Parametric equations 1

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:03.060
Let's say I have a cliff.

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Let's make this cliff, I
don't know, let's say

00:00:05.340 --> 00:00:07.280
it's 50 meters high.

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And on this cliff,
I have a car.

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And this car is not just
sitting on the cliff,

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it's driving off of it.

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A very dramatic problem.

00:00:18.310 --> 00:00:20.860
So let's see, I have
this car here.

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And it's driving off of this
cliff at 5 meters per second.

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And I want to know, what is
the path of this car as it

00:00:33.040 --> 00:00:35.110
falls off of the cliff?

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So let's set up a little
coordinate axis here.

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So let's say that this is
the y-axis right there.

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And then this will
be my x-axis.

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So this is y, this is x.

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Let's say that this is the
point-- well, we know that this

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is a 50 meter high cliff.

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Maybe y equals 0 is sea level.

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So this would be 50 right here.

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And let's say that this point
right here on the cliff,

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that's at x is equal to 10.

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So this point right
here's the point 10, 50.

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And let's say the car's right
at this point, right about

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to drive off the cliff.

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And time is equal to 0.

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So this is that
time equal to 0.

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So t for time.

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Time is equal to zero.

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So my question is, what
happens to this car as

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it drives off the cliff?

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So this is a bit of a physics
problem, and I won't go

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deep into the physics.

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And I won't prove some
of the equations.

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And I encourage you to
watch the [UNINTELLIGIBLE]

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videos, the projectile motion
videos, if you want to know

00:01:45.930 --> 00:01:47.360
where the equations come from.

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But the point here is just to
get the equations and see

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what the graph looks like.

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So if I want to know x as a
function of time-- a suitably

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vibrant color-- so x as a
function of time is

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going to be what?

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Well, we're going to assume
that we're on a planet

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that has no air.

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We're in a vacuum.

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So if we start off in the x
direction at 5 meters per

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second to the right, we won't
be decelerated by air or

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friction or anything else.

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Newton's laws of motion:
an object in motion stays

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in motion, unless it's
affected by a net force.

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And there won't be any net
force in the x direction.

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It's just going to keep
moving to the right at

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5 meters per second.

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And position, or distance,
is just equal to

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velocity times time.

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Our velocity is 5 times time.

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And, of course, it didn't
start at x is equal to 0.

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This is at time equals 0.

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So it started at x
is equal to 10.

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So you want to know-- it's
kind of x of 0, where it

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started off, so plus 10.

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And this should be a little
intuitive for you, right?

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At time is equal to 0, this
term cancels out, where

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x is equal to 10.

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That makes sense.

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At time is equal to 1, we
should be a little bit-- we'll

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be 5 meters further out,
so on and so forth.

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Fair enough.

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That's x as a function
of the parameter time.

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As you probably realize, that
this is a video on parametric

00:03:13.660 --> 00:03:14.920
equations, not physics.

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So it's nice to early on
say the word parameter.

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Parameter.

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And time tends to be the
parameter when people talk

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about parametric equations.

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Although it could be anything.

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It could be radius or angle
or who knows what else?

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So let's figure out what y
is as a function of time.

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So y as a function of time,
it's going to be equal to

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the initial y position,
or y of 0, which is 50.

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We're 50 meters up in the air.

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Plus our initial velocity
in the y direction.

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And we don't actually have
any initial velocity

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in the y direction.

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The car isn't jumping,
or isn't diving.

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It's just moving
horizontally to the right.

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And the cliff is supporting it.

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So it has no y velocity.

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But if you were curious,
it would be the y

00:04:07.460 --> 00:04:09.490
velocity times time.

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But since there's no y velocity
times time, at least initially,

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I'll put nothing there.

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Plus the acceleration
of gravity times

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time squared over 2.

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We want to figure out the sine.

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And just so you know, I mean,
it's nice to touch on the

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physics a little bit, just so
you know where these formulas

00:04:27.940 --> 00:04:30.750
come from and you know the
motivation behind why you would

00:04:30.750 --> 00:04:32.920
even use a parametric equation.

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Gravity goes downwards in this
example, and downwards in

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this example is in the
minus y direction.

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y is decreasing.

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And the real-- and you know,
it's not exact-- but gravity is

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normally 9.8 meters per second
squared in most textbooks.

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But for the sake of simplifying
this, we'll say that it's

00:04:50.560 --> 00:04:54.250
approximately 10 meters
per second squared.

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That's how fast everything
will be accelerated

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downward on this planet.

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Since it has no air, let's
assume it's a planet with

00:04:59.580 --> 00:05:01.510
a little bit more
mass than earth.

00:05:01.510 --> 00:05:05.170
And since it's going downwards,
its direction is negative.

00:05:05.170 --> 00:05:08.180
So in our formula up here, it's
our initial position-- we had

00:05:08.180 --> 00:05:12.630
no velocity times time, so I
won't put that there-- minus 10

00:05:12.630 --> 00:05:15.600
meters per second squared
times t squared over 2.

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And you can watch the
projectile motion videos to

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figure out how I got these
formulas right there.

00:05:21.640 --> 00:05:22.760
But that's not the
point of this.

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The point of this is to graph
what happens to the cars and

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learn a little bit about
parametric equations.

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So what is the path of this car
as it falls off the cliff?

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Let's make a table here.

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So x and y are a function of
this third parameter, t.

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So we're going to set t at
different values and we're

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going to figure out what
x and y are equal to.

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And I'm just going to
arbitrarily pick some t's.

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T is equal to 0.

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T is equal to 1, 2, and 3.

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At time is equal
to 0, what is x?

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x of 0-- this is 0-- x
is equal to 10 meters.

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At time is equal
to 1, what is x?

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And this is x of 1, right?

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If I wanted to write
that notation.

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So 5 times 1 is 5,
plus 10 is 15.

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x of 2?

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5 times 2 plus 10, that's 20.

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And it makes sense.

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Every second we're getting 5
meters more to the right.

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Or x is increasing by 5 meters.

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So when t is equal to
3, 15 plus 10 is 25.

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Easy enough.

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The y is a little bit
more complicated.

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And just to simplify this, it's
the same thing as 5, right?

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10 divided by 2.

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So 50 minus 5t squared.

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So time is equal to 0,
this term cancels out.

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We just have 50 meters
up in the air.

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At time is equal to 1, 1
squared is 1 times 5, is five.

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50 minus 5 is 45 five
meters in the air.

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Is that right?

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Right, yeah, time 1, 50.

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Right.

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And then at time is equal
to 2, 2 squared is 4.

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4 times 5 is 20.

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50 minus 20 is 30.

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And then finally, at time is
equal to 3-- and I just say

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finally because that's the
last number we picked--

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time is equal to 3.

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3 squared is 9.

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9 times 5 is 45.

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50 minus 45 is 5.

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So let's plot these points.

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So the time is equal to 0.

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That's what we got right there.

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At time is equal to 1,
we're at x is equal to 15.

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That's roughly, you see this
is 5, 10, 15-- let me do

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all of them-- 15, 20, 25.

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And then the y-axis-- let me
label that while we're at

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it-- this is roughly
10, 20, 30, 40, 50.

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So at time is equal to
0, we're at 10, 50.

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That's that point right there.

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At time is equal to
1, we're at 15, 45.

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So x is 15, y is 45, which
is right about there.

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So this is t is equal to 1.

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At time is equal to 2, or
at the coordinate 20, 30,

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it's right about there.

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So this is at time
is equal to 2.

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And then at time is equal
to 3, we're at 25, 5.

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So we're right there.

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And if we kept going on,
at some point we're

00:08:31.240 --> 00:08:31.950
going to hit the ground.

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And you can figure out,
actually-- set this equal to 0

00:08:34.860 --> 00:08:36.600
and you figure out the exact
time you hit the ground.

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Actually, let's do that.

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If this is equal to 0, 50,
you get t is equal to

00:08:41.470 --> 00:08:42.470
the square root of 10.

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Which is a little
over 3 seconds.

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Which makes sense, right?

00:08:44.830 --> 00:08:47.220
A little over 3 seconds, we're
going to be hitting the ground.

00:08:47.220 --> 00:08:48.940
But anyway, what's the
path of this car?

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Well, it's going to look
something like this.

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Ooh, it starts getting
accelerated downwards,

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and then plunk!

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It hits the ground at 3
point something seconds.

00:09:00.830 --> 00:09:03.820
Now what was interesting here
is that by setting the

00:09:03.820 --> 00:09:06.600
parameter, not only did we
get the curve-- right?

00:09:06.600 --> 00:09:09.710
We got this curve, which is
kind of half of a parabola,

00:09:09.710 --> 00:09:11.800
half of a downward shaping
parabola-- and we could

00:09:11.800 --> 00:09:14.150
actually eliminate the t,
and just get the equation

00:09:14.150 --> 00:09:14.960
for that parabola.

00:09:14.960 --> 00:09:16.480
And we'll do that
in future videos.

00:09:16.480 --> 00:09:18.360
But what was interesting,
by making it a parametric

00:09:18.360 --> 00:09:20.310
equation, we know the
direction of the car.

00:09:20.310 --> 00:09:22.430
If you just saw this graph
without the car and everything

00:09:22.430 --> 00:09:24.800
else I drew, you wouldn't know
which way the car was falling.

00:09:24.800 --> 00:09:26.920
But now we know that as
t is increasing, we're

00:09:26.920 --> 00:09:27.850
going in that direction.

00:09:27.850 --> 00:09:30.870
So we can draw
some arrows here.

00:09:30.870 --> 00:09:32.600
So because it's a parametric
equation, we can

00:09:32.600 --> 00:09:33.600
draw some arrows.

00:09:33.600 --> 00:09:36.590
And then the most important
thing is we know exactly where

00:09:36.590 --> 00:09:38.910
the car is at any time t.

00:09:38.910 --> 00:09:42.600
You can substitute t is equal
to 1.25 seconds and you'll know

00:09:42.600 --> 00:09:44.520
exactly where the car is.

00:09:44.520 --> 00:09:47.850
So you can plot these points
and you can kind of get a sense

00:09:47.850 --> 00:09:51.250
that as time goes on, we're
getting accelerated downwards.

00:09:51.250 --> 00:09:54.950
And that's why, for every
second further, especially

00:09:54.950 --> 00:09:58.070
the y distance gets
further and further apart.

00:09:58.070 --> 00:10:00.590
Anyway, I just wanted to
give you this example.

00:10:00.590 --> 00:10:03.240
Although this was a good
physics problem, the intention

00:10:03.240 --> 00:10:04.245
wasn't to teach you physics.

00:10:04.245 --> 00:10:07.100
The intention is to give you
the motivation behind why

00:10:07.100 --> 00:10:09.190
parametric equations
even exist.

00:10:13.860 --> 00:10:16.390
These two things are
parametric equations.

00:10:16.390 --> 00:10:22.000
We defined x and y as a
function of a third parameter,

00:10:22.000 --> 00:10:25.910
t, instead of defining y in
terms of x or x in terms of

00:10:25.910 --> 00:10:28.250
y like we've done every
other time since then.

00:10:28.250 --> 00:10:29.240
And this is super useful.

00:10:29.240 --> 00:10:31.030
I mean, you could imagine when
you have really hard physics

00:10:31.030 --> 00:10:33.410
problems where you want to
figure out the three

00:10:33.410 --> 00:10:35.630
dimensional position of
something, then you'll have x

00:10:35.630 --> 00:10:38.420
as a function of t, y as a
function of t, z as

00:10:38.420 --> 00:10:39.400
a function of t.

00:10:39.400 --> 00:10:41.450
All sorts of interesting
problems come out of using

00:10:41.450 --> 00:10:44.020
parametric equations,
not just in physics.

00:10:44.020 --> 00:10:45.980
But anyway, I thought
a good place to start

00:10:45.980 --> 00:10:46.720
is the motivation.

00:10:46.720 --> 00:10:48.380
Because the first time I
learned parametric equations I

00:10:48.380 --> 00:10:51.500
was like, why mess up my nice
and simple world of x's and

00:10:51.500 --> 00:10:54.630
y's by introducing a
third parameter, t?

00:10:54.630 --> 00:10:55.550
This is why.

00:10:55.550 --> 00:10:57.240
Because you can figure
out the path of things.

00:10:57.240 --> 00:10:59.760
You can figure out the
direction of something as it

00:10:59.760 --> 00:11:03.120
moves along a curve, and you
can figure out its exact

00:11:03.120 --> 00:11:07.110
position at any, in
this case, time.


Focus and Directrix of a Parabola 2

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:04.270
In the last video, we showed
that if you had a line, which

00:00:04.270 --> 00:00:09.020
we'll call a directrix--
we draw the directrix.

00:00:09.020 --> 00:00:13.780
That's our directrix and it has
the equation y is equal to k.

00:00:13.780 --> 00:00:17.450
And you have a point, that's
our focus, and it's the

00:00:17.450 --> 00:00:19.510
coordinate a comma b.

00:00:19.510 --> 00:00:22.910
That the locus of all points in
the xy plane that are

00:00:22.910 --> 00:00:27.530
equidistant to this focus and
this directrix has a shape that

00:00:27.530 --> 00:00:29.520
looks something-- we know that
this was a point-- that

00:00:29.520 --> 00:00:32.480
looks something like this.

00:00:32.480 --> 00:00:35.830
And the actually equation, we
actually just took an arbitrary

00:00:35.830 --> 00:00:40.650
xy in the coordinate plane, and
we said that xy has to satisfy

00:00:40.650 --> 00:00:43.900
the condition that its distance
to this focus is equal to--

00:00:43.900 --> 00:00:47.460
that this distance right here
is equal to this distance here.

00:00:47.460 --> 00:00:50.170
And we set that up using the
distance formula to here, and

00:00:50.170 --> 00:00:52.710
then setting that equal to the
distance to the line,

00:00:52.710 --> 00:00:53.670
to the directrix.

00:00:53.670 --> 00:00:55.800
And then we did a bunch of
algebra and we got this

00:00:55.800 --> 00:00:57.440
equation right here.

00:00:57.440 --> 00:01:01.270
For all of the points in the xy
plane that satisfy the

00:01:01.270 --> 00:01:04.870
conditions that its distance to
the focus is equal to the

00:01:04.870 --> 00:01:06.360
distance to this line.

00:01:06.360 --> 00:01:10.040
And we hopefully satisfied
ourselves that this is

00:01:10.040 --> 00:01:12.830
in fact a parabola.

00:01:12.830 --> 00:01:15.100
And although it looks a little
bit hairier than most parabolas

00:01:15.100 --> 00:01:17.690
we see, I think if you look
at it closely, you'll realize

00:01:17.690 --> 00:01:18.820
that it is indeed a parabola.

00:01:18.820 --> 00:01:23.040
For example, the classic
parabola is y is

00:01:23.040 --> 00:01:25.050
equal to x squared.

00:01:25.050 --> 00:01:27.420
How is this the same
thing as this?

00:01:27.420 --> 00:01:30.320
Well in this case, this
coefficient out here

00:01:30.320 --> 00:01:33.530
is just equal to 1.

00:01:33.530 --> 00:01:42.410
So in this case, 1 over 2 b
minus k is equal to 1, right?

00:01:42.410 --> 00:01:44.160
Let me delete that little
thing there, because it

00:01:44.160 --> 00:01:45.900
looks like a squared.

00:01:45.900 --> 00:01:48.790
All right, the coefficient
in front of the x squared

00:01:48.790 --> 00:01:50.950
term is equal to a 1.

00:01:50.950 --> 00:01:52.940
What is a equal to?

00:01:52.940 --> 00:01:57.780
Well this is the same
thing-- actually, let

00:01:57.780 --> 00:01:58.760
me rewrite it that way.

00:01:58.760 --> 00:02:05.520
I could rewrite this equation
right here as y minus 0 is

00:02:05.520 --> 00:02:10.320
equal to 1 times x
minus 0 squared.

00:02:10.320 --> 00:02:13.070
And now hopefully you see that
this has the same pattern as

00:02:13.070 --> 00:02:15.520
this, although the y's and the
x's are just sitting

00:02:15.520 --> 00:02:17.390
on different sides
of the equation.

00:02:17.390 --> 00:02:20.420
So here you see that the 1/2
over b minus k, that's the

00:02:20.420 --> 00:02:23.970
coefficient in front of the
x minus a squared term.

00:02:23.970 --> 00:02:28.290
So that's where we got that
that has to be equal to 1.

00:02:28.290 --> 00:02:34.470
a has to be equal to
0 in this situation.

00:02:37.710 --> 00:02:40.840
This curve right here is equal
to x squared, y is equal

00:02:40.840 --> 00:02:43.040
to x squared, a is 0.

00:02:43.040 --> 00:02:49.650
And then b plus k over 2, this
is going to be equal to that.

00:02:49.650 --> 00:02:52.235
And keep in mind, I actually
swapped the left and

00:02:52.235 --> 00:02:53.350
the right hand sides.

00:02:53.350 --> 00:02:57.050
But this is y minus something,
this is y minus 0.

00:02:57.050 --> 00:02:59.120
So this something has
to be equal to 0.

00:02:59.120 --> 00:03:05.640
So you get b plus k
over 2 is equal to 0.

00:03:05.640 --> 00:03:09.200
And now we should be able to
use this information to figure

00:03:09.200 --> 00:03:14.660
out the actual coordinates of
the focus and the directrix of

00:03:14.660 --> 00:03:17.495
what I would call the classic
parabola, y is equal

00:03:17.495 --> 00:03:18.540
to x squared.

00:03:18.540 --> 00:03:19.500
So keep in mind what we did.

00:03:19.500 --> 00:03:22.930
In the last video, we said what
is the equation of the line

00:03:22.930 --> 00:03:26.740
that is equidistant between
this focus and this directrix?

00:03:26.740 --> 00:03:30.010
And we got this equation,
which we said, hey,

00:03:30.010 --> 00:03:31.070
that's a parabola.

00:03:31.070 --> 00:03:32.530
Now we're going the
other way around.

00:03:32.530 --> 00:03:36.350
We say we have a parabola, and
we're saying that this parabola

00:03:36.350 --> 00:03:40.580
is the set of all points that's
equidistant between some

00:03:40.580 --> 00:03:42.530
focus and some directrix.

00:03:42.530 --> 00:03:45.240
What is that focus and
that directrix for this

00:03:45.240 --> 00:03:46.410
particular parabola?

00:03:46.410 --> 00:03:47.750
That's what we're trying to do.

00:03:47.750 --> 00:03:51.640
And the way we did that was, we
pattern matched this formula to

00:03:51.640 --> 00:03:55.800
this one, to essentially set up
these variables, which

00:03:55.800 --> 00:03:58.620
essentially define a focus
and a directrix, right?

00:03:58.620 --> 00:04:05.290
b is the y-coordinate of the
focus, k defines the horizontal

00:04:05.290 --> 00:04:06.290
line of the directrix.

00:04:06.290 --> 00:04:08.620
So if we can solve for b and
k, we know what the focus

00:04:08.620 --> 00:04:09.390
and the directrix are.

00:04:09.390 --> 00:04:11.100
And for a, although
a is pretty easy.

00:04:11.100 --> 00:04:13.240
So let's do that.

00:04:13.240 --> 00:04:16.500
So if we know that 1 over 2
b minus k is equal to 1--

00:04:16.500 --> 00:04:18.460
let me do that over here.

00:04:18.460 --> 00:04:20.190
I'm going to do it in
particular for this case right

00:04:20.190 --> 00:04:23.090
here, but you don't want to do
that every time, so then we'll

00:04:23.090 --> 00:04:26.760
do a more general formula that
we can just plug in next time

00:04:26.760 --> 00:04:28.430
you see a new parabola.

00:04:28.430 --> 00:04:33.320
So if we multiply both sides of
this equation by 2 b minus k,

00:04:33.320 --> 00:04:39.340
you get 1 is equal to
2 times b minus k.

00:04:39.340 --> 00:04:44.590
Actually I should redivide both
sides by 2, so you get 1/2

00:04:44.590 --> 00:04:49.110
is equal to b minus k.

00:04:49.110 --> 00:04:50.120
Fair enough.

00:04:50.120 --> 00:04:53.420
Let's see, this equation
right here, b plus k

00:04:53.420 --> 00:04:55.310
over 2 is equal to 0.

00:04:55.310 --> 00:04:58.940
That means that b plus k
must be equal to 0, right?

00:04:58.940 --> 00:05:00.610
This numerator has
to be equal to 0.

00:05:00.610 --> 00:05:02.320
So let me write that down.

00:05:02.320 --> 00:05:07.900
b plus k is equal to 0.

00:05:07.900 --> 00:05:10.710
Two linear equations
of to unknowns.

00:05:10.710 --> 00:05:11.590
Let's add them up.

00:05:11.590 --> 00:05:12.520
The k's cancel out.

00:05:12.520 --> 00:05:16.290
You get 2b is equal to 1/2.

00:05:16.290 --> 00:05:21.500
Divide both sides by 2, you
get b is equal to 1/4.

00:05:21.500 --> 00:05:23.180
And now what's k?

00:05:23.180 --> 00:05:25.720
Well, we can just
resubstitute back.

00:05:25.720 --> 00:05:26.790
Let's use this equation.

00:05:26.790 --> 00:05:31.080
We get 0 is equal
to 1/4 plus k.

00:05:31.080 --> 00:05:33.060
You can figure that out by
inspection, but you can

00:05:33.060 --> 00:05:34.720
subtract 1/4 from both sides.

00:05:34.720 --> 00:05:39.540
So this tells us that k
is equal to minus 1/4.

00:05:39.540 --> 00:05:41.690
So let's draw this.

00:05:41.690 --> 00:05:45.660
So if this is my coordinate
axis-- I'll do it in

00:05:45.660 --> 00:05:48.010
a different color.

00:05:48.010 --> 00:05:50.940
That's the x-axis,
that's the y-axis.

00:05:50.940 --> 00:05:55.820
We're dealing with y is equal
to x squared, which you should

00:05:55.820 --> 00:05:58.220
hopefully be reasonably
familiar with at this point.

00:05:58.220 --> 00:06:01.430
It's kind of the classic
U-shaped parabola that

00:06:01.430 --> 00:06:02.720
looks just like that.

00:06:02.720 --> 00:06:05.350
Let me write that down.

00:06:05.350 --> 00:06:08.580
This is y is equal to x
squared, which we could have

00:06:08.580 --> 00:06:15.690
rewritten as y minus 0 is equal
to 1 times x minus 0 squared.

00:06:15.690 --> 00:06:18.200
Same thing, that's this line.

00:06:18.200 --> 00:06:21.810
It's equidistant from a focus
and a directrix, where

00:06:21.810 --> 00:06:24.040
this is the focus.

00:06:24.040 --> 00:06:26.330
And how do I know that the
focus is right there, instead

00:06:26.330 --> 00:06:27.730
of right here or right there?

00:06:27.730 --> 00:06:29.530
Well one, you could kind
of think about it.

00:06:29.530 --> 00:06:33.480
But the other thing to realize
is, what is the vertex

00:06:33.480 --> 00:06:34.080
of this parabola?

00:06:34.080 --> 00:06:35.940
And I think we've gone
over that already.

00:06:35.940 --> 00:06:40.206
But the vertex are the
coordinates given by this

00:06:40.206 --> 00:06:42.210
x value and this y value.

00:06:42.210 --> 00:06:46.290
Or what x values make this
whole term 0 and make

00:06:46.290 --> 00:06:48.060
this whole term 0?

00:06:48.060 --> 00:06:50.140
And that's essentially 0,0.

00:06:50.140 --> 00:06:53.780
So this is the vertex
of this parabola.

00:06:53.780 --> 00:06:56.930
This is the vertex,
0,0 is the vertex.

00:06:56.930 --> 00:06:59.560
I'll say v for vertex.

00:06:59.560 --> 00:07:02.560
In an upward opening parabola,
you can view that as

00:07:02.560 --> 00:07:03.190
the minimum point.

00:07:03.190 --> 00:07:05.820
If it was a downward opening,
it would be the topmost

00:07:05.820 --> 00:07:07.820
point, or the maximum point.

00:07:07.820 --> 00:07:08.910
That's the vertex.

00:07:08.910 --> 00:07:11.380
And we saw from when we solved
this, when we just did our

00:07:11.380 --> 00:07:17.620
pattern matching, that the
x-coordinate of our focus is

00:07:17.620 --> 00:07:21.400
equivalent to, if we just
pattern match right here, x

00:07:21.400 --> 00:07:24.000
minus a squared, here you
have x minus 0 squared.

00:07:24.000 --> 00:07:26.720
This is always going to
be the same as this.

00:07:26.720 --> 00:07:34.220
So the x-coordinate of our
focus is always going to

00:07:34.220 --> 00:07:36.830
be the same thing as the
x-coordinate of our vertex.

00:07:36.830 --> 00:07:40.050
So we figured out that
a was equal to 0.

00:07:40.050 --> 00:07:44.290
So the coordinate here is
0, and then what's b?

00:07:44.290 --> 00:07:46.840
This was the coordinate 0,b.

00:07:46.840 --> 00:07:52.410
We figured out that b is
equal to 1/4, 0, 1/4.

00:07:52.410 --> 00:07:57.270
And then the directrix is
the line y is equal to k.

00:07:57.270 --> 00:08:00.500
In this case, we figure out
k is equal to minus 1/4.

00:08:00.500 --> 00:08:04.670
So our directrix is going
to be right down here.

00:08:04.670 --> 00:08:05.990
Right down here, like that.

00:08:05.990 --> 00:08:10.960
And it's going to be line
y is equal to minus 1/4.

00:08:10.960 --> 00:08:12.960
And that first of all
looks about right.

00:08:12.960 --> 00:08:14.600
And it also gels
with what we know.

00:08:14.600 --> 00:08:17.130
Because this point right here,
our vertex, is equidistant,

00:08:17.130 --> 00:08:22.390
it's 1/4 below our focus and
it's 1/4 above our directrix.

00:08:22.390 --> 00:08:25.810
So at least that little
reality check holds.

00:08:25.810 --> 00:08:27.900
But it's was a particular
circumstance, and you don't

00:08:27.900 --> 00:08:31.270
want to have to go through this
whole situation every time you

00:08:31.270 --> 00:08:35.910
have to figure out the focus or
the directrix of a parabola.

00:08:35.910 --> 00:08:38.850
So let's see if we can come up
with a more general solution.

00:08:38.850 --> 00:08:40.165
And actually what I'm
going to do is, I'm going

00:08:40.165 --> 00:08:41.700
to erase all of this.

00:08:41.700 --> 00:08:46.650
Just because I want to reuse
that, and I haven't committed

00:08:46.650 --> 00:08:48.924
that formula to heart yet.

00:08:48.924 --> 00:08:52.050
Let me erase all of this,
and erase all the work

00:08:52.050 --> 00:08:52.975
we did down here.

00:08:55.930 --> 00:08:56.990
So let's do a general form.

00:08:56.990 --> 00:09:02.480
And actually, I've abused this
so much, let me rewrite it.

00:09:02.480 --> 00:09:05.390
So I'll put the y's on the
left hand side, instead

00:09:05.390 --> 00:09:06.950
of keep telling you that
I've switched them.

00:09:06.950 --> 00:09:13.030
So this is the same thing as--
I'll do it in-- the color

00:09:13.030 --> 00:09:14.650
choosing is the hard part.

00:09:14.650 --> 00:09:17.730
I'll do it in this
light, off-white color.

00:09:17.730 --> 00:09:27.790
So you get y minus b plus k
over 2 is equal to 1 over 2

00:09:27.790 --> 00:09:34.510
times b minus k, times
x minus a squared.

00:09:34.510 --> 00:09:40.000
This is the locus of all points
that are between the directrix,

00:09:40.000 --> 00:09:44.520
y is equal to k-- well, they're
equidistant to the directrix,

00:09:44.520 --> 00:09:49.720
y is equal to k, and
the focus a comma b.

00:09:49.720 --> 00:09:52.270
So let's say that I
have a parabola.

00:09:52.270 --> 00:09:55.080
And I'm going to try to
use different letters, so

00:09:55.080 --> 00:09:57.210
we don't get confused.

00:09:57.210 --> 00:10:01.240
I give you the
parabola y minus y1.

00:10:01.240 --> 00:10:08.850
And let's say this parabola
has a vertex at x1 comma y1.

00:10:08.850 --> 00:10:11.050
So this parabola I'm drawing
has a vertex right there.

00:10:11.050 --> 00:10:16.710
So its formula will be y minus
y1 is equal to some constant,

00:10:16.710 --> 00:10:18.890
let's make that a capital A.

00:10:18.890 --> 00:10:21.170
This capital A is different
than that lowercase a.

00:10:21.170 --> 00:10:23.380
This is what I was going to
embark on in the previous

00:10:23.380 --> 00:10:26.260
video, and then I realized that
I was trying to load that video

00:10:26.260 --> 00:10:28.240
up too much and probably
confusing you.

00:10:28.240 --> 00:10:34.180
So some constant factor, some
type of scaling factor, times

00:10:34.180 --> 00:10:36.780
x minus the x value
of our vertex.

00:10:36.780 --> 00:10:39.930
So x1, all of that, squared.

00:10:39.930 --> 00:10:43.090
So if you're given this
parabola, or if you can get a

00:10:43.090 --> 00:10:48.140
parabola to this form, how do
you figure out the a's, the

00:10:48.140 --> 00:10:50.920
b's, and the k's so you know
the coordinates of the

00:10:50.920 --> 00:10:53.160
directrix and the focus?

00:10:53.160 --> 00:10:55.980
So the easy thing is to figure
out a, because you just

00:10:55.980 --> 00:10:57.500
do a pattern match.

00:10:57.500 --> 00:11:01.480
This is going to
be equal to that.

00:11:01.480 --> 00:11:04.040
So you know that a
is equal to x1.

00:11:04.040 --> 00:11:06.910
So the x-coordinate of our
focus is the same as the

00:11:06.910 --> 00:11:09.250
x-coordinate of our vertex.

00:11:09.250 --> 00:11:12.710
You can pattern match that this
scaling factor right here is

00:11:12.710 --> 00:11:16.870
going to be the same thing
as this 1 over 2 b minus k.

00:11:16.870 --> 00:11:18.860
So let's write that down.

00:11:18.860 --> 00:11:26.450
A is equal to 1
over 2 b minus k.

00:11:26.450 --> 00:11:31.510
And then we have,
finally, this.

00:11:31.510 --> 00:11:32.520
Let me do another color.

00:11:32.520 --> 00:11:33.800
I'll do a magenta.

00:11:33.800 --> 00:11:36.930
The b plus k over 2, we
have y minus that thing.

00:11:36.930 --> 00:11:38.220
We have y minus y1.

00:11:38.220 --> 00:11:40.840
So that's going to
be equal to y1.

00:11:40.840 --> 00:11:48.350
So you have y1 is equal
to b plus k over 2.

00:11:48.350 --> 00:11:50.520
And now we have two equations
with two unknowns.

00:11:50.520 --> 00:11:52.135
Remember, this is
going to be a given.

00:11:52.135 --> 00:11:55.160
The A is going to be given, and
the y1 is going to be given.

00:11:55.160 --> 00:11:57.400
So now we can solve for
b and k in terms of the

00:11:57.400 --> 00:11:58.360
numbers that we have.

00:11:58.360 --> 00:12:00.000
So let's see if we can do that.

00:12:00.000 --> 00:12:04.290
So if we multiply both sides of
this equation times b minus k,

00:12:04.290 --> 00:12:11.480
you get b minus k times
A is equal to 1/2.

00:12:11.480 --> 00:12:15.220
Divide both sides by A,
you get b minus k is

00:12:15.220 --> 00:12:19.260
equal to 1 over 2A.

00:12:19.260 --> 00:12:21.040
And on the right hand side,
let's use this other

00:12:21.040 --> 00:12:21.840
formula right here.

00:12:21.840 --> 00:12:24.800
If we multiply both sides by
2-- I'll use a different

00:12:24.800 --> 00:12:31.010
color-- you get 2 times
y1 is equal to b plus k.

00:12:31.010 --> 00:12:32.470
Two equations with
two unknowns.

00:12:32.470 --> 00:12:36.000
Let's take this and bring it
down here, so you get b plus

00:12:36.000 --> 00:12:40.510
k is equal to 2 times y1.

00:12:40.510 --> 00:12:46.400
And so you add them, and you
get 2b is equal to-- the k's

00:12:46.400 --> 00:12:49.080
cancel out; I'm just adding
these two equations--

00:12:49.080 --> 00:12:55.570
1 over 2A plus 2 y1.

00:12:55.570 --> 00:12:57.980
Or b-- which is,
remember, what was b?

00:12:57.980 --> 00:13:01.250
That was the y-coordinate of
our focus-- is equal to, divide

00:13:01.250 --> 00:13:08.680
everything by 2, is equal
to 1 over 4A plus y1.

00:13:08.680 --> 00:13:09.380
Which is interesting.

00:13:09.380 --> 00:13:14.910
It tells us that we take the
y-coordinate of our vertex

00:13:14.910 --> 00:13:16.980
and we add 1 over 4A to it.

00:13:16.980 --> 00:13:19.700
Which is exactly what
happened last time, right?

00:13:19.700 --> 00:13:22.210
I actually erased what
I did last time.

00:13:22.210 --> 00:13:24.980
But last time, when we
had y equal x squared,

00:13:24.980 --> 00:13:26.830
this value was 0.

00:13:26.830 --> 00:13:29.040
The scaling factor was 1.

00:13:29.040 --> 00:13:32.090
So we saw that the y-coordinate
of our focus was 1/4.

00:13:32.090 --> 00:13:33.730
So this is gelling
with what we know.

00:13:33.730 --> 00:13:34.710
So that's our b.

00:13:39.150 --> 00:13:42.660
And let's see if we
can solve back for k.

00:13:42.660 --> 00:13:47.560
So we know that--
let's do it up here.

00:13:47.560 --> 00:13:52.580
We know that 2 y1 is equal
to b, which is 1 over

00:13:52.580 --> 00:13:57.360
4A plus y1 plus k.

00:13:57.360 --> 00:14:00.360
We can subtract y1 from both
sides, so this goes away and

00:14:00.360 --> 00:14:02.090
we just have a y1 there.

00:14:02.090 --> 00:14:06.640
Subtract 1 over 4A from both
sides and you get y1 minus

00:14:06.640 --> 00:14:10.830
1 over 4A is equal to k.

00:14:10.830 --> 00:14:13.670
So then we are done.

00:14:13.670 --> 00:14:14.550
So this is interesting.

00:14:14.550 --> 00:14:17.640
Right now these might look
a little, these little

00:14:17.640 --> 00:14:18.650
hairy formulas.

00:14:18.650 --> 00:14:20.870
But if we actually graph it,
I think they'll become a

00:14:20.870 --> 00:14:22.720
little bit more intuitive.

00:14:22.720 --> 00:14:29.620
So once again, we had the
parabola y minus y1 is equal to

00:14:29.620 --> 00:14:34.200
A times x minus x1 squared.

00:14:34.200 --> 00:14:36.950
And so the graph of that
parabola will look

00:14:36.950 --> 00:14:37.740
something like this.

00:14:37.740 --> 00:14:40.800
I don't know where it is
relative to the x- and y-axis.

00:14:40.800 --> 00:14:42.500
It will look
something like this.

00:14:42.500 --> 00:14:44.690
It'll be this U shape.

00:14:44.690 --> 00:14:49.120
It could actually be downward
pointing, but I'll just assume

00:14:49.120 --> 00:14:52.490
an upward pointing
parabola for now.

00:14:52.490 --> 00:14:55.110
That's this formula.

00:14:55.110 --> 00:14:59.490
Its vertex, right there,
its vertex is the

00:14:59.490 --> 00:15:02.940
point x1 comma y1.

00:15:02.940 --> 00:15:03.140
Right?

00:15:03.140 --> 00:15:05.660
What y value makes this
expression equal 0?

00:15:05.660 --> 00:15:06.430
It's y1.

00:15:06.430 --> 00:15:07.910
What x value makes
that equal 0?

00:15:07.910 --> 00:15:09.140
It's x1.

00:15:09.140 --> 00:15:11.090
Now notice, what is
this directrix?

00:15:11.090 --> 00:15:19.080
It's the line ok, y equal to
k, but it's 1/4 less than

00:15:19.080 --> 00:15:20.310
this value right here.

00:15:20.310 --> 00:15:24.410
So you literally just go down
1/4-- not just 1/4, 1/4

00:15:24.410 --> 00:15:26.060
times this scaling factor.

00:15:26.060 --> 00:15:29.720
So it's 1/4 A.

00:15:29.720 --> 00:15:31.760
This distance is 1/4 A.

00:15:31.760 --> 00:15:33.215
And then you get your
directrix, right?

00:15:33.215 --> 00:15:37.900
It's y1 minus 1/4 A is
equal to k. y1 minus

00:15:37.900 --> 00:15:40.550
1/4 A is equal to k.

00:15:40.550 --> 00:15:42.270
Or 1 over 4A.

00:15:42.270 --> 00:15:48.900
So your directrix is
going to be right there.

00:15:48.900 --> 00:15:49.600
Let me write this.

00:15:49.600 --> 00:15:55.070
It's y is equal to y1,
which is just this

00:15:55.070 --> 00:15:58.290
level, minus 1 over 4A.

00:15:58.290 --> 00:15:59.310
But this is the intuitive part.

00:15:59.310 --> 00:16:05.250
Just remember, you're going
1 over 4A below the vertex.

00:16:05.250 --> 00:16:09.040
And then the focal point, the x
value of the focal point is

00:16:09.040 --> 00:16:13.040
going to be x1, the same as
the x value of the vertex.

00:16:13.040 --> 00:16:15.890
And then the y value is
just b, and we just go

00:16:15.890 --> 00:16:17.950
1 over 4A above it.

00:16:17.950 --> 00:16:20.420
Which makes sense because we
need to be equidistant from

00:16:20.420 --> 00:16:22.060
the directrix and the focus.

00:16:22.060 --> 00:16:26.070
So this is y1 plus 1 over 4A.

00:16:26.070 --> 00:16:30.620
But the intuitive part is, we
just took 1 over 4A above it.

00:16:30.620 --> 00:16:33.180
So in general, if I gave
you-- so just think

00:16:33.180 --> 00:16:33.930
about this a second.

00:16:33.930 --> 00:16:37.290
If I gave you an equation,
an arbitrary equation now.

00:16:37.290 --> 00:16:49.100
Say, y minus 1 is equal to 2
times x minus 3 squared, we can

00:16:49.100 --> 00:16:54.700
graph this and draw the focus
fairly-- and actually

00:16:54.700 --> 00:16:57.580
it's foci, if there was
more than one focus.

00:16:57.580 --> 00:17:00.540
Let's see, that's the
x-axis, that's the y-axis.

00:17:00.540 --> 00:17:01.930
Where's its vertex?

00:17:01.930 --> 00:17:05.200
The vertex is at the
point 3 comma 1.

00:17:05.200 --> 00:17:09.110
So we go 1, 2, 3 comma 1.

00:17:09.110 --> 00:17:10.690
So that's its vertex.

00:17:10.690 --> 00:17:12.320
It's an upward opening
parabola, because this

00:17:12.320 --> 00:17:13.790
is a positive number.

00:17:13.790 --> 00:17:15.060
Actually, we don't even
have to know that.

00:17:15.060 --> 00:17:17.210
If you just draw the focus and
the directrix, you might be

00:17:17.210 --> 00:17:18.970
able to figure that out.

00:17:18.970 --> 00:17:20.900
But where's the focus point?

00:17:20.900 --> 00:17:25.960
The focal point is going to be
1 over 4A above this point.

00:17:25.960 --> 00:17:30.740
So 1 over 4 times 2
is equal to 1/8.

00:17:30.740 --> 00:17:32.340
So the focal point is
going to be really close.

00:17:32.340 --> 00:17:35.340
It's going to be right up here.

00:17:35.340 --> 00:17:38.210
It's going to be 1/8
above our vertex.

00:17:38.210 --> 00:17:40.970
And then our directrix is
going to be 1/8 below it.

00:17:40.970 --> 00:17:42.130
So it's going to
be right there.

00:17:42.130 --> 00:17:44.000
The directrix is going
to look like this.

00:17:46.580 --> 00:17:47.810
The directrix is going
to be there, the

00:17:47.810 --> 00:17:49.160
focal point is there.

00:17:49.160 --> 00:17:53.280
And then the graph of this
parabola is going to look

00:17:53.280 --> 00:17:56.090
something like this.

00:17:56.090 --> 00:17:59.050
So anyway, hopefully I didn't
confuse you too much.

00:17:59.050 --> 00:18:01.580
And the whole point of doing
all of this is just to realize

00:18:01.580 --> 00:18:04.600
that it's pretty easy to find
the focus and the directrix of

00:18:04.600 --> 00:18:07.360
a parabola if you have its
equation in this form.

00:18:07.360 --> 00:18:11.600
You just take whatever is
multiplying times the x minus

00:18:11.600 --> 00:18:13.570
whatever squared term.

00:18:13.570 --> 00:18:18.270
And you essentially divide
1 by 4 times this value.

00:18:18.270 --> 00:18:21.355
And then you know the distance
from the vertex to the focus,

00:18:21.355 --> 00:18:24.210
and you know the distance from
the vertex to the directrix,

00:18:24.210 --> 00:18:25.530
and you're all done.

00:18:25.530 --> 00:18:27.450
See you in the next video.


Parabola Focus and Directrix 1

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:05.370
Let's say I have a line, let
me make it a straight line.

00:00:05.370 --> 00:00:08.060
And the equation of that line,
since it's running in the

00:00:08.060 --> 00:00:10.320
horizontal direction is
going to be y is equal

00:00:10.320 --> 00:00:11.656
to some constant.

00:00:11.656 --> 00:00:12.970
So let me write that.

00:00:12.970 --> 00:00:17.010
So the equation of this line
right here is y is equal to k.

00:00:17.010 --> 00:00:19.760
And let's say I have
some other point.

00:00:19.760 --> 00:00:23.650
I'll call that-- well, we'll
call that a focus because

00:00:23.650 --> 00:00:25.886
it will be a focus.

00:00:25.886 --> 00:00:29.890
And let's say that point--
let's put it right here-- is

00:00:29.890 --> 00:00:36.300
it the coordinate, let's
call it a comma b.

00:00:36.300 --> 00:00:40.915
And let's think about the set
of all points or the locus of

00:00:40.915 --> 00:00:44.370
all points that are equidistant
to this point right

00:00:44.370 --> 00:00:46.720
here and this line.

00:00:46.720 --> 00:00:48.800
Just so you get the
terminology we'll call

00:00:48.800 --> 00:00:50.770
this line a directrix.

00:00:50.770 --> 00:00:52.500
Let me write that down.

00:00:52.500 --> 00:00:57.050
Directrix, it's probably a word
you've never heard before.

00:00:57.050 --> 00:00:59.640
And frankly, I've never heard
it outside of the context

00:00:59.640 --> 00:01:04.190
of parabolas or conic
sections in general.

00:01:04.190 --> 00:01:05.150
So let's call that a directrix.

00:01:05.150 --> 00:01:08.090
This is our focus.

00:01:08.090 --> 00:01:11.150
And what we want to do is we
want to find all of the points

00:01:11.150 --> 00:01:15.590
in the xy-plane that are
equidistant to this focus

00:01:15.590 --> 00:01:16.560
and this directrix.

00:01:16.560 --> 00:01:19.090
So there's one point that just
by looking at it we could say

00:01:19.090 --> 00:01:22.162
is going to be in our locus.

00:01:22.162 --> 00:01:26.060
So it's going to be right there
because clearly, this point is

00:01:26.060 --> 00:01:28.750
equidistant; it's halfway
between that point

00:01:28.750 --> 00:01:30.380
and this line.

00:01:30.380 --> 00:01:31.500
Now let's see what
other points.

00:01:31.500 --> 00:01:34.900
There will probably be this
point right here because the

00:01:34.900 --> 00:01:38.170
distance from there to there
is this name as the distance

00:01:38.170 --> 00:01:40.410
from if you just drop
a line straight down.

00:01:40.410 --> 00:01:42.330
Remember, we want to get the
shortest distance between

00:01:42.330 --> 00:01:43.380
this point and the line.

00:01:43.380 --> 00:01:48.380
We could have said something
like this distance, but this

00:01:48.380 --> 00:01:50.370
distance wouldn't be the
shortest distance between

00:01:50.370 --> 00:01:51.250
this point and this line.

00:01:51.250 --> 00:01:54.250
So you would go straight
down to the line there.

00:01:54.250 --> 00:01:57.920
Likewise, this point
would be there.

00:01:57.920 --> 00:02:00.960
So that distance is the
same as that distance.

00:02:00.960 --> 00:02:04.420
And you can already, I
think, begin to imagine

00:02:04.420 --> 00:02:05.580
the type of shape this is.

00:02:05.580 --> 00:02:07.960
And if you look at the title of
this video you can probably

00:02:07.960 --> 00:02:08.990
guess where this is going.

00:02:08.990 --> 00:02:14.630
That the shape is going to
look something like this.

00:02:14.630 --> 00:02:18.320
Or something pretty similar to
what we know as a parabola.

00:02:18.320 --> 00:02:19.800
And actually, it
will be a parabola.

00:02:19.800 --> 00:02:22.000
There's no mystery there.

00:02:22.000 --> 00:02:23.740
But what we're doing this in
this video is actually show

00:02:23.740 --> 00:02:24.535
you that is a parabola.

00:02:24.535 --> 00:02:26.570
Show it to you mathematically
instead of this

00:02:26.570 --> 00:02:27.420
little drawing here.

00:02:27.420 --> 00:02:29.390
Where I show you that it kind
of looks like it would be a

00:02:29.390 --> 00:02:34.750
parabola because this distance
right here looks about the same

00:02:34.750 --> 00:02:37.510
as that distance right there.

00:02:37.510 --> 00:02:39.750
I could keep going up and down
the curve and keep doing that,

00:02:39.750 --> 00:02:41.080
but that's not satisfactory.

00:02:41.080 --> 00:02:45.830
Let's actually mathematically
show that the locus of all

00:02:45.830 --> 00:02:49.300
points that are equidistant to
this point and this line-- this

00:02:49.300 --> 00:02:53.320
focus and this directrix--
is in fact, a parabola.

00:02:53.320 --> 00:02:56.870
So let's say I have some point
that's in this locus and let me

00:02:56.870 --> 00:03:01.710
call the point here-- let me do
it in a different color because

00:03:01.710 --> 00:03:04.570
it's the same color as
my directrix-- x, y.

00:03:04.570 --> 00:03:07.980
So I want to find all of the x,
y's-- all of the points that

00:03:07.980 --> 00:03:13.220
satisfy an equation where their
distance to the focus is equal

00:03:13.220 --> 00:03:14.590
to their distance
to the directrix.

00:03:14.590 --> 00:03:17.480
So what's the distance
between x, y and the focus?

00:03:17.480 --> 00:03:19.790
What's this distance
right here?

00:03:19.790 --> 00:03:20.950
d sub 1.

00:03:20.950 --> 00:03:23.076
Well, we just used the
distance formula.

00:03:23.076 --> 00:03:26.656
It's x minus the
x-coordinate of the focus.

00:03:26.656 --> 00:03:30.350
So it's x minus a squared.

00:03:30.350 --> 00:03:33.250
So the difference in
the x's squared.

00:03:33.250 --> 00:03:38.366
Plus the difference in the y's
squared-- y minus b squared.

00:03:38.366 --> 00:03:40.690
The square root of that.

00:03:40.690 --> 00:03:42.810
So this is the distance
to the focus.

00:03:42.810 --> 00:03:44.310
So I'll call that d sub 1.

00:03:44.310 --> 00:03:46.870
That's this distance
right there.

00:03:46.870 --> 00:03:49.830
And we want to find all of the
x and y's where the distance to

00:03:49.830 --> 00:03:53.320
the focus is equal to
the distance to this

00:03:53.320 --> 00:03:54.970
line right there.

00:03:54.970 --> 00:03:59.730
So if we call this distance--
let's call this the d sub 2--

00:03:59.730 --> 00:04:01.480
the distance to the directrix.

00:04:01.480 --> 00:04:05.390
So the distance to the focus is
going to be equal to d sub 2.

00:04:05.390 --> 00:04:07.700
And d sub 2, what's this
going to be equal to?

00:04:07.700 --> 00:04:11.790
Well, it's just the difference
in y because no matter where we

00:04:11.790 --> 00:04:15.250
are we're just going to drop
down straight to the directrix.

00:04:15.250 --> 00:04:20.400
So the difference in y-- I
could just say y minus k.

00:04:20.400 --> 00:04:22.180
That would be the
difference in y.

00:04:22.180 --> 00:04:25.620
But in this case I did make it
so that the coordinate up here

00:04:25.620 --> 00:04:27.690
is higher than the k over here.

00:04:27.690 --> 00:04:30.420
But what if we had a situation
where this point was

00:04:30.420 --> 00:04:31.280
below this line?

00:04:31.280 --> 00:04:33.440
I mean, there's nothing here
that I said that we can only

00:04:33.440 --> 00:04:34.930
deal with points
above this line.

00:04:34.930 --> 00:04:36.930
We haven't proven that
to ourselves yet.

00:04:36.930 --> 00:04:39.020
So to make sure that this
distance is positive we could

00:04:39.020 --> 00:04:41.350
take the absolute value or
we could just square it and

00:04:41.350 --> 00:04:42.910
then take the square root.

00:04:42.910 --> 00:04:45.020
And then that ensures that
we're not dealing with

00:04:45.020 --> 00:04:46.560
negative distances.

00:04:46.560 --> 00:04:50.590
So here we set up the equation
for all of the x's and y's

00:04:50.590 --> 00:04:54.400
where the distance to the focus
is equal to the distance

00:04:54.400 --> 00:04:54.940
to the directrix.

00:04:54.940 --> 00:05:00.070
And let's see if this actually
turns out to be a parabola.

00:05:00.070 --> 00:05:02.750
So the first thing we can do is
we can square both sides and

00:05:02.750 --> 00:05:05.660
get rid of the radicals.

00:05:05.660 --> 00:05:10.010
So we get x minus a squared.

00:05:10.010 --> 00:05:11.160
Radicals aren't good.

00:05:11.160 --> 00:05:13.730
Well, I don't want to make
any social commentary.

00:05:13.730 --> 00:05:20.340
So x minus a squared plus y
minus b squared is equal to

00:05:20.340 --> 00:05:26.620
the square root of the other
side-- y minus k squared.

00:05:26.620 --> 00:05:28.540
And right now, at least, it
looks like there's going to be

00:05:28.540 --> 00:05:31.120
some y squared terms and I mean
there's definitely

00:05:31.120 --> 00:05:31.580
an x squared term.

00:05:31.580 --> 00:05:33.420
That doesn't seem like it's
a parabola right now.

00:05:33.420 --> 00:05:37.000
But let's keep going forward.

00:05:37.000 --> 00:05:40.585
This is x minus a squared.

00:05:40.585 --> 00:05:43.650
And then let me square
these-- actually, expand

00:05:43.650 --> 00:05:45.730
these two binomials.

00:05:45.730 --> 00:05:55.560
So plus y squared minus 2yb
plus b squared is equal to y

00:05:55.560 --> 00:06:01.600
squared minus 2yk
plus k squared.

00:06:01.600 --> 00:06:03.020
Now what can we do
to simplify this?

00:06:03.020 --> 00:06:05.960
Well, we have a y squared
here and we have a y squared

00:06:05.960 --> 00:06:07.460
on the right-hand side.

00:06:07.460 --> 00:06:09.260
So we can subtract y
squared from both

00:06:09.260 --> 00:06:11.242
sides of the equation.

00:06:11.242 --> 00:06:14.520
They cancel out, and then we
got rid of the y squared so all

00:06:14.520 --> 00:06:16.580
of a sudden this is starting to
look a lot more like a parabola

00:06:16.580 --> 00:06:21.260
because this is now an equation
that relates y's

00:06:21.260 --> 00:06:22.170
to an x squared.

00:06:22.170 --> 00:06:25.250
So let's clean this up and
actually make sure that we can

00:06:25.250 --> 00:06:26.860
get it in a form that we
normally associate

00:06:26.860 --> 00:06:28.210
with a parabola.

00:06:28.210 --> 00:06:34.460
So we now have-- I'll just
rewrite it-- x minus a squared

00:06:34.460 --> 00:06:42.260
minus 2yb plus b squared is
equal to minus 2yk

00:06:42.260 --> 00:06:45.780
plus k squared.

00:06:45.780 --> 00:06:46.250
Now what can we do?

00:06:46.250 --> 00:06:48.550
Let's take all the y's
and the constants on one

00:06:48.550 --> 00:06:49.860
side of the equation.

00:06:49.860 --> 00:06:53.660
So if we move both of these
terms onto the right-hand

00:06:53.660 --> 00:06:56.260
side, so we want to
add 2yb to both sides.

00:06:56.260 --> 00:07:01.970
So we get x minus a squared is
equal to-- I'm moving both of

00:07:01.970 --> 00:07:03.845
these terms to the
right-hand side.

00:07:03.845 --> 00:07:07.740
So if we add 2yb to both sides
then we'll have a positive

00:07:07.740 --> 00:07:09.490
2yb on the right hand side.

00:07:09.490 --> 00:07:17.670
So 2yb, and then we'll have
that minus 2yk right there.

00:07:17.670 --> 00:07:21.520
And then we're subtracting
b squared from both sides.

00:07:21.520 --> 00:07:27.720
So it's minus b squared
plus k squared.

00:07:27.720 --> 00:07:31.330
I just arbitrarily decided to
write whatever I transferred

00:07:31.330 --> 00:07:33.710
from the left-hand side to the
right-hand side in magenta

00:07:33.710 --> 00:07:35.590
just so maybe it makes
it a little bit clearer.

00:07:35.590 --> 00:07:38.400
And let's see if we can
simplify this further.

00:07:38.400 --> 00:07:41.420
And remember, just so you know
what I'm doing this whole time.

00:07:41.420 --> 00:07:43.770
In the back of my mind I'm
trying to get it into the

00:07:43.770 --> 00:07:48.170
format that I normally
associate with a parabola.

00:07:48.170 --> 00:07:49.940
And I'll do it in the corner
here, just so you know

00:07:49.940 --> 00:07:51.450
where the algebra's going.

00:07:51.450 --> 00:07:56.000
I'm trying to get into a
format x minus a squared

00:07:56.000 --> 00:08:00.870
is equal to y minus b.

00:08:00.870 --> 00:08:04.360
Because if we have it in
this form, or actually ax.

00:08:04.360 --> 00:08:06.580
Let me write that because I
went too far to the right.

00:08:06.580 --> 00:08:09.910
A times x minus-- well,
actually I already used

00:08:09.910 --> 00:08:13.260
the A so let me write x
minus-- I don't know.

00:08:13.260 --> 00:08:19.830
I'll make up x minus--
I'll call that v squared.

00:08:19.830 --> 00:08:23.120
I'm picking maybe v for
x value of the vertex.

00:08:23.120 --> 00:08:25.010
I'm making up
variables on the fly.

00:08:25.010 --> 00:08:28.930
A times x minus v squared
is equal to y minus b.

00:08:28.930 --> 00:08:31.000
And you know, the actual
letters I chose don't matter.

00:08:31.000 --> 00:08:33.860
But this is a general
form of a parabola.

00:08:33.860 --> 00:08:34.970
And this is where
I'm trying to go.

00:08:34.970 --> 00:08:37.200
If I can get this thing that
we're working on to this

00:08:37.200 --> 00:08:39.410
form then we know we're
dealing with a parabola.

00:08:39.410 --> 00:08:42.130
And then, we can actually
relate what we normally

00:08:42.130 --> 00:08:44.830
associate with a parabola
to these terms up here.

00:08:44.830 --> 00:08:46.120
And obviously, these
are different letters

00:08:46.120 --> 00:08:47.270
than these letters.

00:08:47.270 --> 00:08:49.760
And then we can try to figure
out-- if we have a parabola,

00:08:49.760 --> 00:08:52.590
how do we figure out its
focus and is directrix.

00:08:52.590 --> 00:08:54.110
But anyway, that was a bit
of an aside just so you

00:08:54.110 --> 00:08:54.790
know where we're going.

00:08:54.790 --> 00:08:57.516
So let's try to simplify
this a little bit more.

00:08:57.516 --> 00:08:58.857
So we get--

00:08:58.857 --> 00:09:00.350
[PHONE RINGING]

00:09:00.350 --> 00:09:03.010
--my phone is ringing.

00:09:03.010 --> 00:09:04.200
Let me silent it.

00:09:04.200 --> 00:09:10.000
So we get x minus a squared
is equal to-- let's see.

00:09:10.000 --> 00:09:14.540
If we factor out a 2 and
a y we get-- let me

00:09:14.540 --> 00:09:15.520
factor out a y first.

00:09:15.520 --> 00:09:22.080
We get 2b minus 2k times y.

00:09:22.080 --> 00:09:26.260
And then I want to make
it minus some constant.

00:09:26.260 --> 00:09:27.460
So this is a constant
right here.

00:09:27.460 --> 00:09:29.650
This is one constant squared
plus another constant.

00:09:29.650 --> 00:09:36.750
So I could make this
equal to minus-- let

00:09:36.750 --> 00:09:38.540
me think about this.

00:09:38.540 --> 00:09:42.410
This would be minus b
squared minus k squared.

00:09:42.410 --> 00:09:45.610
If I expand this minus out
you would get minus b

00:09:45.610 --> 00:09:46.870
squared plus k squared.

00:09:46.870 --> 00:09:49.380
So this is the same
thing right there.

00:09:49.380 --> 00:09:49.870
And let's see.

00:09:49.870 --> 00:09:54.840
Can I factor out-- yeah,
I can factor out a 2.

00:09:54.840 --> 00:10:00.540
The whole thing becomes x minus
a squared is equal to 2 times b

00:10:00.540 --> 00:10:07.360
minus k times y minus-- and now
this is the difference of--

00:10:07.360 --> 00:10:10.680
yeah, this is b plus
k times b minus k.

00:10:10.680 --> 00:10:15.970
You might want to review
factoring polynomials if

00:10:15.970 --> 00:10:17.560
that doesn't look familiar.

00:10:17.560 --> 00:10:20.780
But this is b plus
k times b minus k.

00:10:20.780 --> 00:10:23.285
And I want to do that because I
saw this b minus k out there so

00:10:23.285 --> 00:10:27.230
that looked interesting to get
it on both terms right there.

00:10:27.230 --> 00:10:27.740
Now let's see.

00:10:27.740 --> 00:10:32.060
If we divide everything times--
let's divide everything by 1

00:10:32.060 --> 00:10:36.660
over 2 b minus k because I just
want a y here, so I have some

00:10:36.660 --> 00:10:37.800
coefficient in front of the y.

00:10:37.800 --> 00:10:41.330
So let's divide everything
by 1 over 2 b minus k.

00:10:44.420 --> 00:10:46.430
Let me do it down here
so I have some space.

00:10:46.430 --> 00:10:57.660
1 over 2 b minus k times x
minus a squared is equal to--

00:10:57.660 --> 00:11:01.120
I'm dividing this by 2 b
minus k, so that goes away.

00:11:01.120 --> 00:11:08.150
So it's y minus-- so if I
divide this by 2 b minus k the

00:11:08.150 --> 00:11:18.780
b minus k's cancel out and I'm
just left with b plus k over 2.

00:11:18.780 --> 00:11:22.340
And I think I have gotten it
in the form that I wanted.

00:11:22.340 --> 00:11:25.480
In this case, the a that
I wrote up here is now

00:11:25.480 --> 00:11:29.550
1 over 2 b minus k.

00:11:29.550 --> 00:11:32.150
This v is this a
that I did here.

00:11:32.150 --> 00:11:35.400
The y is the y and then my
constant term is out here.

00:11:35.400 --> 00:11:38.800
So I have hopefully shown to
you that the locus of all

00:11:38.800 --> 00:11:43.180
points that are equidistant to
this point-- which is really

00:11:43.180 --> 00:11:44.230
just an arbitrary point.

00:11:44.230 --> 00:11:47.110
I said it's the point a,
b-- and some line, is

00:11:47.110 --> 00:11:48.440
in fact, a parabola.

00:11:48.440 --> 00:11:51.960
And that's one of the ways that
people define a parabola.

00:11:51.960 --> 00:11:54.300
And we'll actually see soon
that when you actually change

00:11:54.300 --> 00:11:57.930
this relation-- in this case,
we found that all of the points

00:11:57.930 --> 00:12:00.980
that are equidistant to the
focus and the directrix.

00:12:00.980 --> 00:12:04.760
If we changed that ratio, if we
find all of the points that are

00:12:04.760 --> 00:12:08.010
1/2 as far away from the
directrix as the focus, we

00:12:08.010 --> 00:12:09.830
start getting another
conic section.

00:12:09.830 --> 00:12:11.410
If we said twice as far
we would get another

00:12:11.410 --> 00:12:12.060
conic section.

00:12:12.060 --> 00:12:15.320
So this is a very interesting
way of relating-- sometimes the

00:12:15.320 --> 00:12:17.850
conic sections in the first
video I showed how they're all

00:12:17.850 --> 00:12:19.970
related to cutting the
three-dimensional cone.

00:12:19.970 --> 00:12:22.610
But they're also related in
this way and eventually I want

00:12:22.610 --> 00:12:25.130
to relate this to the
three-dimensional cone so you

00:12:25.130 --> 00:12:26.790
can see how it all
fits together.

00:12:26.790 --> 00:12:29.810
Which is always the fun
thing about mathematics.

00:12:29.810 --> 00:12:33.790
So this so far, all we've shown
to you is if we do take the

00:12:33.790 --> 00:12:36.780
locus of all points that are
equidistant to a point and a

00:12:36.780 --> 00:12:38.770
directrix that it
is a parabola.

00:12:38.770 --> 00:12:40.570
Now, the interesting
thing to do is let's

00:12:40.570 --> 00:12:43.830
say I have a parabola.

00:12:43.830 --> 00:12:46.490
Let me give a general form.

00:12:46.490 --> 00:12:48.810
And I'll try to use different
letters than I used up here.

00:12:48.810 --> 00:12:54.052
So let's say I have a
parabola y minus--

00:12:54.052 --> 00:12:56.920
let's call it y sub 1.

00:12:56.920 --> 00:13:01.070
Where it's just a particular
point on the parabola.

00:13:01.070 --> 00:13:05.090
Y minus y sub 1 or you could
call it some constant-- is

00:13:05.090 --> 00:13:09.585
equal to-- let's call it, I
want to use a capital A just

00:13:09.585 --> 00:13:12.090
because that tends to be--
I'll use a capital A and just

00:13:12.090 --> 00:13:14.340
know that it's different
than this lowercase A.

00:13:14.340 --> 00:13:18.960
Some capital A times
x minus x sub 1.

00:13:18.960 --> 00:13:19.200
squared.

00:13:19.200 --> 00:13:22.140
Whenever you see a y sub 1 or
an x sub 1, that means you're

00:13:22.140 --> 00:13:23.770
actually dealing with
a constant term.

00:13:23.770 --> 00:13:26.560
If you see just an x or y that
generally means that you're

00:13:26.560 --> 00:13:27.770
dealing with a variable term.

00:13:27.770 --> 00:13:28.570
So these are constants.

00:13:28.570 --> 00:13:30.240
I could have put letters there.

00:13:30.240 --> 00:13:32.000
I could have put a and b
there, but then that would've

00:13:32.000 --> 00:13:33.070
gotten confused with that.

00:13:33.070 --> 00:13:35.970
But let's see if we can relate
these letters because this is

00:13:35.970 --> 00:13:37.270
something that we normally see.

00:13:37.270 --> 00:13:40.690
And then, we want to find
the focus or the directrix

00:13:40.690 --> 00:13:41.870
of a parabola like this.

00:13:41.870 --> 00:13:44.950
How can we relate these to
these values right here?

00:13:44.950 --> 00:13:48.860
How does k and a and b relate
to these right there?

00:13:48.860 --> 00:13:49.810
So let's see if we
can set that up.

00:13:49.810 --> 00:13:54.600
So if we just pattern
match that is A.

00:13:54.600 --> 00:13:58.250
If we say x sub 1 is lowercase
a, actually that's a straight

00:13:58.250 --> 00:14:00.770
up pattern match, so we can
already write that on the side.

00:14:00.770 --> 00:14:03.830
x sub 1 is equal to a.

00:14:03.830 --> 00:14:06.210
So whatever x value I have
here, this is also the x

00:14:06.210 --> 00:14:07.750
value of the focus point.

00:14:07.750 --> 00:14:10.210
Which makes sense because in
this case, we already learned

00:14:10.210 --> 00:14:11.235
this in previous video.

00:14:11.235 --> 00:14:15.660
But we learned that the vertex
of this parabola is the

00:14:15.660 --> 00:14:19.160
point x sub 1 comma y sub 1.

00:14:19.160 --> 00:14:20.370
Anyway, I don't want
to confuse you.

00:14:20.370 --> 00:14:21.690
Actually, you know
what I'm going to do?

00:14:21.690 --> 00:14:24.980
I don't want to go too far off
into this topic, and I think

00:14:24.980 --> 00:14:26.520
this video has already
gotten long enough.

00:14:26.520 --> 00:14:28.110
I will continue this
in the next video.

00:14:28.110 --> 00:14:29.612
See you soon.


Partial fraction expansion 3

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.250
There's one more case of
partial fraction expansion

00:00:04.250 --> 00:00:06.960
or decomposition problems
that you might see, so I

00:00:06.960 --> 00:00:08.100
thought I would cover it.

00:00:08.100 --> 00:00:10.960
And that's the situation where
you have a repeated factor

00:00:10.960 --> 00:00:11.790
in the denominator.

00:00:11.790 --> 00:00:15.060
So let's see, I've constructed
a little problem here.

00:00:15.060 --> 00:00:16.540
It's 6x squared.

00:00:16.540 --> 00:00:18.790
Let me make sure
my pen is right.

00:00:18.790 --> 00:00:27.460
6x squared minus 19x plus 15.

00:00:27.460 --> 00:00:33.880
All of that over x minus 1
times-- and this is the

00:00:33.880 --> 00:00:36.640
thing that makes this one
especially interesting--

00:00:36.640 --> 00:00:40.500
x minus 2 squared.

00:00:40.500 --> 00:00:42.560
So you might say, gee, this
is a little bit different.

00:00:42.560 --> 00:00:43.660
Because what do I do here?

00:00:43.660 --> 00:00:47.840
I have this first degree
factor, but it shows up twice.

00:00:47.840 --> 00:00:49.860
It doesn't make sense,
for example, it wouldn't

00:00:49.860 --> 00:00:51.020
make sense to do this.

00:00:51.020 --> 00:00:59.240
A over x minus 1 plus B over x
minus 2 plus C over x minus 2.

00:00:59.240 --> 00:01:01.610
Because if you did this, the
B and the C would just add

00:01:01.610 --> 00:01:03.670
together, because they have
the same denominator.

00:01:03.670 --> 00:01:05.690
You could just view
them as one variable.

00:01:05.690 --> 00:01:07.670
You wouldn't have to have two
separate variables here.

00:01:07.670 --> 00:01:09.940
So this wouldn't make sense
as a partial fraction

00:01:09.940 --> 00:01:11.160
expansion of this.

00:01:11.160 --> 00:01:14.820
You might want to maybe square
this and view it as a second

00:01:14.820 --> 00:01:17.420
degree term, and do it like you
did in the previous example.

00:01:17.420 --> 00:01:20.540
But then you wouldn't have
fully decomposed this problem.

00:01:20.540 --> 00:01:22.900
And so the answer here-- and
I'll just show you how to do

00:01:22.900 --> 00:01:25.830
it, I'll maybe leave it to you
a little bit to think about why

00:01:25.830 --> 00:01:30.700
it works-- is to decompose it
almost like this, but instead

00:01:30.700 --> 00:01:33.180
of having C over x minus
2, you're going to have C

00:01:33.180 --> 00:01:35.005
over x minus 2 squared.

00:01:35.005 --> 00:01:36.720
And I'll try to give
you a little intuition

00:01:36.720 --> 00:01:38.150
about why that happens.

00:01:38.150 --> 00:01:43.490
So the decomposition here is
going to be A over x minus 1

00:01:43.490 --> 00:01:53.430
plus B over x minus 2 plus
C over x minus 2 squared.

00:01:53.430 --> 00:01:55.890
And the intuition here is
that if you were to-- let's

00:01:55.890 --> 00:01:57.270
ignore this term right here.

00:01:57.270 --> 00:02:00.790
But if you were to add these
two terms right there, you

00:02:00.790 --> 00:02:07.790
would get a rational function
that would end up having

00:02:07.790 --> 00:02:10.200
something times x
plus something.

00:02:10.200 --> 00:02:13.390
And so it would be consistent
with what we did in the second

00:02:13.390 --> 00:02:16.960
partial fraction video, where
if you would have a second

00:02:16.960 --> 00:02:20.360
degree term on the bottom, and
the numerator, when you add

00:02:20.360 --> 00:02:23.300
just these two parts-- let
me clarify what I'm saying.

00:02:23.300 --> 00:02:24.750
And this is just for intuition.

00:02:24.750 --> 00:02:30.990
If you were to add just these
two parts here, you would get

00:02:30.990 --> 00:02:34.680
something times x plus
something, all of that

00:02:34.680 --> 00:02:38.710
over x minus 2 squared.

00:02:38.710 --> 00:02:42.110
And that's consistent with what
we did in the previous video,

00:02:42.110 --> 00:02:44.030
where we said if we have a
second degree term in the

00:02:44.030 --> 00:02:46.610
denominator-- which this really
is if you were to expand this--

00:02:46.610 --> 00:02:48.930
you should have a first degree
term in the numerator.

00:02:48.930 --> 00:02:49.830
So I'll leave that there.

00:02:49.830 --> 00:02:52.570
That's a little bit of a
nuance, and it's good to

00:02:52.570 --> 00:02:54.570
think about why this works.

00:02:54.570 --> 00:02:57.500
But with that said, let's
just solve this problem.

00:02:57.500 --> 00:03:02.890
Let me erase some of this
stuff that I've written.

00:03:02.890 --> 00:03:06.860
Let me erase all of that.

00:03:06.860 --> 00:03:07.980
All right.

00:03:07.980 --> 00:03:09.590
Ready to work on this problem.

00:03:09.590 --> 00:03:11.640
So given that this is the
expansion, now we just have

00:03:11.640 --> 00:03:14.630
to solve for A, B, and C.

00:03:14.630 --> 00:03:17.820
So if we were to add these
three fractions, the common

00:03:17.820 --> 00:03:27.380
denominator is x minus 1
times x minus 2 squared.

00:03:29.950 --> 00:03:33.860
This is a factor of that, so
the least common multiple of

00:03:33.860 --> 00:03:35.290
this and this is just this.

00:03:35.290 --> 00:03:37.320
So it's just x minus 2 squared
there, and that's why we

00:03:37.320 --> 00:03:39.200
just did this to begin with.

00:03:39.200 --> 00:03:42.920
And the numerator is going
to be-- let me do it in

00:03:42.920 --> 00:03:45.646
a different color-- A.

00:03:45.646 --> 00:03:47.460
What do we have to
multiply A by?

00:03:47.460 --> 00:03:50.860
A times x minus 2
squared, right?

00:03:50.860 --> 00:03:52.330
If we canceled these out,
you're just left with

00:03:52.330 --> 00:03:54.440
A over x minus 1.

00:03:54.440 --> 00:04:02.400
Plus B times x minus
1 times x minus 2.

00:04:02.400 --> 00:04:02.620
Right?

00:04:02.620 --> 00:04:05.670
If we cancel out the x minus
1's and one of these x minus

00:04:05.670 --> 00:04:09.890
2's from down here, you're just
left with B over x minus 2.

00:04:09.890 --> 00:04:14.650
Plus C times x minus 1.

00:04:14.650 --> 00:04:18.260
If that cancels with that,
we're left with C over

00:04:18.260 --> 00:04:19.850
x minus 2 squared.

00:04:19.850 --> 00:04:24.360
And all of this is going to be
equal to-- that up there in

00:04:24.360 --> 00:04:28.410
blue is going to be equal to--
I have to write it smaller-- 6x

00:04:28.410 --> 00:04:36.440
squared minus 19x plus 15,
all of that over x minus 1

00:04:36.440 --> 00:04:39.390
times x minus 2 squared.

00:04:39.390 --> 00:04:41.370
And like we've done in
every problem so far, the

00:04:41.370 --> 00:04:44.060
denominators are the same, so
we can just set the numerators

00:04:44.060 --> 00:04:48.480
equal to each other and try to
solve for A, B, and C.

00:04:48.480 --> 00:04:50.450
Let me write that.

00:04:50.450 --> 00:04:52.070
And I'm really just
rewriting it.

00:04:52.070 --> 00:05:00.550
A times x minus 2 squared plus
B times x minus 1 times x minus

00:05:00.550 --> 00:05:07.160
2 plus C times x minus 1
is going to be equal to 6x

00:05:07.160 --> 00:05:11.260
squared minus 19x plus 15.

00:05:11.260 --> 00:05:12.140
There you go.

00:05:12.140 --> 00:05:13.530
And now we have to
solve for them.

00:05:13.530 --> 00:05:15.730
And we can do it just the way
we done the previous problems.

00:05:15.730 --> 00:05:19.610
We can pick choices for x that
make things cancel out nicely.

00:05:19.610 --> 00:05:21.660
So if we wanted to solve
for A, we want to make the

00:05:21.660 --> 00:05:23.380
B and the C cancel out.

00:05:23.380 --> 00:05:25.270
We just have to pick
x is equal to 1.

00:05:25.270 --> 00:05:26.910
Because if x is equal
to 1, this becomes

00:05:26.910 --> 00:05:28.840
0, so C disappears.

00:05:28.840 --> 00:05:31.230
This becomes 0, so this whole
term disappears, and we're

00:05:31.230 --> 00:05:32.360
just left with this and this.

00:05:32.360 --> 00:05:33.450
So let's do that.

00:05:33.450 --> 00:05:39.420
So when x is equal to 1, we
get-- let me pick another

00:05:39.420 --> 00:05:47.560
color-- A times, 1 minus 2 is
minus 1, squared is just 1.

00:05:47.560 --> 00:05:48.430
So it's A times 1.

00:05:48.430 --> 00:05:49.920
So we can just
leave the A there.

00:05:49.920 --> 00:05:52.400
This becomes 0, because
this term right here

00:05:52.400 --> 00:05:54.030
will be 0 if x is 1.

00:05:54.030 --> 00:05:56.540
This become 0, because
that term would be 0

00:05:56.540 --> 00:05:57.710
if x is equal to 1.

00:05:57.710 --> 00:06:05.010
So A is equal to 6 times
1-- so 6-- minus 19 times

00:06:05.010 --> 00:06:09.200
1-- minus 19-- plus 15.

00:06:09.200 --> 00:06:09.850
And what is that?

00:06:09.850 --> 00:06:16.210
Minus 19 plus 15, that's minus
4, plus 6, that is equal to 2.

00:06:16.210 --> 00:06:17.390
There you go.

00:06:17.390 --> 00:06:20.890
Now let's try to solve for--
let's see, what can we do?

00:06:20.890 --> 00:06:23.830
If we make x equal to 2,
the A will disappear.

00:06:23.830 --> 00:06:26.760
This whole term will disappear,
because this would be 0.

00:06:26.760 --> 00:06:29.220
So we could use that
to solve for C.

00:06:29.220 --> 00:06:36.560
So if we say that x is equal
to 2, then we get this term

00:06:36.560 --> 00:06:38.270
would be 0, because that's 0.

00:06:38.270 --> 00:06:41.930
This whole expression is 0,
because x minus 2 would be 0.

00:06:41.930 --> 00:06:45.660
And we're just left with C
times 2 minus 1, so that's

00:06:45.660 --> 00:06:51.040
just 1-- so C is equal
to 6 times 4, right?

00:06:51.040 --> 00:06:52.180
2 squared.

00:06:52.180 --> 00:07:00.910
So it's 24, minus 38--
19 times 2-- plus 15.

00:07:00.910 --> 00:07:07.340
Let's see, 24 plus 15 is 39,
minus 38, that is equal to 1.

00:07:07.340 --> 00:07:09.250
We're in the home stretch.

00:07:09.250 --> 00:07:12.430
And now for B, there's no
obvious way to make the other

00:07:12.430 --> 00:07:14.600
two cancel out without
making the B cancel out.

00:07:14.600 --> 00:07:16.570
But we've solved for everything
else, so we can really just

00:07:16.570 --> 00:07:19.790
pick an arbitrary value for x
that'll make things

00:07:19.790 --> 00:07:21.200
easy to compute.

00:07:21.200 --> 00:07:24.870
So if we just pick x is equal
to 0, that's always something

00:07:24.870 --> 00:07:27.760
that can clear out a lot of
the hairiness of an algebra

00:07:27.760 --> 00:07:29.980
problem. x is equal
to 0, not 3.

00:07:29.980 --> 00:07:31.760
x is equal to 0.

00:07:31.760 --> 00:07:33.630
Let me do that in a
different color.

00:07:33.630 --> 00:07:35.690
If x is equal to 0,
then what do we have?

00:07:35.690 --> 00:07:38.350
We have A, but A is 2, right?

00:07:38.350 --> 00:07:41.570
2 times 0 minus 2 squared.

00:07:41.570 --> 00:07:46.050
So that's minus 2 squared,
so that's 2 times 4.

00:07:46.050 --> 00:07:50.690
Plus B-- that's what we're
trying to solve for; we already

00:07:50.690 --> 00:07:54.200
solved for everything--
B times minus 1, right?

00:07:54.200 --> 00:07:56.140
0 minus 1 is minus 1.

00:07:56.140 --> 00:08:03.620
Times minus 2, plus C times
minus 1 is equal to-- well,

00:08:03.620 --> 00:08:08.660
if x is 0, then this is 0,
this is 0, is equal to 15.

00:08:08.660 --> 00:08:10.190
Then we just solve for B.

00:08:10.190 --> 00:08:16.240
We get 8 plus 2B, right?

00:08:16.240 --> 00:08:20.280
Minus 1 times minus
2 is plus 2.

00:08:20.280 --> 00:08:21.770
Oh, and we shouldn't have
written the C here,

00:08:21.770 --> 00:08:22.560
we know what C is.

00:08:22.560 --> 00:08:23.580
C is 1.

00:08:23.580 --> 00:08:24.960
So this is just a 1.

00:08:24.960 --> 00:08:30.260
So 1 times minus 1 is
minus 1, is equal to 15.

00:08:30.260 --> 00:08:36.990
And let's see, we have 2B is
equal to, lets see, 16 is

00:08:36.990 --> 00:08:41.760
equal to 8, B is equal to 4,
dividing both sides by 2.

00:08:41.760 --> 00:08:43.080
So we're done.

00:08:43.080 --> 00:08:47.580
So the partial fraction
decomposition of this right

00:08:47.580 --> 00:08:54.490
here is A, which we've
solved for, which is 2.

00:08:54.490 --> 00:09:03.440
So it equals 2 over x minus 1
plus B, which is 4-- plus 4

00:09:03.440 --> 00:09:12.820
over x minus 2, plus C, which
is 1, over x minus 2 squared.

00:09:12.820 --> 00:09:15.780
And what we did in this with
the repeated factor is

00:09:15.780 --> 00:09:17.690
true if we went to a
higher degree term.

00:09:17.690 --> 00:09:22.970
So if we had blah blah blah,
some polynomial there, and it

00:09:22.970 --> 00:09:27.770
was all over x minus-- I
don't know, some number.

00:09:27.770 --> 00:09:31.420
x minus a to the 10th power.

00:09:31.420 --> 00:09:33.770
Well, to the--
well, yeah, sure.

00:09:33.770 --> 00:09:34.980
To the 10th power.

00:09:34.980 --> 00:09:40.300
If we wanted to decompose this
partial fraction, or expand it,

00:09:40.300 --> 00:09:45.630
it would be A over x minus
a-- it's a different a.

00:09:45.630 --> 00:09:46.930
I'm just showing this.

00:09:46.930 --> 00:09:51.210
Plus B times x minus a
squared, plus, and you

00:09:51.210 --> 00:09:52.220
go blah blah blah.

00:09:52.220 --> 00:09:56.270
You'd have 10 terms, plus-- I
don't know what the 10th letter

00:09:56.270 --> 00:09:59.150
of the alphabet is; maybe
it's H or I or something.

00:09:59.150 --> 00:10:00.510
Maybe it's J.

00:10:00.510 --> 00:10:05.160
J over x minus a to the 10th.

00:10:05.160 --> 00:10:07.550
So you can use that property in
general, though you'll very

00:10:07.550 --> 00:10:09.960
seldom see something like this,
because computationally

00:10:09.960 --> 00:10:11.950
it would take you
forever to do it.

00:10:11.950 --> 00:10:14.440
The hairiest problem that you
might see with three variables

00:10:14.440 --> 00:10:18.290
on an actual exam is probably
something like I showed you in

00:10:18.290 --> 00:10:21.120
the previous example, or what
I just showed you right now.

00:10:21.120 --> 00:10:22.950
Anything hairier than
that, you'd probably end

00:10:22.950 --> 00:10:23.720
up using a computer.

00:10:23.720 --> 00:10:25.270
But you should know how to do
it, because if you have a

00:10:25.270 --> 00:10:27.530
computer and you're solving a
real world problem that is a

00:10:27.530 --> 00:10:30.490
lot more complicated than this,
you should know how to do it.

00:10:30.490 --> 00:10:34.170
Anyway, hope you found
that a bit helpful.


Partial fraction expansion 2

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:02.790
Let's see if we can tackle a
more complicated partial

00:00:02.790 --> 00:00:04.950
fraction decomposition problem.

00:00:04.950 --> 00:00:15.386
I have 10x squared plus 12x
plus 20, all of that over

00:00:15.386 --> 00:00:18.630
x to the third minus 8.

00:00:18.630 --> 00:00:23.050
The first thing to do with any
of these rational expressions

00:00:23.050 --> 00:00:25.920
that you want to decompose is
to just make sure that the

00:00:25.920 --> 00:00:29.180
numerator is of a lower degree
than the denominator, and if

00:00:29.180 --> 00:00:32.260
it's not, then you just do the
algebraic long division like

00:00:32.260 --> 00:00:33.350
we did in the first video.

00:00:33.350 --> 00:00:34.660
But here, you can do
from [UNINTELLIGIBLE]

00:00:34.660 --> 00:00:37.290
the highest degree term here is
a second-degree term, here it's

00:00:37.290 --> 00:00:38.990
a third-degree term,
so we're cool.

00:00:38.990 --> 00:00:40.470
This is a lower degree
than that one.

00:00:40.470 --> 00:00:42.570
If it was the same or higher,
we would do a little

00:00:42.570 --> 00:00:43.790
long division.

00:00:43.790 --> 00:00:46.500
The next thing to do, if we're
going to decompose this into

00:00:46.500 --> 00:00:49.480
its components, we have to
figure out the factors of the

00:00:49.480 --> 00:00:54.090
denominator right here, so that
we can use those factors as the

00:00:54.090 --> 00:00:57.440
denominators in each of the
components, and a third-degree

00:00:57.440 --> 00:01:00.690
polynomial is much, much, much
harder to factor than

00:01:00.690 --> 00:01:03.850
a second-degree
polynomial, normally.

00:01:03.850 --> 00:01:05.990
But in this case, there's
something that should hopefully

00:01:05.990 --> 00:01:08.370
pop out at you-- if it doesn't
immediately, hopefully what I'm

00:01:08.370 --> 00:01:10.890
about to say will make it pop
out at you in the future-- is

00:01:10.890 --> 00:01:12.880
you should always think about
what number, when you

00:01:12.880 --> 00:01:18.030
substitute into a polynomial,
will make it equal to 0.

00:01:18.030 --> 00:01:21.420
And in this case, what to the
third power minus 8 equals 0?

00:01:21.420 --> 00:01:23.310
And hopefully 2
pops out at you.

00:01:23.310 --> 00:01:25.310
And this is something
you can only do really

00:01:25.310 --> 00:01:27.840
through inspection or
through experience.

00:01:27.840 --> 00:01:30.450
And you'll immediately see 2 to
the third minus 8 is 0, so 2 is

00:01:30.450 --> 00:01:39.890
a 0 of this, or 2 makes this
expression 0, and that tells us

00:01:39.890 --> 00:01:42.680
that x minus 2 is a factor.

00:01:42.680 --> 00:01:48.940
So we can rewrite this right
here as 10x squared plus 12x

00:01:48.940 --> 00:01:55.060
plus 20 over x minus 2
times something something

00:01:55.060 --> 00:01:55.810
something something.

00:01:55.810 --> 00:01:57.680
We don't know what that
something is yet.

00:01:57.680 --> 00:02:00.420
And I just want to hit the
point home of why this

00:02:00.420 --> 00:02:02.990
is true, or what the
intuition vibe has true.

00:02:02.990 --> 00:02:06.600
If 2 makes this 0, 2 should
also make the factored

00:02:06.600 --> 00:02:07.700
expression 0.

00:02:07.700 --> 00:02:10.260
And we know that 2 would make
this factored expression 0,

00:02:10.260 --> 00:02:14.920
because when you put a 2 right
here, this factor become 0, so

00:02:14.920 --> 00:02:16.750
it'll make the whole thing 0.

00:02:16.750 --> 00:02:19.140
And so that's why, that's the
intuition where, if you

00:02:19.140 --> 00:02:21.760
substitute a number here and it
makes this 0, you do x minus

00:02:21.760 --> 00:02:24.490
that number here, and we
know that that will be

00:02:24.490 --> 00:02:25.590
a factor of the thing.

00:02:25.590 --> 00:02:29.300
Well, anyway, the next step if
we really want to decompose

00:02:29.300 --> 00:02:32.590
this rational expression is to
figure out what this part of it

00:02:32.590 --> 00:02:36.790
is, and the way to do that is
with algebraic long division.

00:02:36.790 --> 00:02:39.770
We essentially just divide x
minus 2 into x to the third

00:02:39.770 --> 00:02:42.620
minus 8 to get this,
so let's do that.

00:02:42.620 --> 00:02:50.790
So you get x minus 2 goes into
x to the third-- and actually,

00:02:50.790 --> 00:02:55.050
what I'm going to do is, I'm
going to write-- I leave space

00:02:55.050 --> 00:02:58.330
for the second-degree term,
which is 0, the first-degree

00:02:58.330 --> 00:03:01.940
term, and then minus 8 is the
constant term, so minus 8-- I

00:03:01.940 --> 00:03:06.170
like to put all my degree terms
in their appropriate columns.

00:03:06.170 --> 00:03:08.570
You don't have to when you
do algebraic long division.

00:03:08.570 --> 00:03:12.590
So x minus 2 goes into x to the
third, or x goes into x to

00:03:12.590 --> 00:03:13.370
the third, how many times?

00:03:13.370 --> 00:03:15.330
You just have to look at
the highest degree terms.

00:03:15.330 --> 00:03:18.530
Well it goes into it x squared
times-- I'll put it in the x

00:03:18.530 --> 00:03:22.680
squared column-- x squared
times x minus 2, x squared

00:03:22.680 --> 00:03:26.930
times x is x to the third, x
squared times minus 2

00:03:26.930 --> 00:03:30.540
is minus 2x squared.

00:03:30.540 --> 00:03:35.680
And now we want to subtract
that from that, so I could

00:03:35.680 --> 00:03:38.000
just-- or I could add the
negative, I always find that

00:03:38.000 --> 00:03:42.020
easier-- so these cancel out,
so I'm left with 2x

00:03:42.020 --> 00:03:44.620
squared minus 8.

00:03:44.620 --> 00:03:47.400
How many times does x minus 2
go into 2x squared minus 8?

00:03:47.400 --> 00:03:50.610
You just look at the highest
degree terms, x goes into 2x

00:03:50.610 --> 00:04:00.980
squared 2x times, so plus 2x,
2x times x is 2x squared, 2x

00:04:00.980 --> 00:04:05.700
times minus 2 is minus 4x.

00:04:05.700 --> 00:04:08.000
Subtract that from that, so we
could-- or we could just add

00:04:08.000 --> 00:04:16.560
the negative-- and then we have
4x minus 8, x goes into 4x four

00:04:16.560 --> 00:04:23.560
times, 4 times x minus 2 is
4x minus 8, and there's no

00:04:23.560 --> 00:04:26.830
remainder, and this is further
confirmation that x minus 2

00:04:26.830 --> 00:04:30.720
definitely was a factor of x to
the third minus 8 and the other

00:04:30.720 --> 00:04:32.950
factor is this right here.

00:04:32.950 --> 00:04:37.290
x squared plus 2x plus 4 times
x minus 2 is equal to x

00:04:37.290 --> 00:04:39.470
to the third minus 8.

00:04:39.470 --> 00:04:41.080
So now we can write
our original problem.

00:04:45.590 --> 00:04:50.330
Actually, let me write-- well,
let me write it down here.

00:04:50.330 --> 00:04:53.180
What was it, it was 10-- let me
switch to another color-- it

00:04:53.180 --> 00:05:06.110
was 10x squared-- I have a bad
memory-- plus 12x plus 20, over

00:05:06.110 --> 00:05:11.610
x minus 2 times-- what was
that-- x squared

00:05:11.610 --> 00:05:12.960
plus 2x plus 4.

00:05:17.025 --> 00:05:21.180
So all the work we did so far
is just to factor out that x to

00:05:21.180 --> 00:05:25.450
the third minus 8, but now we
can actually do some partial

00:05:25.450 --> 00:05:29.230
fraction expansion, or partial
fraction decomposition.

00:05:29.230 --> 00:05:31.020
So this is going to be equal
to-- and this is the

00:05:31.020 --> 00:05:33.630
interesting point-- this is
where we diverge or advance

00:05:33.630 --> 00:05:35.240
a little bit from what we
did in the first video.

00:05:35.240 --> 00:05:40.840
This is going to be equal to a
constant over this denominator,

00:05:40.840 --> 00:05:45.260
over x minus 2, plus-- and this
is a little different than what

00:05:45.260 --> 00:05:49.785
you were going to see, what you
saw in the last video-- plus an

00:05:49.785 --> 00:05:52.790
x-term, some coefficient times
an x-term, plus c, so

00:05:52.790 --> 00:05:55.340
bx plus c, over this.

00:05:55.340 --> 00:05:58.510
x squared plus 2x plus 4.

00:05:58.510 --> 00:06:00.550
And you might say, Sal, how
did you know to do that?

00:06:00.550 --> 00:06:02.180
Well, what you do is
you look at the degree

00:06:02.180 --> 00:06:02.990
of the denominator.

00:06:02.990 --> 00:06:05.420
This is a first-degree
denominator, and you say, OK.

00:06:05.420 --> 00:06:09.930
The degree of the numerator in
this part of the fraction, I

00:06:09.930 --> 00:06:11.960
guess you could call it, is
going to be 1 less than that.

00:06:11.960 --> 00:06:14.440
So this first degree-- so this
is going to be a 0-degree or a

00:06:14.440 --> 00:06:18.300
constant term-- Here
the degree is 2.

00:06:18.300 --> 00:06:21.640
The degree is 2, so the degree
of its numerators is going to

00:06:21.640 --> 00:06:25.730
be 1, and since its degree is 1
it could still have a constant

00:06:25.730 --> 00:06:29.660
term, which is a 0-degree
term, so you get bx plus c.

00:06:29.660 --> 00:06:32.110
And you know, maybe it does end
up just being a constant, in

00:06:32.110 --> 00:06:35.060
which case we can solve for
b and it'll just be 0.

00:06:35.060 --> 00:06:36.600
And actually, there's another
thing you might be asking, is

00:06:36.600 --> 00:06:39.470
hey, why don't we factor this
further, and you can try,

00:06:39.470 --> 00:06:41.980
but if you look at it by
inspection, you could kind of

00:06:41.980 --> 00:06:44.030
in your head do the quadratic
formula and you'll see that you

00:06:44.030 --> 00:06:45.720
only get imaginary roots here.

00:06:45.720 --> 00:06:48.830
So this actually isn't more--
this isn't factorable

00:06:48.830 --> 00:06:50.070
in the reals anymore.

00:06:50.070 --> 00:06:52.140
There no more real 0's to this.

00:06:52.140 --> 00:06:54.770
So we have factored this
as much as we can.

00:06:54.770 --> 00:06:59.030
So the key now to complete our
partial fraction expansion is

00:06:59.030 --> 00:07:01.500
to just solve for the a, b's,
and c's, and we'll do it

00:07:01.500 --> 00:07:05.660
exactly the same way we
did it in the last video.

00:07:05.660 --> 00:07:07.710
So what we want to do
is essentially add

00:07:07.710 --> 00:07:09.090
these two things.

00:07:09.090 --> 00:07:12.050
So if we add-- let me write
this on the left hand side-- so

00:07:12.050 --> 00:07:21.890
10x squared plus 12x plus 20,
over x minus 2 times x squared

00:07:21.890 --> 00:07:27.170
plus 2x plus 4, is equal to--
if we add these two things, we

00:07:27.170 --> 00:07:31.010
want to get a common
denominator, which is x minus 2

00:07:31.010 --> 00:07:36.340
times x squared plus 2x plus 4,
just multiply the two things,

00:07:36.340 --> 00:07:40.290
and that'll be, when we add the
common denominator, then it's a

00:07:40.290 --> 00:07:51.200
times this, x squared plus 2x
plus 4, plus bx plus c, times

00:07:51.200 --> 00:07:54.080
this, times x minus 2.

00:07:54.080 --> 00:07:56.630
Well the denominators are the
same, so the numerators have to

00:07:56.630 --> 00:08:05.030
equal each other, so we have--
I'll rewrite it-- 10x squared

00:08:05.030 --> 00:08:15.500
plus 12x plus 20 is equal to a
times x squared plus 2x plus 4

00:08:15.500 --> 00:08:22.200
plus bx plus c, all of
that times x minus 2.

00:08:22.200 --> 00:08:25.170
This might look like a hairy
problem, and if you do it kind

00:08:25.170 --> 00:08:27.775
of the slow way that I do in
the last video, it would take

00:08:27.775 --> 00:08:30.240
you forever, it would be three
equations and three unknowns,

00:08:30.240 --> 00:08:34.130
very very hairy, but we can do
as a system pretty much the

00:08:34.130 --> 00:08:35.080
same way we did it
the first one.

00:08:35.080 --> 00:08:37.600
Let's pick x values that
simplify this and allow us to

00:08:37.600 --> 00:08:41.910
solve for one of these a, b's,
or c's at a time, and the

00:08:41.910 --> 00:08:44.430
immediate thing is what can I--
what x value can I substitute

00:08:44.430 --> 00:08:47.030
that will cancel out at
least two of the variables,

00:08:47.030 --> 00:08:48.460
two of the a, b, or c,?

00:08:48.460 --> 00:08:52.410
And if I put x is equal to 2,
then this thing becomes 0,

00:08:52.410 --> 00:08:54.260
which will make this whole
expression 0, and I'll just be

00:08:54.260 --> 00:08:58.040
left with an a and the x's,
which I'll know is 2, because

00:08:58.040 --> 00:08:59.930
I'm picking x to be equal
to 2, so let's do that.

00:08:59.930 --> 00:09:05.220
So if x is equal to 2,
then what do I get?

00:09:05.220 --> 00:09:11.270
I get 10x squared, so I get 10
times 4, that's 2 squared, plus

00:09:11.270 --> 00:09:18.850
12 times 2, which is 24, plus
20, is equal to a times 2

00:09:18.850 --> 00:09:24.020
squared is 4, plus 4, plus 4.

00:09:24.020 --> 00:09:26.980
And then all of this just
becomes 0, because I

00:09:26.980 --> 00:09:28.860
picked x is equal to 2.

00:09:28.860 --> 00:09:34.780
So this is what, 10 times 4 is
40, plus 44, that's 84, is

00:09:34.780 --> 00:09:43.230
equal to 12a, divide both sides
by 12, we get a is equal to 7.

00:09:43.230 --> 00:09:45.920
We're making progress.

00:09:45.920 --> 00:09:46.820
So now let's see.

00:09:46.820 --> 00:09:49.530
We know what a is, can we
pick any other x values that

00:09:49.530 --> 00:09:51.570
will make the b disappear?

00:09:51.570 --> 00:09:52.720
Or the c.

00:09:52.720 --> 00:09:55.050
Well, we can make the b
disappear if we make x is equal

00:09:55.050 --> 00:09:57.470
to 0, so let's try that out.

00:09:57.470 --> 00:10:02.910
So if we say-- let me pick a
new, vibrant color, a yellow--

00:10:02.910 --> 00:10:08.840
if we make x equal to 0, then
the left hand side of this,

00:10:08.840 --> 00:10:11.780
this is 0,0, we just left
with-- we're just left with 20,

00:10:11.780 --> 00:10:18.640
is equal to a, which is just 7,
times 0 plus 0 plus 4, 7 times

00:10:18.640 --> 00:10:24.160
4, plus-- bx plus c, the bx
just disappears because b times

00:10:24.160 --> 00:10:32.970
0 is 0, plus c times 0 minus 2,
c times minus 2-- scroll down a

00:10:32.970 --> 00:10:39.700
little bit-- so we get 20 is
equal to 28 minus 2c, subtract

00:10:39.700 --> 00:10:43.930
28 from both sides, minus 8 is
equal to minus 2c,

00:10:43.930 --> 00:10:47.580
c is equal to 4.

00:10:47.580 --> 00:10:49.560
We're almost done.

00:10:49.560 --> 00:10:53.780
Now let me rewrite our equation
up here, with the a's, and

00:10:53.780 --> 00:10:55.290
let me see if I can do this.

00:10:55.290 --> 00:10:55.630
All right.

00:10:55.630 --> 00:11:06.580
So now I have 10x squared plus
12x plus 20 is equal to 7, 7

00:11:06.580 --> 00:11:17.340
times x squared plus 2x plus 4
plus bx plus c is 4,

00:11:17.340 --> 00:11:19.040
times x minus 2.

00:11:19.040 --> 00:11:21.270
Almost ran out of space.

00:11:21.270 --> 00:11:24.280
So now we have x's and b's, so
to solve for b, we can really

00:11:24.280 --> 00:11:27.310
just substitute any value of x
that makes our math reasonably

00:11:27.310 --> 00:11:29.080
easy that doesn't make
the b disappear.

00:11:29.080 --> 00:11:33.160
So we can't pick x is equal to
0, but a good number is 1, so

00:11:33.160 --> 00:11:41.140
if x is equal to 1, we have 10
plus 12 plus 20 is equal to 7

00:11:41.140 --> 00:11:48.650
times 1, plus 2 plus 4, that's
7, plus b times 1 is just b,

00:11:48.650 --> 00:11:54.300
plus 4, times 1 minus
2, is minus 1.

00:11:54.300 --> 00:12:04.400
So this is 22, 42 is equal to
49 minus b minus 4-- see, you

00:12:04.400 --> 00:12:09.950
can, I don't want to mess up
the math-- so we can say 42 is

00:12:09.950 --> 00:12:17.060
equal to 45 minus b, subtract
45 from both sides, you get

00:12:17.060 --> 00:12:22.110
minus 3 is equal to minus b,
and we get b is equal to 3,

00:12:22.110 --> 00:12:24.650
divide both sides by minus 1.

00:12:24.650 --> 00:12:28.940
And so on our original problem
up here, we now know that b is

00:12:28.940 --> 00:12:37.150
equal to 3, c was equal to
4, and a is equal to 7.

00:12:37.150 --> 00:12:40.510
So the partial fraction
decomposition of this, we're

00:12:40.510 --> 00:12:48.200
now done, is 7 over x minus
2 plus 3x plus 4, over x

00:12:48.200 --> 00:12:50.830
squared plus 2x plus 4.

00:12:50.830 --> 00:12:53.400
Well, that was a pretty tiring
problem, and you can see, the

00:12:53.400 --> 00:12:56.150
partial fraction decomposition
becomes a lot more complicated

00:12:56.150 --> 00:12:59.090
when you have a higher degree
denominators, but hopefully you

00:12:59.090 --> 00:13:01.180
found that a little bit useful.