Solve by completing the square: Non-integer solutions | Algebra I | Khan Academy

Let's say we're told that zero is equal to x squared plus six x plus three. What is an x, or what our x is that would satisfy this equation? Pause this video and try to figure it out.

All right, now let's work through it together. So the first thing that I would try to do is see if I could factor this right-hand expression. I have some expression that's equal to zero. So, if I could factor it, that might help solve.

So, let's see: can I think of two numbers that, when I add them, I get 6, and when I take their product, I get positive 3? Well, if I'm thinking just in terms of integers, 3 is a prime number. It only has 2 factors: 1 and 3. And let's see, 1 plus 3 is not equal to 6. So, it doesn't look like factoring is going to help me much.

So, the next thing I'll turn to is completing the square. In fact, completing the square is a method that can help us solve if there are x values that would satisfy this equation. The way I do it, I'll say 0 is equal to... Let me rewrite the first part: x squared plus 6x. Then, I'm going to write the plus 3 out here, and my goal is to add something to this equation—or to the right-hand expression—right over here. Then, I'm going to subtract that same thing, so I'm not really changing the value of the right-hand side.

I want to add something here that I'm later going to subtract so that what I have in parentheses is a perfect square. Well, the way to make it a perfect square— and we've talked about this in other videos when we introduced ourselves to completing the square—is we'll look at this first degree coefficient right over here, this positive 6, and say, okay, half of that is positive 3. If we were to square that, we would get 9.

So, let's add a 9 there, and then we could also subtract a 9. Notice we haven't changed the value of the right-hand side expression; we're adding 9 and we're subtracting 9. Actually, the parentheses are just there to help make it a little bit more visually clear to us, but you don't even need the parentheses. You would essentially get the same result.

But then what happens if we simplify this a little bit? What I just showed you—let me do it in this green-blue color—this thing can be rewritten as x plus 3 squared. That's why we added 9 there; we said, all right, we're going to be dealing with a 3 because 3 is half of 6, and if we squared 3, we get a 9 there.

Then, this second part right over here, 3 minus 9, that's equal to negative 6. So, we could write it like this: 0 is equal to x plus 3 squared minus 6.

Now, what we can do is isolate this x plus 3 squared by adding 6 to both sides. So let's do that. Let's add 6 there, let's add 6 there, and what we get on the left-hand side, we get 6 is equal to... on the right-hand side, we just get x plus 3 squared.

Now, we can take the square root of both sides and we could say that the plus or minus square root of 6 is equal to x plus 3. And if this doesn't make full sense, just pause the video a little bit and think about it. If I'm saying that something squared is equal to 6, that means that the something is either going to be the positive square root of 6 or the negative square root of 6.

And so now, we can, if we want to solve for x, just subtract 3 from both sides. So, let's subtract 3 from both sides. What do we get? We get on the right-hand side, we're just left with an x, and that's going to be equal to negative 3 plus or minus the square root of 6. And we are done.

Obviously, we could rewrite this as say x could be equal to negative 3 plus the square root of 6, or x could be equal to negative 3 minus the square root of 6.