Int Algebra: Solving by Completing the Square

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Quadratic Equations and Inequalities
Solving by Completing the Square
Solving by Completing the Square Page [1 of 2]
Okay, now I have something serious that I want to talk to you about and this is important, so I really want you to listen
to me.
You know, we saw earlier, you might remember that if I give you something like x2 = 25 – of course, you could bring
this over to the other side, you can factor the difference of two perfect squares and find the answer, + 5. And that’s
the way that I like you to think about doing it. Of course the lazy person’s approach would be just to take the square
roots of both sides, but you have to remember plus or minus square roots, because there are these two answers. So
you could just then go from here and say, “Well, square rooting both sides, I’d have x = + 25 ,” which would equal +
5, and there’s your answer.
Okay, well, that works pretty well whenever you have something squared equals a number. You just take plus or
minus the square roots of both sides. If I, for example, say to you, “x2 = 9,” well you could say, “x = + 9 , + 3.” Not a
problem. If I say to you, “x2 = 2,” even that’s not a problem, because you could say, “x = + 2 .” The point is if you
have something that’s just a perfect square here and it equals something, I can just take + the square roots of both
sides and be done with you.
However, let’s consider the following extremely annoying example: x2 + 6x + 1 = 0. Let me tell you that this is not
going to be pretty. The first thing that I would, if I were me, and I am, is to try to factor. You can’t pull anything out,
there’s no common factor there. The trinomials, let me just try it – x and x, both the same sign, both positive. Now, I
need to put a number in here that multiplies to give 1 and combines to give 6. Well, 1 and 1 or
2
1
and 2, or
something, none of those work. So, in fact, this is something that I can’t factor, using this technique. And that really is
annoying, because if it can’t be factored, how in the world am I going to solve this?
Well, what I'm going to do is I’m going to remember this idea that I just told you about, which is if I have a perfect
square equal a number, I can take + the square root of both sides. Well, one maybe tempted to say, “Oh, okay, what
I’ll do is I’ll bring this stuff over to the other side and then take square roots, because I’d have a perfect square here.”
But the problem is that then I’ll have x’s there, too. So then I’ll have x’s under a square root and, let me tell you, as
much as I hate having x’s with squares, I hate x’s under square roots that much more. So what in the world can I
possibly do?
Well, the fantasy is going to be to try to take these people here and turn that into a perfect square. So I want to take
these and turn this into a perfect square. This technique, by the way, is called completing the square. And why? It’s
because it’s as though you have something like this and you want to now make it into a perfect square. So you want
to add extra stuff and, all of a sudden, it’s a perfect square. So I can take the square roots of both sides. So this is
called completing the squares. I start off with something that’s not a square and I just tack on a little bit more, add
something and, all of a sudden, it becomes square. See it?
Now, here is the method for completing the square. Step one is to move any constant you may have, any number
that doesn’t have an x, to the right. I know this doesn’t feel right. I’m telling you, this is going to be serious business,
because you’ll always want to have things equal zero so you can factor. But we tried factoring and it failed. Now let’s
move on. So to complete the square, I’m going to take this and move it to the other side. So I'm going to move all
constants over to the right-hand side. So now it looks like that. But actually, now what I'm going to do is complete the
square. I’m going to add something in to make the thing a square. So what am I going to add in to make the thing a
square? Well, here’s what I’m going to first of all do; first of all, when I write these things, I put a huge space, because
I'm going to put something in there. Now, to keep this equal sign legitimate, whatever I add or subtract here, I have to
add or subtract the same thing here. And now, here’s what you do: you take a look at the coefficient in front of the x
term. Notice that’s a plus 6. Take half of it – so take the plus 6, take half of it – what does that equal? That would
equal plus 3, and now square it. And that equals 9. That is the number that’s going to complete the square. Let me
say that again. The procedure is always the same. Take whatever the coefficient is in front of the x term and take
half of it, take that answer and square it and that’s what you get.
Quadratic Equations and Inequalities
Solving by Completing the Square
Solving by Completing the Square Page [2 of 2]
So remember I’m always starting off with something that looks like this, has the x2 out in front. So now, I’m going to
add 9 to that side, but to keep this thing balanced, I have to add it on that side, too. And now, what do I see? Well
now, don’t start moving things over. You see sometimes people, after doing all this work, they say, “Okay, I’ll move
everything over and try to factor.” If you move everything over and try to factor, you’re going to get exactly this,
because these will cancel out and you're going to get the same thing. So then you're going to undo what you're doing.
Don’t do that. Instead keep this segregated from this.
Okay, now this thing should factor and, if we did it correctly and carefully, it should be a perfect square. So let’s see if
it really is a perfect square: x, x, same sign positive, 3, 3. Let’s check. 3x + 3x = 6x, 3 × 3 = 9, perfect square. And,
on this side, we have an 8. So what I see is (x + 3)2 = 8. Well, how can I solve this? Well now that I have a perfect
square, I can take + square roots. And so what I see is the following: I’d see x + 3 = + 8 . Now, I want x alone, so
I’ll bring this 3 over to the other side, so I subtract it. So I see x = -3 + 8 . Now, actually, 8 we can simplify that a
little teeny bit, if we wanted to, because remember that 8 = 4 × 2. So the 8 would equal 4 × 2 . 4 = 2 and
so I’d have 2 2 . So I could actually write this as 2 2 . So I could say that x = -3 + 2 2 .
So there’s two answer here, and let me just make sure you see what they are. x = -3 + 2 2 or x = -3 - 2 2 . That’s
what this + means, it means you write everything down first with a plus, and then everything down next with a minus.
These are two different answers and they turn out both to solve or to satisfy that original equation, the one that went
x2 + 6x + 1 = 0. You can plug those in and see. Now you can see why I couldn’t factor it. You see there’s all sorts of
square root stuff there. And the way I did it was to complete the square. The method was, since I have something
that looks like this, I bring this over, take half of this and square it.
Okay, next up I'm going to show you an example where we have to use a slightly different approach, because there’ll
be something in front of the x2. This method is going to work when you have nothing in front of the x2. If there’s
something in front of the x2, you have to do one little teeny thing. I’ll show you what that is next.