in this lesson we're going to focus on the zero product property And the basic idea behind the zero product property is if you're multiplying two numbers a and b And if a and b equals zero then either a or b must be zero because 0 * anything is zero Now let's think about this Let's say we have two numbers that multiply to zero The only way this could happen is if one of those numbers is zero For instance if we have eight if the first number is eight the second number has to be zero It really doesn't matter what the first number is If it's 6 12 15 as long as the second number is zero the whole thing is going to be zero If the first number is zero it doesn't matter what the second number is It could be -4 7 12 0 * anything is zero So whenever you have two things that multiply to zero one of those things has to be zero Now this property is very useful when solving quadratic equations Typically when you factor a tromial you might get something that looks like this And you need to solve for x using a zero product property We can do that If either x - 3 or x + 2 is equal to zero this entire expression will be equal to zero So using the zero product property we can break this single equation into two parts We can set x - 3 equal to zero and we can set x + 2 equal to zero because if just one of these two factors equals zero the whole thing is going to be zero And so that's why we can do this It's because of the zero product property Now once we have these two equations we can go ahead and solve it To get the answer for the first one we just got to add three to both sides and we'll get x is equal to pos3 For the second one we need to subtract both sides by two and we'll get x is equal to -2 So that's one of the applications of using the zero product property It's very useful when solving quadratic equations especially when you're factoring For the sake of practice go ahead and calculate the value of x for these two equations The first one is going to be 3x * x - 7 is equal to 0 And for the second one it's going to be 2x - 3 * 3x - 5 = z So use the zero product property to calculate the value of x for each of these two equations Now for the first one what we can do is we can set each factor equal to zero So we could set 3x equal to zero and x - 7 = 0 For the first one we need to divide both sides by 3 3x / 3 is simply x 0 / 3 is zero So the first answer is just x is equal to zero For the second equation we simply need to add 7 to both sides and we get the answer x is equal to 7 So if we were to plug in zero or seven into the original equation it's going to work For instance if we plug in 0 into each x value we'll have 3 * 0 * 0 - 7 3 * 0 is 0 0 - 7 is -7 0 * -7 is 0 So this works Now if we plug in 7 it will work as well 3 * 7 and then 7 - 7 that's going to be zero 3 * 7 is 21 7 - 7 is 0 21 * 0 is 0 So using the zero product property we can solve for x whenever it's in factored form Now let's try the other example Now we're going to follow the same process We're going to set each factor equal to zero So we'll break it up into two equations 2x - 3 is equal to 0 and 3x - 5 is equal to zero So let's begin by adding three to both sides So we'll get 2x= 3 And then we'll divide both sides by two So the first answer that we get is x= 32 If we were to plug this in to the original equation this part will equal zero which means the whole equation will equal zero Now for the second one we're going to add five to both sides So we'll get 3x is equal to 5 And then we'll divide both sides by 3 So we'll get x= 5 or 3 So if you were to plug in any one of these two x values into the original equation the whole thing is going to equal zero So that's how you can solve equations using the zero product property First it needs to be in factored form like this And then you could set each factor equal to zero and then solve for