 Consider the following quadratic equation, x minus two times x plus one equals four. This is a very tempting question to give to learners here about quadratic equations because the left hand side is factored, it's a binomial times a binomial and that's nearly always our goal, right? Yay, we're trying to solve this by factoring so it already did it for me. But that would be great if the left hand side was equal to zero, right? Because then the zero product property would apply. But the problem is the right hand side is equal to four and so you might be tempted to be like, well, I'll just factor four then. But four has lots of different factorizations. You have two times two, sure. You have one times four. You also have one half times eight. You have five times four fifths. That's another factorization. And so it's like, dang, yo, there's a lot of ways of factoring four. This is what's so significant about zero. This is why we use the zero product property is that when it comes to factoring zero, there's essentially only one factorization, zero times something else who cares what that other thing is. For any non-zero number though, the factorization gets a little bit complicated. So in this regard, if we want to solve this quadratic equation, we have to destroy the factorization we have. We have to kind of tear down the old empire to build the new one here. So foil out the left hand side, you're gonna get x squared plus x minus two x minus two is equal to four. Combining like terms to put it in standard form, we're gonna get x squared minus x. I'm also gonna move the four to the other side. So we're gonna get minus six right here. Fortunately, this is a monic quadratic equation, the leading coefficients one. So I'm gonna look for a magic pair and that'll give me the factorization. I need factors of negative six that add up to negative one. Six does factor as three and two, which differ only by one. So let's make sure we get the sign in the right spot. If I have a negative three and a positive two, then negative three times two gives me negative six, but negative three plus two gives me a negative one. There's your factorization and then your solutions will be three and negative two. So you might be tempted to say the solutions are like positive two and negative one, but nope, nope, nope, nope. We need to put this thing into standard form before we factor it. Because the right hand side was not zero, we really can't benefit from this factorization whatsoever. So I don't want you to be tempted by this one. Just having a factorization is not good enough. The factorization needs to equal zero. If the product doesn't equal zero, we can't use the zero product property and there's no such thing as a four product property or a seven product property. It needs to be the zero product property.