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Alright, what I like to do is show you how to complete the square.
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And I'm just going to work with a very basic and proper...
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Tracer Rodney, could you please call it such a center too?
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Alright, so when I'm completing the square, there's a couple things on the work out.
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I'm just going to kind of do step by step when I'll explain,
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and then I like you to do it on your own, what exactly I'm doing.
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So when we're completing the square, what complete square helps us do
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is that one thing gets us into the standard form of a quadratic equation.
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So it puts us in the standard form, which is very helpful for trying to identify the vertex.
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It also helps, there's also another way to solve for X.
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So instead of doing the quadratic formula, or if you can't factor a problem,
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then what you can do is you can do a, you know, you can complete the square.
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So this problem actually looks like it can be factored.
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You know, if I was going to factor this, right?
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So this could be easily factored, but I'm going to show you complete the square.
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So you can see that, I'm still going to get the same answers.
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Alright, so X equal to negative 3 and X equals 1.
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So you know what the answers are going to be by factoring.
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And let's complete the square.
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So the first thing to complete the square, what I'm going to do is I need to get it to a perfect square triangle.
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So what I'm going to do is I'm going to put parentheses around my X.
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Alright, so I'm going to do that now with your function.
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And the reason why I'm doing this is because I want to isolate my X guys.
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And what I'm going to do is I want to get this to be a perfect square triangle.
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I don't really care about my constant right now.
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It's just over there.
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And the other important thing is I need to make sure there's a 1 in front of this number.
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So if there's a 2 or if there's a negative 1, whatever's in front of there, I'm going to have to factor out.
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Now, what you're going to want, the reason why I put parentheses just from the X is you only want to factor that out out of here.
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You don't need to factor it out of this negative 3.
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We just leave that constant kind of on its own.
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So if I did have something here, let's say I had a 2 here, I would have to factor a 2 out of the X squared and a 2 out of the X.
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Fortunately for this first problem, I had nothing in front of the X. So I'm good.
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So now to get this to be a perfect square triangle meal, what I'm going to have to do is we'll call our B over 2 squared.
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And if you remember a quadratic formula is in the form of, or you know, a squared form is A X squared plus B X plus C.
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So my B in this equation is 2. So I have 2 over 2 squared.
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2 divided by 2 is 1. 1 squared is equal to 1.
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So what I'm going to do is I'm going to add that 1 into my equations.
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Now I have f of X equals ag squared plus 2x plus 1 minus 3.
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Now I added a 1 on the right side. So if I'm going to add a 1 to make an equation, you know, still the same.
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You have to add 1 and subtract 1.
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So if you do the same thing, you'll find you'll be able to.
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Okay. Did you add it? Make sure you add it inside the parentheses.
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And then whatever you add, make sure you subtract.
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You've got to keep an equation the same, right?
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If I say 3 is equal to 3, the only thing that's true.
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But if you add a 1 to this side, now it's not true anymore, right?
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Now 3 is equal to 4. So if I'm going to add a 1, I'll subtract the 1.
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Now why would we do that? We don't. We would never do that.
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But you need to show that. This is why it makes sense.
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You can't just add and expect things to be true. Add it's correct.
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But so why did I add it inside the parentheses?
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Why would I do this? Or what's even the point to do this?
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Well, this will call a perfect square trinomial.
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And what's so important about this is this can be factored down when I like factor this.
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Or you know, breaks into binomials. This is actually a binomial squared.
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So I can actually rewrite this as x plus 1 times x plus 1.
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And if I was to expand that, I would get x squared plus 2x plus 1.
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Now, an easy way to look at this is how do I always, how can I get this to be a binomial squared?
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Well, if you kind of look at it as x plus or minus b over 2 squared.
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And this is what we want to get. We want this start end point that we want to get to.
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We want to get a binomial squared.
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So I look at this and I say what was b over 2?
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b over 2 was 1.
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So I look at this and I say, well, if I did x plus 1 squared,
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1 times 1 is 1. And then if I was about to do this, you have 1x plus 1x, which is 2x.
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If this middle term was negative, then I'll have x minus 1.
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Makes sense? So I'm going to leave this as x plus 1 squared.
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And then negative 3 minus 1 is negative 4.
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Now, what I've just done is complete the squared.
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So now, this is what we call standard form.
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And if I want to find the vertex of this equation, it's so easy.
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The vertex is the opposite of inside the function, negative 1 and negative 4.
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So it's really easy to find the vertex.
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The other thing that's really helpful about this is it's really easy to find your zeros.
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Or your x-intercepts.
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So if I want to find the zeros, what I need to do is I need to put my output value as 0.
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So I need to see 0 equals x plus 1 squared minus 4.
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And now, what I need to do is I can solve for x.
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I'll let you finish writing this a little bit. Let you take it in.
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Then you might go fast.
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How's it going to take me? Good.
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So now, it's self-rex.
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We've got to get to x by itself, right?
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On new subtraction first.
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Right? Now I do this root.
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So therefore I have square root of 4 equals x plus 1 minus the 1.
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x equals negative 1 plus or minus the square root of 2.
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Now remember, our two answers were negative 3 and negative 4, right?
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I'm sorry, it's critical 4.
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So I have x equals negative 1 plus or minus square root of 4 is 2.
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So negative 1 plus 2 is negative 1 and negative 1 minus 2 is negative 3.
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Are those the same two answers I got when I defector? Yes, they are.
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So some people like, you know, so when it's equal, when you defector a problem, it's very easy, just to factor.
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However, if it's a heart, if it's a problem, you cannot factor completing a square and that's set at equal to 0.
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It's a very easy way to find the zeros I'm going to function.
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Ready for a test?