- Solve by completing the square: Integer solutions
- Solve by completing the square: Non-integer solutions
- Solve equations by completing the square
- Worked example: completing the square (leading coefficient ≠ 1)
- Completing the square
- Solving quadratics by completing the square: no solution
- Proof of the quadratic formula
- Solving quadratics by completing the square
- Completing the square review
- Quadratic formula proof review
We can use the strategy of completing the square to solve quadratic equations even when the solutions aren't integers. Created by Sal Khan.
- [Instructor] 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, it'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 six, and when I take their product, I get positive three? Well, if I'm thinking just in terms of integers, three is a prime number, it only has two factors one and three. And let's see one plus three is not equal to six, 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 if there are x values that would satisfy this equation, completing the square will help us solve it. And the way I do it, I'll say zero is equal to, let me rewrite the first part, x squared plus six x, and then I'm gonna write the plus three out here. And my goal is to add something to the right-hand expression, right over here, and then I'm gonna subtract that same thing, so I'm not really changing the value of the right-hand side. And I wanna 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 six, and say, okay, half of that is positive three, and if we were to square that, we would get nine. So let's add a nine there. And then we could also subtract a nine. Notice, we haven't changed the value of the right-hand side expression, we're adding nine and we're subtracting nine. And actually, the parentheses are just there to help it make 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? Well, what I just showed you, let me do it in this green blue color, this thing can be rewritten as x plus three squared. That's why we added nine there, we said, all right, we're gonna be dealing with a three 'cause three is half of six. And if we squared three, we get a nine there. And then this second part, right over here, three minus nine, that's equal to negative six. So we could write it like this, zero is equal to x plus three squared minus six. And now what we can do is isolate this x plus three squared by adding six to both sides, so let's do that. Let's add six there, let's add six there. And what we get on the left-hand side, we get six is equal to, on the right-hand side, we just get x plus three squared. And now we can take the square root of both sides, and we could say that the plus or minus square root of six is equal to x plus three. 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 six, that means that the something is either going to be the positive square root of six, or the negative square root of six. And so now we can, if we wanna solve for x, we can just subtract three from both sides. So let's subtract three from both sides, and what do we get? We get on the right-hand side, we just are left with an x, and that's going to be equal to negative three plus or minus the square root of six. And we are done. And obviously we could rewrite this and say, x could be equal to negative three plus the square root of six, or x could be equal to negative three minus the square root of six.