 This video is going to talk about the zero property to solve equations. So the zero property says that we have A times B is equal to zero. And the only way to multiply and get zero is something to be zero. So either A is equal to zero or B is equal to zero. So here's an example. X times X plus one. Those are factors and equal to zero. So we set X equals zero, that factor equal to zero. And then the X plus one, we set that factor equal to zero. X equals zero is just zero. But if I subtract one from both sides, I get X equal negative one. And if I check it, you can see that we have, if I put zero in for both those X's, zero times zero plus one. That would be zero times one, but anything times zero is zero. And if we put negative one in for both those X's, negative one times negative one plus one, well there's my zero times negative one. So again, I multiply it by zero, so it's equal to zero. So let's practice. We have two factors here set equal to zero. So we are ready to set our factors equal to zero. So the first one would be X plus five is equal to zero. And if we continue working, we subtract five from both sides. And we find out that X is equal to negative five. That's one of our answers. And then we set two X minus nine equal to zero. So we add nine to both sides. So two X will be equal to our nine. And we divide by two and find out that X is equal to nine over two, or you might say four point five. So what happens though if we don't have factors? Well, if we don't have factors, we need to factor because the property said A times B. It didn't say plus. There was no addition or subtraction in there to give me terms. So we need to factor this. So let's use our little X factor. A is one times B. C, which is 54, gives us 54. And the middle term is negative 15. So factors of negative 54 that will add up to negative 15. It's a positive 54. That means both my numbers are going to be negative. And it happens to be negative nine and negative six at multiply to 54 and add to negative 15. So we have our factors of X minus nine and X minus six. Remember, we can do that because our A was one. So we can just go right to our factors. And that's equal to zero. And now that I have it factored, I'm going to consider, well, what if X minus nine were zero? Then adding nine to both sides, we find out that X is equal to nine. And then we ask ourselves, what happens if X minus six is equal to zero? Well, let's solve for X. So we add six to both sides. And X is equal to positive six. All right, now we have another one to factor. This one has a huge numbers, but I'm looking at that and I can see that it has the greatest common factor of three. That leaves me with Y squared. Negative four Y times three would be negative 12. And negative 32 times three would be negative 96. And that's equal to zero. So I've got my three and it's a Y squared now. So I just need to find factors of 32 that are going to add up to negative four. It's a negative 32 adding up to negative four. So that means I have a negative and a positive. And it would be negative eight and positive four. That gives me negative 32 and I multiply and negative four when I add. Y minus eight Y plus four. And that's equal to zero. So our factors all equal to zero. Now there's no variable here, so we don't have to worry about the three. But when we have a variable, we have to go and find it. So X minus eight is equal to zero. Add eight to both sides. And we find out that Y is equal to eight. And we do the other factor, Y plus four equals zero. Subtract the four from both sides. And now we have Y equal to negative. That's a negative four. Let's see if I can erase part of that. Make it look a little nicer. Y is equal to negative four. All right, now what's the problem with this one is that we don't have it set equal to zero. Because now remembering the property again, it was equal to zero. This must be zero before we can factor. So we need to take the seven X to the other side. We need to subtract it from the two X squared. So we have our two X squared. We need to subtract the seven X from it. And then when we subtract it from itself, we get zero. Now we're ready to factor. So X is the common factor. And that leaves us with two X minus seven. And that's all still equal to zero. All we did was factor the one side. And if we set our factors equal to zero, when it's just X, we find out that X equals zero. And two X minus seven is equal to zero. Add the seven to both sides. And I'm not going to physically do it this time. But adding seven to both sides will give us two X equals seven. And then dividing both sides by two, X is going to be seven over two. So X is either zero or X is seven over two. Now we have two terms on one side and one term on the other. But remember, we would like to keep our Y squared or X squared term, whatever the variable happens to be, positive. Which means I want to take that six Y, actually that six Y squared. And I want to subtract Y from it. And that'll be equal to two. I'm going to move them one at a time. Now I want to subtract the two from both sides. So I have six Y squared minus one Y minus two equal to zero. And I'm ready to factor. So if I have my X factors here, I have six times negative two is negative twelve. And B is negative one. And I need to have opposite signs. So I'll put my negative to the left. And I have negative four positive three because a bigger number needed to be negative to add to negative one. I have to do four terms because A was greater than one. So we have six Y squared. And then we have minus four Y and plus three Y. And then we have our last term, which was negative two. And remember, that's equal to zero. So we factor here. Two Y is a common factor. And then we have three Y that'll give me six Y squared minus two, which will give me the minus four Y. And three Y and negative two have nothing in common, but a positive one. And then that just leaves me with three Y minus two. And it's still equal to zero. I'm not completely factored yet. So I'm going to come in here and do my greatest common factor, which is three Y minus two. And then I'm going to have my other factor, which is the two Y and the plus one from the outsides of each of those terms. And then it's still equal to zero. But it's now I have factors and now I'm ready to. I cannot set them equal to zero until I have factors equal to zero. So three Y minus two equals zero. If I add the two to both sides, it cancels on the left hand side and adds two to that side on the right hand side. And if I divide by three, Y is going to be equal to two thirds. And if I take my two Y plus one equal to zero, if I subtract one from both sides, it ends up two Y equal. Negative one. Divide by two and Y is equal to. Negative one.