The Discriminant of a Quadratic Equation - Concept
The discriminant is part of the quadratic formula which lies underneath the square root. The quadratic equation discriminant is important because it tells us the number and type of solutions. This information is helpful because it serves as a double check when solving quadratic equations by any of the four methods (factoring, completing the square, using square roots, and using the quadratic formula).
Discriminant of a quadratic equation is part of the quadratic formula. It's actually the part that lies underneath the square root. So, discriminate you'll hear is b squared minus 4ac, which hopefully looks familiar because you know the quadratic formula. And really what the discriminate does is it tells us what type of solution and the number of solutions our quadratic equations are going to have. It doesn't tell us what they are. It just tells us the type and the number. OK?
How this works is basically there's four scenarios. I prefer not to memorize them, but I am going to go through them each and then you can use logic or memorize them to sort of figure them out.
Alright. So, what the discriminant can be? There are different options. First of which is it's going to be greater than zero and a perfect square. So, what I mean by that is like 16, 25, any number greater than zero and a perfect square.
So, the discriminant is what's underneath the square root so if it's a perfect square you're going to be able to take the square root of it and our square root is gone from our quadratic formula. What that tells us then is that we have two rational solutions. Perfect square. You can take the square root. The square root goes away.
Alright, discriminant is greater than zero and not a perfect square. So, that would be say like 10, 20, something like that where we can't take a square root. What that tells us is we then throw it underneath a square root sign. Our square root isn't going to go away.
We still have a square root in there of a number that we can take the square root of so what we are going to end up with is two irrational. So, we have a square root and we have plus a square root, minus the square root. So, we have two irrational solutions.
Discrimanent is equal to zero. OK, what that does in terms of our quadratic formula is it makes that whole square root go away. So, you have plus or minus the square root of zero, disappears and we are just left with negative b over 2a.
So, in this case, we have one rational solution, one fractional solution. And the last scenario for our discriminant is it's less than zero. OK, that means a negative number. Discriminant is negative that means what's going into the square root is negative, which means we have two imaginary solutions.
Square root of a negative is an imaginary number. And so we are not going to have any sort of real solutions; we are just going to have imaginary solutions. OK.
So, discriminant is what's underneath the square root in the quadratic formula and it tells us about the number and type of solutions for that quadratic equation.
You can go ahead and memorize these four different things. In general, I just prefer to use logic, OK? Know what the discriminant is, know that it's underneath the square root, and then you know how a square root behaves enough to be able to deduce these anytime you need to.