find the specific answers.
The example below shows an example of a quadratic equation with a positive discriminant:
When the Discriminant is Zero
If the discriminant is 0, that means you have a 0 under the square root in the quadratic formula. What happens when you have a 0 under a square root? The square root of 0 is just 0. When this happens, the plus or minus part of the quadratic formula essentially just goes away. This will leave you with only 1 real solution.
If there's only one real solution, that means that the graph will only have 1 x-intercept. The parabola will hit the x-axis right at its maximum or minimum point.
Here's an example of a quadratic equation with a discriminant of 0:
A Negative Discriminant
If the discriminant is negative, that means there is a negative number under the square root in the quadratic formula. You may have learned in the past that you "can't take the square root of a negative number." The truth is that you can take the square root of a negative number, but the answer is not real. The square root of a negative number will involve the imaginary number i. This means that if you have a negative discriminant, you will get two complex solutions.
If the solutions are both complex, you will not see them on the graph. The graph will either be too high or too low and will not cross the x-axis. There will be not be any x-intercepts.
Here's an example of a quadratic equation with a negative discriminant:
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