Int Algebra: Completing the Square: An Example

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Quadratic Equations and Inequalities
Solving by Completing the Square
Completing the Square: Another Example Page [1 of 2]
So whenever you have a quadratic equation that starts with x2 and stuff, you can always solve it by completing the
square. Let me now show you what you would do if you actually had something in front of the x2.
Suppose we have 2x2 + 4 – 1 = 0. Now, if you want to complete the square, what you want to do is reduce it to the
other case I talked about, when, in fact, there was only a 1 in front of this. So whenever you have stuff in front of this,
the first step is just to divide everything through, both sides so you don’t change anything, by that coefficient. And that
way you’ve reduced it down to something with an x2 and you can use the method I just explained. So what I'm going
to do is I'm going to divide both sides by 2. That means I have to divide every single term by 2. So here, it cancels
out, which is the whole point of this. Here, I have +2x, and here it’s a little bit annoying, I have
2
? 1 , because I have
2
-1
. And that equals, well, zero over anything is just still zero. Okay, great!
Well, now back to the case I talked about with a 1 in front of here. So now I can actually use the completing the
square method I described, which was, let me remind you, you bring all the constant terms over to this side, take the
thing in front of the x, take half of it and square it and add it to both sides. So let’s try that right now. x2 + 2x, and I
always leave a big space here. I bring this over, it becomes now a positive
2
1
. Now what I want to do is add
something to both sides. What do I do? I take this thing in front of the x, take half of it, it should be 1, and square 1.
Squaring 1 is just 1. So I’d add it to both sides, plus 1, plus 1. You have to add to both sides, in order to keep this
thing balanced.
If you were to do that technique with the original thing, with the 2 there, in fact, this will not factor nicely into a perfect
square. So always remember to first clear away that coefficient by dividing everything through by the coefficient and
getting it down to this. And then it’ll work.
Okay, let’s see what happens. Well, this should factor into a perfect square. It should be that, and you can check it
and see that x times x = x2. The inside term is x, the other term is x, that’s 2x. 1 × 1 = 1. What do we have on this
side? Here we have 1
2
1
, and what’s 1
2
1
? 1
2
1
is just
2
3
. That’s where we are right now.
Okay, well how do we solve? Well now I’ve got a perfect square, so I can take plus or minus square roots of both
sides. So let’s try that. If I take plus or minus square roots of both sides, what I see is x + 1 = +
2
3
. Now, that
square root is on everything, including that little 2, which is pathetically drawn on the bottom there. Now I want to
bring the 1 over, so I subtract the 1. And so what I see is x = -1 +
2
3
.
Now, that can’t be simplified anymore, although some people might like you to rationalize the denominator. So if you
want to do that, let me just show you what that would be really fast, remind you how that goes. That would be
2
3
.
I’d multiply top and bottom by 2 and that would give me 6 and on the bottom I’d have 2 × 2 = 2. So that’s
an equal thing to this, but it has no denominator with a square root,
2
6
. So you could say the answer is x = -1 +
2
6
, or the other answer is x = -1 -
2
6
.
Quadratic Equations and Inequalities
Solving by Completing the Square
Completing the Square: Another Example Page [2 of 2]
So there are two answers to the original quadratic equation. That one, by the way, I remind you, is way over here.
And it turns out these two answers satisfy that. Notice that again these aren’t nice numbers, like 3 and –5, and that’s
because this thing couldn’t be factored.
So now we’re seeing that we can even solve quadratic equations where we can’t even factor it. Well, we could take
this basic principle, this principle of completing the square, and if we do it in general with not particular numbers, but
just with any old things in there, we’d actually produce a formula that would always solve a quadratic equation. And
this is known as the quadratic formula. And we’ll see that next time.