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>> Hi, this is Julie Harland
and I'm YourMathGal.
Please visit my website
at yourmathgal.com
or you could search for any
of my videos organized
by topic.
This is the Equation
of a Circle Part 5
and we do the following
problem on this video.
All right, here's our problem.
Write the equation
of the circle
in standard form,
state the center and radius
and graph.
Well, it's nice that--
is that we're told
that this is a circle
so that's easy.
This is the equation
of a circle in general form
so it doesn't tell us the
center or the radius
and that's exactly why we're
being asked to put it
in standard form first.
So what we do is we're going
to use the completing the
square, we're going
to put the X squared
in X term together
so we have X squared plus 4X
and we're going
to have add something
to get a perfect square.
There, we're going
to put the Y squared
and then negative 6Y together
and add something
to make it a perfect square
and we're going
to put the constant
on the other side by adding 12
to both sides all right,
so if we add 12 to both sides,
we have equals 12.
Now, both of these things
that I'm going to be adding
on the left
to make it a perfect square,
I like to always write
down plus blank plus another
blank so I don't forget I'm
going to have to add something
to the right side as well.
So I'm going to come down,
I'm going to figure
out what the perfect squares
would be.
All right,
so I have X squared plus 4X.
So remember how this works,
you have an X here
and since it's a plus 4,
you're going to--
for the coefficient of X,
we're going
to take half of that.
So what's half of plus 4?
It's plus 2.
2 squared is what goes
on the blank
so what we're doing is we are
adding 4 to both sides.
All right, we also have
to complete the square
of the Ys, right?
So we have Y squared minus 6Y
plus something.
Trying to figure that out,
so again, we've going
to put Y. The coefficient is
minus 6, we take half of that
that would be minus 3.
So to make it a perfect
square, that would be 3
squared, 9 so I have to add
to make that a perfect square
and make it equal
to the same thing
as Y minus 3 quantity squared
so I'm adding 9 to both sides.
So what do I have
on the right-hand side?
I really have 12 plus 4 plus 9
which is 25
and I've just written
that equation in standard form
or you can write it
as equal 5 squared,
if you write it
like that then you could see
the radius as well,
it's going to be 5, right?
I'm running 25
as something squared.
So what is the center
of the circle?
Negative 2, right?
You have to put a negative 2
for X to make what's
in parenthesis, you know,
that's one way of thinking
about it and 3 will be the
Y value.
So this is my H,
K that's my center
of the circle
and my radius is 5.
So if I want
to graph this circle all
right, it's a lot easier now
to graph.
[ Pause ]
I'm going to have
to put small things here.
All right, here we go.
So 8, that's 8.
1, 2, 3, 4, 5, 6, 7--
it's best if you have real
graph papers
so it looks little better
than mine.
All right,
so we've got negative 2,
3 as the center.
So I'm just going
to put a little dot here
but remember
that it is not part
of the circle.
To get the circle,
you've got to go the radius
in all four directions.
So we're going to 5
to the right, that's part
of the circle.
Okay, always going
from the center,
the center is negative 2, 3,
we're going to go up 5,
north of the left 5
and we're going to go down 5,
and we could get a
rough sketch.
Now, another thing you can do
is figure out where it crosses
the X and Y-axis.
If they ask you to do that,
let's figure that out.
One of the X-intercepts,
you're going
to let Y equals 0, correct?
Let's just get a practice
with algebra.
So you can either plug in 0
into the standard form
of the equation
or you could plug in it
into the equation
up here, right?
The general form, all right?
I don't know which is going
to be easier.
Let me do it in a general form
in the top one.
So what happens if we put
in 0 for Y?
Then what I get?
We have X squared plus 0
squared plus 4X minus--
I'm putting the 0 for Y,
6 times 0 minus 12 equals 0.
So what does that give me?
That gives me X squared plus
4X minus 12 equals 0
and if I can factor,
that would be awesome.
I think I can.
I think it's X plus 6
and X minus 2 equals 0,
is that correct?
I think so.
So X is negative 6
or 2 so that's two
ordered pairs.
One of the ordered pairs would
be negative 6,
0 and the other one would be
2, 0.
Now, why would you do this?
Only if you want
to be a little bit more
accurate when you're graphing
the circles.
So now, we'll just know two
more points,
I've got negative 6, 0--
1, 2, 3, 4, 5, there's one
of that ordered pairs
and I've got 2, 0.
So in other words,
it just helps me
in doing the sketch a little
bit more.
And also, it sort of symmetric
across the center line right
here, you can sort
of imagine the center line
across where the center
of the circle is.
So this is going
to be another one right
up here.
There's going
to be another one right
up here.
It just helps you make
some guides.
You could also do the same
thing by plugging in 0 for X
into this equation
and then you're going
to figure out what the
Y-intercepts are, okay?
So let's go ahead and now,
we're going to plug in--
[ Noise ]
Here we go.
To get the Y-intercepts,
we're going
to let X equals 0, right?
So let's go up here,
we're going to put in 0
for X. Now, I'm going to--
when it's--
it's going to be 0,
I'm just going to like,
you know, make it disappear.
So by putting 0 squared,
that's 0.
So I end up with Y squared--
this term is gone,
the plus 4X, right?
And then I've got minus 6X--
I mean, 6Y minus 12 equals 0
and let's see does
that factor--
I don't think
that one is going
to be as easy.
So for this one, you--
it's not that simple to do,
you would have
to use the quadratic formula
to find out those Y
coordinates, okay?
So I'm not going to do it.
You could use the quadratic
formula to solve
for Y. The other thing you
could do is you could have
plugged it into the Y--
you could have plugged in X
in right here.
So let's do it there,
see if it's any easier.
All right, so we're going
to plug in 0 for X
into that equation
right there.
So I'm going to get you--
when you put in 0 for X,
everything would be--
you're going
to get 2 squared 'cause 0 plus
2 is 2.
So you're going
to get 2 squared plus Y minus
3 squared equals 25
and 2 squared is 4.
So by just subtracting 4
from both sides
to get Y minus 3 squared is--
what's 25 minus 4?
21 and I'm just going
to take the square roots
of both sides
and you could you see
if I would use the quadratic
formula, I will get the same
thing, 3 plus
or minus the square root
of 21.
So if I did this,
I'd actually have to take
out my calculator
and I'm going
to approximate this so we want
to put in the square root
of 21 which is approximately
4.6, okay?
So let's see,
if you did 3 plus 4.6 you'd
have 7.6, right?
And if you did 3 minus 4.6 it
would be negative 1.6.
So what does that mean?
On your Y-axis,
you would have 7.6.
Well, it's really close
up here, or you would have
negative 1.6 which is
about right here,
that gives you not too may
more points.
So usually, you don't want
to go through all that hassle
but you should know how
to do it anyway.
You should know if you--
if somebody asked you what the
X and Y-intercepts were,
you should be able to go
through that process.
So the simple thing,
of course,
is just to do these 4 points
here and sketch it, you know,
fairly well and you're done
with the problem
which is all I really ask for.
So I'm just kind
of catching you
up on your Algebra here,
right?
So the instructions
where write the equation
in standard form,
state the center and radius
in graph and we did it.
But if the directions again,
what are the X
and the Y-intercepts?
We actually computed them
so let's just write
down what they really are,
the X-intercepts,
those were the easy ones,
remember negative 6,
0 and 2, 0?
And the Y-intercepts were the
crazy ones,
that's when you put in 0
for X, what do we get?
Now, I don't want the
approximate,
it's 3 plus radical 21, right?
Which does approximate to 7.6
but the actual Y-intercept is
0, 3plus square root of 21
and the other one is 3 minus
square root of 21
for the Y coordinate.
So sometimes, they'll also ask
for the X-intercepts
and Y-intercepts and just--
as a reminder,
that's how you would do it,
you put in 0 for X
to get the Y-intercepts,
put in 0 for Y
to get the X-intercept.
All right,
so onto the next video,
I will work
with some more problems
like this.
[ Silence ]
Please visit my web site
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.