 One of the features of higher mathematics is our focus shifts from finding a solution to being able to say something about the solution whether or not we can actually find it and One of the first places we see that is in what's called the discriminant of a quadratic equation This is part of a much larger topic known in mathematics as the theory of equations And so one place to begin with is with the quadratic formula The quadratic formula allows us to solve any quadratic equation However, not all solutions are created equal For example, let's consider this equation x squared plus 2x minus five equals zero. We'll use the quadratic formula We'll check the fine print the quadratic formula only works when we have equation equals zero, which we do A is the coefficient of x squared B is the coefficient of x and C is the constant term So that gives us a B and C equal to and we'll do a bunch of simplifications and calculations So first we'll replace a with one B with two and C with negative five And then we'll do a bunch of computations Now it's useful to remember that this plus minus symbol here means that we're either going to add The square root of 24 or we'll subtract the square root of 24 and it's convenient to rewrite this as two solutions And let's take one more step, which will prepare us for something We're going to do a little bit later on we have a Fraction where the numerator is a sum or a difference and that means we can split this apart into two fractions So our first fraction negative 2 plus square root of 24 the whole thing over 2 becomes negative 2 over 2 plus square root 24 over 2 and Likewise our second fraction can also be split up and We can simplify this a little bit this minus 2 over 2 is just negative 1 and so we get our final two solutions Well, that was fun. Let's do another one. How about x squared minus 6x plus 9 equals zero So again, we'll pull in our quadratic formula a is our coefficient of x squared That's one B is our coefficient of x. That's minus 6 C is our constant term. That's 9. We'll drop those into the quadratic formula We'll simplify and again, we'll split our fraction into two fractions and We get our solutions x equals 3 or x equals 3 Now an ordinary person would say that this equation has one solution x equal to 3 But a mathematician is not an ordinary person. We say this equation has two solutions They just happen to be the same and As we'll see there is a good reason for saying that this equation has two solutions that are the same How about this equation x squared minus 4x plus 8? So again, we'll use our quadratic formula A is the coefficient of x squared. That's 1 B is the coefficient of x. That's negative 4 and C is the constant term. That's 8 We'll drop those into the quadratic formula We'll do a little bit of arithmetic We'll take the square root of negative 6 to 11. Well, well, wait, wait, wait, wait You can't take the square root of a negative number at least not yet a Little later on we'll see what we can do with this But for right now the important thing to observe here is that the quadratic formula gives us no real solution Let's take a moment and look at what we have the equation a x squared plus b x plus c can be solved using the quadratic formula But there are three possibilities First we might get two solutions that are different Second we might get two solutions that are actually the same number and Finally, we might get no real solutions Now, why does that happen? Let's take a look at our solutions in more detail When the quadratic formula gave us two distinct solutions when we took the square root We were taking the square root of a positive number On the other hand when the quadratic formula gave us two solutions that were the same thing We were taking the square root of zero and when the quadratic formula didn't work at all We were taking the square root of a negative number And so that says that this thing inside the square root is going to be important so if b squared minus 4ac is positive then there's going to be two distinct real solutions if b squared minus 4ac is equal to zero then there are two equal real solutions A normal person would say there is one real solution, but mathematicians aren't normal people And if b squared minus 4ac is less than zero there are going to be no real solutions Put together this says that this value of b squared minus 4ac seems to be important. So let's give it a name We'll call it the discriminant and so the discriminant of the quadratic expression ax squared plus bx plus c is the value b squared minus 4ac One useful thing to keep in mind is that we're only looking for the discriminant and something about the real solutions to 3x squared minus 7x plus 8 Wait a minute This isn't an equation. There's no equals So let's go ahead and put an equals in so we can talk about solutions to the equation Now since we're not actually looking for the solutions, but we do want to describe them It's sufficient to find the discriminant. So let's return our definition a b and c have the same meaning that they have for the quadratic formula So we'll substitute in those values We do a little arithmetic And we find the value of the discriminant is negative 47 The reason the discriminant is important is because in the quadratic formula We'll be taking the square root of the discriminant In this case the discriminant is negative and since we can't take the square root of a negative number We say that there are no real solutions We can actually go a little bit farther if b squared minus 4ac is positive We can actually take a square root But if it's not a perfect square then the square root will not be rational And so our solutions based on the quadratic formula will not be rational numbers So knowledge is power Let's say as much as we can about the real solutions to 3x squared minus 3x minus 20 equal to zero So we'll find our discriminant again a is the coefficient of x squared. That's three b is the coefficient of x negative three c is our constant negative 20 We'll substitute those into our discriminant formula So we find our discriminant is positive and if our discriminant is positive then there are two distinct real solutions The second thing to recognize is that 249 is not a perfect square So when we take the square root in the quadratic formula We'll end up with an irrational number and our two solutions will also be irrational