Solving by completing the square: rational solutions

0:00- [Instructor] So, let's say we wanna solve the following equation, x squared minus 2/3 x plus 1/2 is equal to 5/12, and we wanna solve it by completing the square. So, like always, pause this video and have a go at this by yourself before we do this together.

0:21All right, now let's do this together. Now, when we complete the square, what we wanna do is get it in the form where we have x plus or minus some constant squared is equal to some other constant squared. And one way we could do that in this example is, we can move all the constants to one side, so let's move 'em all to the right-hand side.

0:44So, if I wanna get rid of this 1/2, I can subtract 1/2 from the left side. If I do it from the left side, I, of course, have to do it from the right side. So, I could subtract 1/2 there, and 1/2 is the same thing as 6/12, so let me write that down as 6/12. Well, negative 1/2 is the same thing as negative 6/12, so I'm subtracting the same thing from both sides, and then what do I get?

1:06On the left-hand side, I am going to get x squared minus 2/3x. These cancel out, and on the right hand-side, if I have 5/12 minus 6/12, that is negative 1/12. Now, what I wanna do is have some type of a perfect square on the left-hand side, and this is essentially where we are completing the square.

1:30And in other videos, we have seen that if we take half of this coefficient here and add the square of that here, then we're gonna have a perfect square. And of course, if we add anything to the left-hand side, we have to add it to the right hand side too. So, if we say half of this, that's going to be negative 1/3, and then if we want to square that, that's going to be positive 1/9.

1:54All I did is I took half of negative 2/3, which is negative 1/3, and I squared it to get 1/9. Now, if I add 1/9 to the left-hand side, I, of course,. also have to add it to the right-hand side, so plus 1/9. Now, we can see that we have on the left, all of this is taking this form.

2:15This is going to be x minus 1/3 squared. You can multiply that out and verify that that's exactly what we get over here, and that's why we added 1/9 here. We took half of this, and then squared it and added that. Now, on the right-hand side, what do we have? So, let's find a common denominator.

2:35Let's see, 12 and 9. It looks like 36. So, I'll do that over here. This is the same thing, I'll use those same colors. Negative 1/12th as a 36th, that's negative 3 over 36. And then plus 1/9, that's plus 4 over 36. Plus 4 over 36. So, negative 3 over 36 plus 4 over 36 is going to be equal to, if I add these two together, it's equal to 1 over 36.

3:05Well, 1 over 36 is the same thing, I could rewrite this as, well, actually, lemme just keep it this way, is equal to 1 over 36. And so, what we see here is that x minus 1/3, x minus 1/3 could be the positive square root of 1 over 36, which would be 1/6, or x minus 1/3 could be equal to the negative of the square root, which would be negative 1/6.

3:35'Cause if this whole thing was negative 1/6 or 1/6, when you square it, you get 1/36th. So, let's go with this left-hand scenario right here. We just add 1/3 to both sides, so let's just do that. And 1/3 is the same thing as 2/6, so let me do that as 2/6. And so, these cancel out, and on this side, I get x is equal to 1/6 plus 2/6 is 3/6, which is the same thing as 1/2.

4:03I ran outta space there. Same thing as 1/2, or over here, I'm going to add 1/3 to both sides. Let me just do it this way. 1/3 is the same thing as plus 2/6. That's still 1/3, and I am going to get x is equal to negative 1/6 plus 2/6 is equal to positive 1/6. So, there we have it, x could be equal to 1/2, or I dunno if you can, let's see, I'm sure you could see this x.

4:36X is equal to 1/2, or x is equal to 1/6. And you could validate that that works with that original equation.