 So in this lecture so far, we've learned how we can solve quadratic equations by using factorizations and the zero product property. And I really like this technique. It can be really fast and very effective, but it depends on one thing. Can we find a magic pair? That's what's necessary to do this reverse foil process. And there's two issues of that. One, if the numbers get big enough, there actually could be a lot of different factorizations. So there's a lot of possibilities to check as you're looking for the magic pair. And then there's also the unfortunate reality that sometimes the magic pair doesn't exist. We cannot find it. Maybe the hydra ate it already before Hercules got there. I don't know. But the magic pair doesn't exist and we can't find something that doesn't exist. So what we're going to do is start introducing an alternative method for solving quadratic equations that can be very useful. And it's the idea of using square roots. Like we talked about with inverse functions, the square root is essentially the inverse operation of the squaring function. It's not exactly true because square is not one to one. But as long as we keep track of the fact it's not one to one, we can use the square root function to help us out here. So for example, if we have the equation x squared equals P, we can set that equal to zero and we can't factor in the following way. So we have x minus the square root of P times x plus the square root of P. I mean this is really just using the difference of squares here where we're recognizing that even if P is not a perfect square as an integer, it is a perfect square as a real number. Every number will just factor as the square root of P times the square root of P. And so therefore the solution is going to be x equals plus or minus the square root of P. The idea is if you have x squared equals P, you can take the square root of both sides performing the inverse operation to the square. But because the square is not one to one, we have to compensate for the fact it's not one to one. There's a positive solution and a negative solution. As much like we did with absolute values, as long as we remember there's a positive negative, then we'll take the square root of both sides and we can go from there. Now if your P value is greater than or equal to zero, you'll take the square root and you'll get a real solution. If P is positive, there'll be two solutions, plus and minus the square root of P. If P was zero, you'll just get one solution, which is zero. But what happens if P is negative? If P is negative, you'll be taking the square root of a negative number, which is not a real number, but it is a complex number. So you won't have any real solutions, but you will have complex solutions that might involve some imaginary components. So consider this one right here, x squared equals five. Well to solve this, we can take the square root of both sides, in which case you're going to get plus or minus the square root of five. That would be the solution to this equation. Now we probably should write them out separately, the square root of five and the negative square root of five. There are in fact two distinct solutions here. How about this next one, x equals negative nine. Well if you take the square root of both sides, you're going to get x equals plus or minus the square root of nine. When you take the square root of a negative number, that just means it's imaginary. You get the, you get i times the square root of nine, which is square root of nine is three. So you're going to get plus or minus three i. In other words, three i and negative three i. And so even though the solution is non real, we're going to allow it in this chapter. We're going to, we're going to consider all complex solutions to quadratic equations. What about, what about part C right here equals 16? Well the fact that we're squaring a binomial as opposed to just x itself doesn't really hinder us much. I mean, this might be relevant if we're graphing something, maybe your function just got shifted to the right by two, whatever. Well, if we take the square root of both sides, the left hand side, when you take the square root, it'll cancel with the square giving just x minus two. And on the right hand side, we're going to get the square root of 16. Just remember to take the plus or minus 16 there. This gives us plus or minus four. And so then to solve for x, we just have to add two to both sides. And we get x equals two plus or minus four, which there's two different numbers here. There's going to be two plus four and you're going to get two minus four. So we see that x equals six and negative two. Nothing too big about that. Not a big deal. Kind of like what happened when we did stuff with absolute value. Now, this was really convenient that this thing was factored as x minus two squared. Even though it wasn't equal to zero is equal to 16. All right. The reason why we were able to get good past this is we weren't trying to say that, oh, if the factors are if the product 16, I know that the factors have to be four and four or two and eight or something. We actually want to stay able to take the square root of both sides. Alternatively, if we had this thing multiplied out, it would look like x squared minus four x plus four equals 16. And, you know, if you move this the other side, we could have tried to solve this by factoring x squared minus four x minus that'd be minus 12 equals zero. You could try to factor that and you would see that factors of negative 12 are going to be positive six and positive two, which gives us the solutions of six and negative two. We could have solved that one by factoring, but it was kind of convenient that we had that perfect score on the left hand side. You're just able just to take the square root. It was a lot faster. So when you look at this one right here, x squared plus six x plus nine equals 12. You know what, we could subtract 12 from both sides, put in standard form, x squared plus six x minus three equals zero. We could try to do that, but it's like, huh, I tried to factors of negative three to add up to be six. I don't think I can do it. I could take three minus one. That's only two. Or I could do negative three plus one, which is a negative two. That's not going to work. We could find a magic pair right there. But if you look on the left hand side, x squared plus six x plus nine, I can't help but notice that this is in fact a perfect square trinomial, a perfect square. And as such, it would factor as, because notice your first term is a perfect square. It's x squared. Your last term is a perfect squared, which is three squared, three times x. If you double that, you get a six x. This is a perfect square. It would factor as x plus three squared is equal to 12. And so even though I can't solve this with a magic pair, I could try to solve this by this method of squaring, square rooting, I should say. Take the square root of both sides. We get x plus three equals plus or minus the square root of 12. Now the square root of 12, 12 is not a perfect square, but it does have perfect square divisors. 12 is four times three. So this becomes x plus three is equal to plus or minus two square root of three. And so then when you subtract three from both sides, you get that x equals negative three plus or minus two root three. So you're going to see that the solution to this, to this quadratic equation, it has irrational solutions. It has this square root of three. It's not a whole number. It's not even a fraction. It's this irrational number. And so we're only going to be able to solve quadratic equations by factoring. That is the magic pairs will only exist basically when the answers are whole numbers or fractions. Integers are rational numbers. But if you have irrational solutions, you won't be able to solve them by factoring. But this idea of the square root, but it only works if you have this perfect square right here. Now it turns out that is not a huge restriction to us whatsoever. Because if we do not have a square root, we can force it to be a square root. We can give the quadratic equation an ultimatum. You be a square root or else. And it turns out we always get what we want. And we'll talk about this more in the next lecture as we talk about completing the square.