Subscribe! http://www.freemathvideos.com Welcome, ladies and gentlemen. So what I'd like to do is show you how to determine the center and the radius of a circle when given an equation in standard form. And you can see that in our previous video, we found the center and the radius when we were given an equation that was in a format like this.
And it was pretty basic to find the center in the radius. We just had to find the h and k, and then find the radius. And you know, kind of minimal math was needed.
However, if you look at these two equations, we don't really have a way to see h and k, nor even r. So what we have to do is we have to do some manipulation.
And one important thing of noticing about this equation of a circle is that we have binomials squared. We have x minus h squared. Y minus k squared. Whereas these we just have the x squared and the y squared, right? We don't have a binomial squared-- in parentheses, x minus something, y minus something, and then that squared.
So that's what we want to create. We need to create a binomial squared. To do that, we need to create a perfect square trinomial. And the process of taking any equation or expression and creating a perfect square trinomial that can be factored down to a binomial squared is called completing the square.
Now, the first process in completing the square-- you kind of notice that we have x and y. So what I'm going to do is we're going to complete the square, and I'm basically going to group the x's together and group the y's together, and then put r over to the right hand side. So the first thing I'm going to do is subtract the one on both sides. And I have x squared plus 2x plus y squared equals 3.
Now, I'm going to want to group these together, and I'm just going to kind of draw like this, just put some kind of parentheses around there. Because basically what I want to do for the x's and for the y's is I need to create a perfect square trinomial.
Now there is a little caveat to that. I don't have anything with this y squared, right? And actually, since on that y minus there's no other term, I can actually create a perfect square trinomial pretty easily, and I'll show you what I mean by that-- or a binomial squared.
However, for this, I have this 2x. So I need to somehow rewrite x squared plus 2x as a perfect square trinomial. And the process again, as I mentioned, is completing the square. To do that, we're going to take b divided by 2 and square it.
Well, if you remember, b comes from ax squared plus bx plus c, where b is your coefficient of your linear term. So we're going to look at-- down at my second equation-- notice that b is 2. So therefore I get 2 divided by 2 squared. 2 divided by 2 is 1. 1 squared is 1.
Now, I'm going to take that 1 and I am going to insert it inside this parentheses. So now I have x squared plus 2x plus 1, plus I have this y squared. I don't really need the parentheses, so I'm going to leave that out there. Equals 3.
Now, since I added a 1 to the left side of the equation, I have to make sure I add a 1 to the right side of the equation. So I'm going to add a 1 over there. Now I have created a perfect square trinomial. By completing the square, by finding the value c that completes the square, I now have created a perfect square trinomial.
Now, all I need to do is factor this down to a binomial square. And you can basically just ask yourself, what two numbers multiply to give you 1, add to give you 2? Well, that is x plus 1 times x plus 1, which is the same expression multiplied by itself, so we write it as a binomial squared.
OK? Now again, you might say well, all right, so you did x. That's great, but you didn't do y. Well, remember when we just have a y squared by itself, we can just rewrite it. We can just rewrite k as 0. So I can write this as x plus 1 squared plus y plus 0-- or minus 0. Doesn't really matter-- squared equals 4.
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