Solving equations with zero product property

Let's say that we've got the equation (2x - 1) times (x + 4) is equal to (0). Pause this video and see if you can figure out the (x) values that would satisfy this equation, or essentially our solutions to this equation. All right, now let's work through this together.

So at first, you might be tempted to multiply these things out, or there's multiple ways that you might have tried to approach it. But the key realization here is that you have two things being multiplied, and it's being equal to zero. So you have the first thing being multiplied is (2x - 1). This expression is being multiplied by (x + 4), and to get it to be equal to zero, one or both of these expressions needs to be equal to zero.

Let me really reinforce that idea. If I had two variables, let's say (a) and (b), and I told you (a) times (b) is equal to zero, well, can you get the product of two numbers to equal zero without at least one of them being equal to zero? And the simple answer is no. If (a) is seven, the only way that you would get zero is if (b) is zero, or if (b) was five, the only way to get zero is if (a) is zero.

So you see from this example either—let me write this down—either (a) or (b) or both. Because (0) times (0) is (0) or both must be zero. The only way that you get the product of two quantities and you get zero is if one or both of them is equal to zero. I really want to reinforce this idea; I'm going to put a red box around it so that it really gets stuck in your brain, and I want you to think about why that is. Try to come up with two numbers. Try to multiply them so that you get (0), and you're going to see that one of those numbers is going to need to be (0).

So we're going to use this idea right over here. Now, this might look a little bit different, but you could view (2x - 1) as our (a) and you could view (x + 4) as our (b). So either (2x - 1) needs to be equal to (0) or (x + 4) needs to be equal to (0), or both of them needs to be equal to (0). So I could write that as (2x - 1) needs to be equal to (0) or (x + 4) is equal to (0).

And so let's solve each of these. If (2x - 1) could be equal to (0), we'll see you could add (1) to both sides and we get (2x) is equal to (1). Divide both sides by (2), and this is just straightforward solving a linear equation. If this looks unfamiliar, I encourage you to watch videos on solving linear equations on Khan Academy. But you'll get (x) is equal to one-half as one solution.

This is interesting because we're going to have two solutions here. Or over here, if we want to solve for (x), we can subtract (4) from both sides, and we would get (x) is equal to negative (4). So it's neat in an equation like this; you can actually have two solutions. (x) could be equal to one-half, or (x) could be equal to negative four.

I think it's pretty interesting to substitute either one of these in. If (x) is equal to one-half, what is going to happen? Well, this is going to be (2) times one-half minus (1). (2) times one-half minus (1)—that's going to be our first expression, and then our second expression is going to be one-half plus (4). So what's this going to be equal to? Well, (2) times one-half is (1); (1 - 1) is (0). So I don't care what you have over here: (0) times anything is going to be equal to (0).

So when (x) equals one-half, the first thing becomes (0), making the product equal (0). Likewise, if (x) equals negative (4), it's pretty clear that this second expression is going to be (0), and even though this first expression isn't going to be (0) in that case, anything times (0) is going to be (0).

Let's do one more example here. So let me delete out everything that I just wrote here. And so let's—I'm going to involve a function. So let's say someone told you that (f(x)) is equal to ((x - 5)(5x + 2)), and someone said find the zeros of (f(x)). Well, the zeros are what are the (x) values that make (f(x)) equal to zero. When does (f(x)) equal (0)? For what (x) values does (f(x)) equal (0)? That's what people are really asking when they say find the zeros of (f(x)).

So to do that, well, when does (f(x)) equal (0)? Well, (f(x)) is equal to (0) when this expression right over here is equal to (0), and so it sets up just like the equation we just saw: ((x - 5)(5x + 2)) when does that equal (0)? And like we saw before, this has—well, this is just like what we saw before, and I encourage you to pause the video and try to work it out on your own.

So there are two situations where this could happen, where either the first expression equals (0) or the second expression, or maybe in some cases you'll have a situation where both expressions equal (0). So we could say either (x - 5) is equal to (0) or (5x + 2) is equal to (0). I'll write an (or) right over here.

Now, if we solve for (x), you add (5) to both sides of this equation, you get (x) is equal to (5). Here, let's see— to solve for (x), you can subtract (2) from both sides; you get (5x) is equal to negative (2), and you can divide both sides by (5) to solve for (x), and you get (x) is equal to negative two-fifths.

So here are our two zeros. You input either one of these into (f(x)). If you input (x) equals (5), if you take (f(5)), if you try to evaluate (f(5)), then this first expression is going to be (0), and so a product of (0) and something else, it doesn't matter that this is going to be (27); (0) times (27) is (0).

And if you take (f) of negative two-fifths, it doesn't matter what this first expression is; the second expression right over here is going to be (0). (0) times anything is (0).