Adding and subtracting rational expressions 3

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:01.830
Find the difference.

00:00:01.830 --> 00:00:04.290
Express the answer as a
simplified rational

00:00:04.290 --> 00:00:06.890
expression, and state
the domain.

00:00:06.890 --> 00:00:08.930
We have two rational
expressions, and we're

00:00:08.930 --> 00:00:10.710
subtracting one from
the other.

00:00:10.710 --> 00:00:13.910
Just like when we first learned
to subtract fractions,

00:00:13.910 --> 00:00:17.200
or add fractions, we have to
find a common denominator.

00:00:17.200 --> 00:00:19.080
The best way to find a common
denominator, if were just

00:00:19.080 --> 00:00:22.330
dealing with regular numbers, or
with algebraic expressions,

00:00:22.330 --> 00:00:24.780
is to factor them out, and
make sure that our common

00:00:24.780 --> 00:00:27.515
denominator has all of the
factors in it-- that'll ensure

00:00:27.515 --> 00:00:30.970
that it's divisible by the
two denominators here.

00:00:30.970 --> 00:00:32.759
This guy right here is
completely factored-- he's

00:00:32.759 --> 00:00:34.250
just a plus 2.

00:00:34.250 --> 00:00:37.610
This one over here, let's see if
we can factor it: a squared

00:00:37.610 --> 00:00:39.440
plus 4a plus 4.

00:00:39.440 --> 00:00:43.070
Well, you see the pattern that
4 is 2 squared, 4 is 2 times

00:00:43.070 --> 00:00:47.890
2, so a squared plus 4a plus 4
is a plus 2 times a plus 2, or

00:00:47.890 --> 00:00:49.790
a plus 2 squared.

00:00:49.790 --> 00:00:53.870
We could say it's a plus 2 times
a plus 2-- that's what a

00:00:53.870 --> 00:00:56.560
squared plus 4a plus 4 is.

00:00:56.560 --> 00:00:58.990
This is obviously divisible
by itself-- everything is

00:00:58.990 --> 00:01:03.280
divisible by itself, except, I
guess, for 0, is divisible by

00:01:03.280 --> 00:01:07.040
itself, and it's also divisible
by a plus 2, so this

00:01:07.040 --> 00:01:10.210
is the least common multiple of
this expression, and that

00:01:10.210 --> 00:01:13.480
expression, and it could be
a good common denominator.

00:01:13.480 --> 00:01:16.070
Let's set that up.

00:01:16.070 --> 00:01:19.530
This will be the same thing as
being equal to this first term

00:01:19.530 --> 00:01:26.480
right here, a minus 2 over
a plus 2, but we want the

00:01:26.480 --> 00:01:29.935
denominator now to be a plus 2
times a plus 2-- we wanted it

00:01:29.935 --> 00:01:31.620
to be a plus 2 squared.

00:01:31.620 --> 00:01:34.530
So, let's multiply this
numerator and denominator by a

00:01:34.530 --> 00:01:37.510
plus 2, so its denominator is
the same thing as this.

00:01:37.510 --> 00:01:40.450
Let's multiply both the
numerator and the denominator

00:01:40.450 --> 00:01:43.750
by a plus 2.

00:01:43.750 --> 00:01:46.445
We're going to assume that a
is not equal to negative 2,

00:01:46.445 --> 00:01:48.270
that would have made this
undefined, and it would have

00:01:48.270 --> 00:01:49.970
also made this undefined.

00:01:49.970 --> 00:01:51.950
Throughout this whole thing,
we're going to assume that a

00:01:51.950 --> 00:01:54.800
cannot be equal to negative 2.

00:01:54.800 --> 00:01:57.360
The domain is all real numbers,
a can be any real

00:01:57.360 --> 00:02:00.120
number except for negative 2.

00:02:00.120 --> 00:02:03.780
So, the first term is that--
extend the line a little bit--

00:02:03.780 --> 00:02:05.390
and then the second term doesn't
change, because its

00:02:05.390 --> 00:02:08.340
denominator is already the
common denominator.

00:02:08.340 --> 00:02:12.230
Minus a minus 3 over-- and we
could write it either as a

00:02:12.230 --> 00:02:15.660
plus 2 times a plus 2, or
as this thing over here.

00:02:15.660 --> 00:02:17.040
Let's write it in the factored
form, because it'll make it

00:02:17.040 --> 00:02:21.820
easier to simplify later on:
a plus 2 times a plus 2.

00:02:21.820 --> 00:02:25.400
And now, before we-- let's set
this up like this-- now,

00:02:25.400 --> 00:02:28.890
before we add the numerators,
it'll probably be a good idea

00:02:28.890 --> 00:02:31.410
to multiply this out right
there, but let me write the

00:02:31.410 --> 00:02:35.050
denominator, we know what
that is: it is a plus 2

00:02:35.050 --> 00:02:37.240
times a plus 2.

00:02:37.240 --> 00:02:40.890
Now this numerator: if we have
a minus 2 times a plus 2,

00:02:40.890 --> 00:02:42.000
we've seen that pattern
before.

00:02:42.000 --> 00:02:45.540
We can multiply it out if you
like, but we've seen it enough

00:02:45.540 --> 00:02:47.960
hopefully to recognize that this
is going to be a squared

00:02:47.960 --> 00:02:49.370
minus 2 squared.

00:02:49.370 --> 00:02:52.880
This is going to be
a squared minus 4.

00:02:52.880 --> 00:02:55.250
You can multiply it out, and the
middle terms cancel out--

00:02:55.250 --> 00:02:57.850
the negative 2 times a cancels
out the a times 2, and you're

00:02:57.850 --> 00:03:02.230
just left with a squared minus
4-- that's that over there.

00:03:02.230 --> 00:03:07.330
And then you have this: you have
minus a minus 3, so let's

00:03:07.330 --> 00:03:11.440
be very careful here-- you're
subtracting a minus 3, so you

00:03:11.440 --> 00:03:13.630
want to distribute the negative
sign, or multiply

00:03:13.630 --> 00:03:15.860
both of these terms
times negative 1.

00:03:15.860 --> 00:03:20.760
So you could put a minus a here,
and then negative 3 is

00:03:20.760 --> 00:03:24.530
plus 3, so what does
this simplify to?

00:03:24.530 --> 00:03:33.050
You have a squared minus a
plus-- let's see, negative 4

00:03:33.050 --> 00:03:42.220
plus 3 is negative 1, all
of that over a plus 2

00:03:42.220 --> 00:03:44.080
times a plus 2.

00:03:44.080 --> 00:03:46.285
We could write that as
a plus 2 squared.

00:03:49.800 --> 00:03:53.230
Now, we might want to factor
this numerator out more, to

00:03:53.230 --> 00:03:55.760
just make sure it doesn't
contain a common factor with

00:03:55.760 --> 00:03:56.450
the denominator.

00:03:56.450 --> 00:03:59.200
The denominator is just 2a
plus 2 is multiplied by

00:03:59.200 --> 00:04:00.040
themselves.

00:04:00.040 --> 00:04:02.230
And you can see from inspection
a plus 2 will not

00:04:02.230 --> 00:04:05.600
be a factor in this top
expression-- if it was, this

00:04:05.600 --> 00:04:08.490
number right here would be
divisible by 2, it's not

00:04:08.490 --> 00:04:09.260
divisible by 2.

00:04:09.260 --> 00:04:12.780
So, a plus 2 is not one of the
factors here, so there's not

00:04:12.780 --> 00:04:16.910
going to be any more
simplification, even if we

00:04:16.910 --> 00:04:20.019
were able to factor this thing,
and the numerator out.

00:04:20.019 --> 00:04:20.250
So

00:04:20.250 --> 00:04:21.010
we're done.

00:04:21.010 --> 00:04:23.720
We have simplified the rational
expression, and the

00:04:23.720 --> 00:04:28.960
domain is for all a's, except
for a cannot, or, all a's

00:04:28.960 --> 00:04:31.950
given that a does not equal
negative 2-- all a's except

00:04:31.950 --> 00:04:33.330
for negative 2.

00:04:33.330 --> 00:04:36.210
And we are done.


Adding and Subtracting Rational Expressions 1

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.420
Find the perimeter of a
rectangle with the length and

00:00:03.420 --> 00:00:04.760
width given below.

00:00:04.760 --> 00:00:07.260
Express your answer as
a simplified rational

00:00:07.260 --> 00:00:10.550
expression, and state
the domain.

00:00:10.550 --> 00:00:12.960
All right, so they give us the
length-- the length is this

00:00:12.960 --> 00:00:15.690
rational expression, and the
width is that right there.

00:00:15.690 --> 00:00:17.820
If we just draw ourselves
a rectangle-- let's draw

00:00:17.820 --> 00:00:20.470
ourselves a rectangle up
here, so we don't use

00:00:20.470 --> 00:00:22.060
valuable real estate.

00:00:22.060 --> 00:00:26.120
If that's the width, and that
is the length, the perimeter

00:00:26.120 --> 00:00:28.720
of this rectangle is going to
be the width plus the length

00:00:28.720 --> 00:00:31.235
times 2, because you have two
widths right here, and you

00:00:31.235 --> 00:00:31.960
have two lengths.

00:00:31.960 --> 00:00:36.970
So you could view that as w plus
w plus l plus l, which is

00:00:36.970 --> 00:00:42.710
the same thing as 2w plus 2l,
which is the same thing as 2

00:00:42.710 --> 00:00:46.120
times the width plus
the length.

00:00:46.120 --> 00:00:49.690
So, we have to find out what two
times the width plus the

00:00:49.690 --> 00:00:51.540
length is as a rational
expression, so

00:00:51.540 --> 00:00:53.110
let's write that down.

00:00:53.110 --> 00:01:01.710
The perimeter is going to be
equal to 2 times the width

00:01:01.710 --> 00:01:04.069
plus the length, or we could say
the length plus the width,

00:01:04.069 --> 00:01:04.890
either way.

00:01:04.890 --> 00:01:11.240
So the width is x minus 3y over
x plus 4, and then the

00:01:11.240 --> 00:01:19.110
length is plus 2x plus
5y over x plus 4.

00:01:19.110 --> 00:01:21.900
Let's add these two rational
expressions: we have the same

00:01:21.900 --> 00:01:24.360
denominator, so we can just
add the numerators.

00:01:24.360 --> 00:01:28.050
So this is going to be equal to
2 times-- the denominator

00:01:28.050 --> 00:01:31.700
is going to be x plus 4, and we
have the same denominator.

00:01:31.700 --> 00:01:33.640
This problem wasn't too hairy,
we didn't have to come up with

00:01:33.640 --> 00:01:35.290
a common denominator.

00:01:35.290 --> 00:01:39.230
We're going to have x minus
3y plus 2x plus 5y in the

00:01:39.230 --> 00:01:41.080
numerator, so let's
just simplify.

00:01:41.080 --> 00:01:45.400
We have our x terms-- you have
an x plus a 2x, so that's

00:01:45.400 --> 00:01:49.570
going to give us a 3x in the
numerator, and then you have a

00:01:49.570 --> 00:01:54.080
negative 3x, and a 5y
in the numerator.

00:01:54.080 --> 00:02:00.050
Negative 3 plus 5 is plus
2y in the numerator.

00:02:00.050 --> 00:02:02.010
And then we can multiply out
this 2-- we can distribute

00:02:02.010 --> 00:02:07.760
this 2, and then we end up with,
we have 2 times 3x is

00:02:07.760 --> 00:02:17.680
6x, 2 times 2y is 4y plus 4y,
all of that over x plus 4.

00:02:17.680 --> 00:02:21.520
And there's really no way to
simplify this anymore-- here,

00:02:21.520 --> 00:02:24.760
we actually had the 2 factored
out, and 3x plus 2y, no common

00:02:24.760 --> 00:02:27.610
factors, it's not going to
cancel with x plus 4.

00:02:27.610 --> 00:02:31.170
And when they say state the
domain, we just have to think

00:02:31.170 --> 00:02:35.600
about the x values that aren't
part of the domain, that would

00:02:35.600 --> 00:02:38.890
make either of these
expressions, or this final

00:02:38.890 --> 00:02:40.470
expression, undefined.

00:02:40.470 --> 00:02:42.720
Actually, all they care about
is the final expression.

00:02:42.720 --> 00:02:45.070
And the thing that would make
the final expression undefined

00:02:45.070 --> 00:02:47.350
is if this denominator is
equal to 0, and the

00:02:47.350 --> 00:02:50.540
denominator is going to be equal
to zero when x is equal

00:02:50.540 --> 00:02:52.150
to negative 4.

00:02:52.150 --> 00:02:55.340
So we can just constrain the
domain and say x cannot be

00:02:55.340 --> 00:02:57.600
equal to negative 4, that we are
not defining, that this is

00:02:57.600 --> 00:02:59.600
undefined at x equals
negative 4.

00:02:59.600 --> 00:03:02.950
So the simplified, rational
expression for perimeter: 6x

00:03:02.950 --> 00:03:04.730
plus 4y over x plus 4.

00:03:04.730 --> 00:03:07.070
x cannot be negative 4.


Adding and subtracting rational expressions 2

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:01.430
Find the sum.

00:00:01.430 --> 00:00:03.710
Express the answer as a
simplified rational

00:00:03.710 --> 00:00:05.250
expression, and state
the domains.

00:00:05.250 --> 00:00:07.865
We have these two rational
expressions, or two fractions,

00:00:07.865 --> 00:00:08.370
if you will.

00:00:08.370 --> 00:00:10.790
And whenever we add fractions,
we need to find a common

00:00:10.790 --> 00:00:13.040
denominator, and the common
denominator has to be

00:00:13.040 --> 00:00:15.740
something that's divisible by
both of these denominators.

00:00:15.740 --> 00:00:18.360
In general, we want to find
the least common, or the

00:00:18.360 --> 00:00:21.360
smallest, multiple of these
numbers, or the smallest

00:00:21.360 --> 00:00:23.180
number that's divisible
by both.

00:00:23.180 --> 00:00:25.160
When you look at it immediately,
it might pop out

00:00:25.160 --> 00:00:29.930
at you that 6 is divisible by
3, and x to the fourth is

00:00:29.930 --> 00:00:33.110
definitely divisible by x
squared, so 6x to the fourth

00:00:33.110 --> 00:00:35.260
is definitely divisible
by 3x squared.

00:00:35.260 --> 00:00:38.190
Of course, it's divisible by
itself, so this actually is

00:00:38.190 --> 00:00:42.630
the least common multiple-- this
is the smallest number,

00:00:42.630 --> 00:00:45.410
or the smallest expression, I
guess, that is divisible by

00:00:45.410 --> 00:00:49.180
both 6x to the fourth and 3x
squared, so let's make that

00:00:49.180 --> 00:00:50.530
the common denominator.

00:00:50.530 --> 00:00:56.080
So this sum is going to be
equal to 5 over 6x fourth

00:00:56.080 --> 00:01:01.320
plus-- and now, what we want
to do is write this with 6x

00:01:01.320 --> 00:01:03.500
fourth as the denominator, so
let me just write it again.

00:01:03.500 --> 00:01:08.010
So plus 7 over 3x squared.

00:01:08.010 --> 00:01:10.975
So how do we make a 3x squared
into a 6x to the fourth?

00:01:10.975 --> 00:01:13.880
Well we're going to have to
multiply it times 2 to make

00:01:13.880 --> 00:01:16.270
the 3 into a 6, and then we're
going to have to multiply it

00:01:16.270 --> 00:01:18.350
times another x squared, so
we're going to have to

00:01:18.350 --> 00:01:20.720
multiply it by 2x squared.

00:01:20.720 --> 00:01:23.070
Now we can't just multiply only
the denominator by 2x

00:01:23.070 --> 00:01:25.210
squared-- that'll fundamentally
change the value

00:01:25.210 --> 00:01:26.220
of this expression.

00:01:26.220 --> 00:01:30.220
We can only multiply it by 1,
so let's multiply it by 2x

00:01:30.220 --> 00:01:34.020
squared over 2x squared-- and
we're assuming here that is x

00:01:34.020 --> 00:01:35.335
is not equal to 0.

00:01:35.335 --> 00:01:36.860
x does not equal to 0.

00:01:36.860 --> 00:01:40.690
And that was actually a safe
assumption to make, that 0 is

00:01:40.690 --> 00:01:43.530
not a member of our domain right
from the get go, because

00:01:43.530 --> 00:01:46.230
that would've made either of
these expressions undefined.

00:01:46.230 --> 00:01:50.180
If we assume x is not equal to
0, we can multiply by 2x

00:01:50.180 --> 00:01:54.230
squared over 2x squared, and
then that will give us the

00:01:54.230 --> 00:01:59.410
expression 5 over 6x to the
fourth, plus-- this become 7

00:01:59.410 --> 00:02:01.470
times 2 is 14.

00:02:01.470 --> 00:02:07.920
Fourteen x squared over 3 times
2 is 6, x to the squared

00:02:07.920 --> 00:02:10.690
times x squared is x to the
fourth, so now we have a

00:02:10.690 --> 00:02:11.640
common denominator.

00:02:11.640 --> 00:02:14.890
The common denominator is 6x to
the fourth, and we can just

00:02:14.890 --> 00:02:18.490
add the numerators, so it's 5
plus 14x squared-- or, I like

00:02:18.490 --> 00:02:22.260
to write the higher degree
term first-- or 14x

00:02:22.260 --> 00:02:24.580
squared plus 5.

00:02:24.580 --> 00:02:29.670
And we are assuming that x does
not equal 0, because this

00:02:29.670 --> 00:02:32.840
would make the expression
undefined.

00:02:32.840 --> 00:02:35.880
That's about as simple as we can
make it-- we can't divide.

00:02:35.880 --> 00:02:39.450
14 is divisible by 2, and so
6, but 5 isn't, so we can't

00:02:39.450 --> 00:02:44.440
divide everything by 2, and
then, there's x squared x to

00:02:44.440 --> 00:02:47.110
the fourth, but 5 has no x term
on it, so we can't divide

00:02:47.110 --> 00:02:49.290
everything by x or
a power of x.

00:02:49.290 --> 00:02:52.540
We're done: it's 14x squared
plus 5 over 6x to the fourth,

00:02:52.540 --> 00:02:55.640
and x cannot be equal to zero.


Multiplying and dividing rational expressions 3

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.510
Divide and express as a
simplified rational.

00:00:03.510 --> 00:00:05.090
State the domain.

00:00:05.090 --> 00:00:06.580
We start off with
this expression.

00:00:06.580 --> 00:00:09.760
We actually have one rational
expression divided by another

00:00:09.760 --> 00:00:10.960
rational expression.

00:00:10.960 --> 00:00:13.850
And like we've seen multiple
times before, these rational

00:00:13.850 --> 00:00:16.640
expressions aren't defined when
their denominators are

00:00:16.640 --> 00:00:17.920
equal to 0.

00:00:17.920 --> 00:00:23.150
So p plus 5 cannot be equal to
0, or if we subtract both

00:00:23.150 --> 00:00:26.070
sides of this-- we can't call
it an equation, but we could

00:00:26.070 --> 00:00:29.070
call it a not-equation-- by
negative 5, if we subtract

00:00:29.070 --> 00:00:34.400
negative 5 from both sides, you
get p cannot be equal to--

00:00:34.400 --> 00:00:36.380
these cancel out-- negative 5.

00:00:36.380 --> 00:00:38.010
That's what that tells us.

00:00:38.010 --> 00:00:41.130
Over here, we could do the same
exercise 4p plus 20

00:00:41.130 --> 00:00:43.060
also cannot be equal to 0.

00:00:43.060 --> 00:00:46.390
If it was, this expression
would be undefined.

00:00:46.390 --> 00:00:48.300
Subtract 20 from both sides.

00:00:48.300 --> 00:00:50.700
4p cannot be equal
to negative 20.

00:00:50.700 --> 00:00:52.330
Divide both sides by 4.

00:00:52.330 --> 00:00:54.410
p cannot be equal
to negative 5.

00:00:54.410 --> 00:00:57.860
So in both situations, p being
equal to negative 5 would make

00:00:57.860 --> 00:01:02.840
either of these rational
expressions undefined.

00:01:02.840 --> 00:01:16.700
So the domain here is the set
of all reals such-- or p is

00:01:16.700 --> 00:01:22.010
equal to the set of all reals
such that p does not equal

00:01:22.010 --> 00:01:24.870
negative 5, or essentially all
numbers except for negative 5,

00:01:24.870 --> 00:01:25.800
all real numbers.

00:01:25.800 --> 00:01:29.260
We've stated the domain, so now
let's actually simplify

00:01:29.260 --> 00:01:30.870
this expression.

00:01:30.870 --> 00:01:34.160
When you divide by a fraction or
a rational expression, it's

00:01:34.160 --> 00:01:36.320
the same thing as multiplying
by the inverse.

00:01:36.320 --> 00:01:39.360
Let me just rewrite this
thing over here.

00:01:39.360 --> 00:01:47.390
2p plus 6 over p plus 5 divided
by 10 over 4p plus 20

00:01:47.390 --> 00:01:51.050
is the same thing as multiplying
by the reciprocal

00:01:51.050 --> 00:01:56.840
here, multiplying by
4p plus 20 over 10.

00:01:56.840 --> 00:01:59.470
I changed the division into a
multiplication and I flipped

00:01:59.470 --> 00:02:00.870
this guy right here.

00:02:00.870 --> 00:02:05.530
Now, this is going to be equal
to 2p plus 6 times 4p plus 20

00:02:05.530 --> 00:02:07.350
in the numerator.

00:02:07.350 --> 00:02:08.770
I won't skip too many steps.

00:02:08.770 --> 00:02:09.419
Let me just write that.

00:02:09.419 --> 00:02:16.480
2p plus 6 times 4p plus 20 in
the numerator and then p plus

00:02:16.480 --> 00:02:21.070
5 times 10 in the denominator.

00:02:21.070 --> 00:02:22.940
Now, in order to see if we can
simplify this, we need to

00:02:22.940 --> 00:02:25.690
completely factor all of the
terms in the numerator and the

00:02:25.690 --> 00:02:26.850
denominator.

00:02:26.850 --> 00:02:30.590
In the numerator, 2p plus 6, we
can factor out a 2, so the

00:02:30.590 --> 00:02:38.750
2p plus 6 we can rewrite
it as 2 times p plus 3.

00:02:38.750 --> 00:02:43.240
Then the 4p plus 20, we
can rewrite that.

00:02:43.240 --> 00:02:50.870
We can factor out a 4 as--
so 4 times p plus 5.

00:02:50.870 --> 00:02:54.030
Then we have our p plus 5 down
there in the denominator.

00:02:54.030 --> 00:02:55.630
We have this p plus 5.

00:02:55.630 --> 00:02:57.280
We can just write it down
in the denominator.

00:03:00.230 --> 00:03:03.960
Even 10, we can factor that
further into its prime

00:03:03.960 --> 00:03:06.540
components or into its
prime factorization.

00:03:06.540 --> 00:03:12.710
We can factor 10
into 2 times 5.

00:03:12.710 --> 00:03:14.610
That's the same thing as 10.

00:03:14.610 --> 00:03:15.690
Let's see what we
can simplify.

00:03:15.690 --> 00:03:18.690
Of course, this whole time, we
have to add the caveat that p

00:03:18.690 --> 00:03:20.610
cannot equal negative 5.

00:03:20.610 --> 00:03:22.540
We have to add this restriction
on the domain in

00:03:22.540 --> 00:03:25.240
order for it to be the same
expression as the one we

00:03:25.240 --> 00:03:26.730
started off with.

00:03:26.730 --> 00:03:27.680
Now, what can we cancel out?

00:03:27.680 --> 00:03:29.240
We have a 2 divided by a 2.

00:03:29.240 --> 00:03:30.330
Those cancel out.

00:03:30.330 --> 00:03:32.550
We have a p plus 5 divided
by a p plus 5.

00:03:32.550 --> 00:03:34.800
We know that p plus 5 isn't
going to be equal to 0 because

00:03:34.800 --> 00:03:37.910
of this constraint, so we
can cancel those out.

00:03:37.910 --> 00:03:39.050
What are we left with?

00:03:39.050 --> 00:03:46.980
In the numerator, we have 4
times p plus 3, and in the

00:03:46.980 --> 00:03:52.610
denominator, all we have is that
green 5, and we're done!

00:03:52.610 --> 00:03:55.550
We could right this as 4/5 times
p plus 3, or just the

00:03:55.550 --> 00:03:56.720
way we did it right there.

00:03:56.720 --> 00:03:59.060
But we don't want to forget
that we have to add the

00:03:59.060 --> 00:04:03.260
constraint p cannot be equal
to negative 5, so that this

00:04:03.260 --> 00:04:06.980
thing is mathematically
equivalent to

00:04:06.980 --> 00:04:09.640
this thing right here.


Multiplying and dividing rational expressions 2

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.100
Multiply and express as
a simplified rational.

00:00:04.100 --> 00:00:05.710
State the domain.

00:00:05.710 --> 00:00:08.340
Let's multiply it, and then
before we simplify it, let's

00:00:08.340 --> 00:00:09.370
look at the domain.

00:00:09.370 --> 00:00:13.880
This is equal to, if we just
multiplied the numerators, a

00:00:13.880 --> 00:00:22.210
squared minus 4 times a plus 1,
all of that over-- multiply

00:00:22.210 --> 00:00:28.010
the denominators-- a squared
minus 1 times a plus 2.

00:00:31.560 --> 00:00:35.270
Now, the a squared minus 4 and
the a squared minus 1 might

00:00:35.270 --> 00:00:36.650
look familiar to us.

00:00:36.650 --> 00:00:39.880
These are the difference in
squares, a special type of

00:00:39.880 --> 00:00:42.040
binomial that you could
immediately, or hopefully

00:00:42.040 --> 00:00:43.005
maybe immediately recognize.

00:00:43.005 --> 00:00:45.950
It takes the form a squared
minus b squared, difference of

00:00:45.950 --> 00:00:49.570
squares, and it's always going
to be equal to a plus b

00:00:49.570 --> 00:00:51.530
times a minus b.

00:00:51.530 --> 00:00:54.590
We can factor this a squared
minus 4 and we can also factor

00:00:54.590 --> 00:00:57.010
this a squared minus 1, and
that'll help us actually

00:00:57.010 --> 00:01:01.370
simplify the expression or
simplify the rational.

00:01:01.370 --> 00:01:10.400
This top part, we can factor
the a squared minus 4 as a

00:01:10.400 --> 00:01:16.300
plus 2-- 2 squared is 4-- times
a minus 2, and then that

00:01:16.300 --> 00:01:17.550
times a plus 1.

00:01:20.680 --> 00:01:26.650
Then in the denominator, we can
factor a squared minus 1--

00:01:26.650 --> 00:01:28.540
let me do that in a
different color.

00:01:28.540 --> 00:01:33.290
a squared minus 1 we can
factor as a plus 1

00:01:33.290 --> 00:01:35.110
times a minus 1.

00:01:35.110 --> 00:01:36.620
If you ever want to say, hey,
why does this work?

00:01:36.620 --> 00:01:38.640
Just multiply it out and you'll
see that when you

00:01:38.640 --> 00:01:42.120
multiply these two things, you
get that thing right there.

00:01:42.120 --> 00:01:44.020
Then in the denominator, we
also have an a plus 2.

00:01:48.270 --> 00:01:50.400
We've multiplied it, we've
factored out the numerator, we

00:01:50.400 --> 00:01:51.980
factored out the denominator.

00:01:51.980 --> 00:01:53.640
Let's rearrange it
a little bit.

00:01:53.640 --> 00:01:56.690
So this numerator, let's put the
a plus 2's first in both

00:01:56.690 --> 00:01:57.820
the numerator and
the denominator.

00:01:57.820 --> 00:02:02.550
So this, we could get a plus 2
in the numerator, and then in

00:02:02.550 --> 00:02:06.460
the denominator, we also
have an a plus 2.

00:02:06.460 --> 00:02:11.490
In the numerator, we took
care of our a plus 2's.

00:02:11.490 --> 00:02:13.270
That's the only one that's
common, so in the numerator,

00:02:13.270 --> 00:02:15.200
we also have an a minus 2.

00:02:15.200 --> 00:02:16.690
Actually, we have an
a plus 1-- let's

00:02:16.690 --> 00:02:18.970
write that there, too.

00:02:18.970 --> 00:02:20.240
We have an a plus 1
in the numerator.

00:02:20.240 --> 00:02:21.510
We also have an a plus
1 in denominator.

00:02:24.310 --> 00:02:27.770
In the numerator, we have
an a minus 2, and in the

00:02:27.770 --> 00:02:30.680
denominator, we have
an a minus 1.

00:02:30.680 --> 00:02:32.550
So all I did is I rearranged
the numerator and the

00:02:32.550 --> 00:02:34.740
denominator, so if there was
something that was of a

00:02:34.740 --> 00:02:37.270
similar-- if the same expression
was in both, I just

00:02:37.270 --> 00:02:39.420
wrote them on top of each
other, essentially.

00:02:39.420 --> 00:02:42.430
Now, before we simplify, this is
a good time to think about

00:02:42.430 --> 00:02:46.120
the domain or think about the
a values that aren't in the

00:02:46.120 --> 00:02:49.200
domain, the a values that would
invalidate or make this

00:02:49.200 --> 00:02:51.550
expression undefined.

00:02:51.550 --> 00:02:53.880
Like we've seen before, the a
values that would do that are

00:02:53.880 --> 00:02:57.570
the ones that would make the
denominator equal 0.

00:02:57.570 --> 00:03:00.320
So the a values that would make
that equal to 0 is a is

00:03:00.320 --> 00:03:01.960
equal to negative 2.

00:03:01.960 --> 00:03:02.920
You could solve for i.

00:03:02.920 --> 00:03:06.040
You could say a plus 2 is equal
to 0, or a is equal to

00:03:06.040 --> 00:03:07.400
negative 2.

00:03:07.400 --> 00:03:09.040
a plus 1 is equal to 0.

00:03:09.040 --> 00:03:12.300
Subtract 1 from both sides.
a is equal to negative 1.

00:03:12.300 --> 00:03:14.920
Or a minus 1 is equal to 0.

00:03:14.920 --> 00:03:19.060
Add one to both sides, and
you get a is equal to 1.

00:03:19.060 --> 00:03:21.080
For this expression right here,
you have to add the

00:03:21.080 --> 00:03:30.520
constraint that a cannot equal
negative 2, negative 1, or 1,

00:03:30.520 --> 00:03:34.030
that a can be any real number
except for these.

00:03:34.030 --> 00:03:35.110
We're essentially stating
our domain.

00:03:35.110 --> 00:03:38.730
We're stating the domain is all
possible a's except for

00:03:38.730 --> 00:03:41.970
these things right here, so we'd
have to add that little

00:03:41.970 --> 00:03:44.010
caveat right there.

00:03:44.010 --> 00:03:47.000
Now that we've done that,
we can factor it.

00:03:47.000 --> 00:03:49.715
We have an a plus 2
over an a plus 2.

00:03:49.715 --> 00:03:51.700
We know that a is not going to
be equal to negative 2, so

00:03:51.700 --> 00:03:54.000
that's always going
to be defined.

00:03:54.000 --> 00:03:56.220
When you divide something
by itself, that is

00:03:56.220 --> 00:03:57.250
going to just be 1.

00:03:57.250 --> 00:03:59.900
The same thing with the a plus
1 over the a plus 1.

00:03:59.900 --> 00:04:01.330
That's going to be 1.

00:04:01.330 --> 00:04:03.790
All you're going to be left
with is an a minus 2

00:04:03.790 --> 00:04:05.590
over a minus 1.

00:04:05.590 --> 00:04:11.200
So the simplified rational is a
minus 2 over a minus 1 with

00:04:11.200 --> 00:04:15.790
the constraint that a cannot
equal negative 2,

00:04:15.790 --> 00:04:18.079
negative 1, or 1.

00:04:18.079 --> 00:04:20.610
You're probably saying, Sal,
what's wrong with it equaling,

00:04:20.610 --> 00:04:21.970
for example, negative 1 here?

00:04:21.970 --> 00:04:25.145
Negative 1 minus 1, it's only
going to be a negative 2 here.

00:04:25.145 --> 00:04:26.460
It's going to be defined.

00:04:26.460 --> 00:04:30.690
But in order for this expression
to really be the

00:04:30.690 --> 00:04:35.400
same as this expression up here,
it has to have the same

00:04:35.400 --> 00:04:35.950
constraints.

00:04:35.950 --> 00:04:37.530
It has to have the
same domain.

00:04:37.530 --> 00:04:42.450
It cannot be defined at negative
1 if this guy also is

00:04:42.450 --> 00:04:43.930
not defined at negative 1.

00:04:43.930 --> 00:04:47.070
And so these constraints
essentially ensure that we're

00:04:47.070 --> 00:04:49.070
dealing with the same
expression, not one that's

00:04:49.070 --> 00:04:50.740
just close.


Multiplying and dividing rational expressions 1

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.850
Multiply and express as
a simplified rational.

00:00:03.850 --> 00:00:05.270
State the domain.

00:00:05.270 --> 00:00:06.500
We'll start with the domain.

00:00:06.500 --> 00:00:11.240
The only numbers that will make
this expression undefined

00:00:11.240 --> 00:00:13.560
are the ones that would make
the denominator equal to 0,

00:00:13.560 --> 00:00:16.500
and those are the situation, or
that situation would occur,

00:00:16.500 --> 00:00:19.750
when either a, b, x,
or y is equal to 0.

00:00:19.750 --> 00:00:22.750
If any of those are equal to 0,
then we have an undefined

00:00:22.750 --> 00:00:23.930
expression.

00:00:23.930 --> 00:00:34.200
We could say the domain is
all real a's, b's, x,

00:00:34.200 --> 00:00:39.740
and y's, except 0.

00:00:39.740 --> 00:00:45.080
Or we could be specific: except
a, b, x, and y can't

00:00:45.080 --> 00:00:45.840
equal to 0.

00:00:45.840 --> 00:00:55.410
Or you could write this: given
that a, b, x, and y does not

00:00:55.410 --> 00:00:57.210
equal 0, that none of them
can be equal to 0.

00:00:57.210 --> 00:01:00.310
These are just multiple ways
of stating the same thing.

00:01:00.310 --> 00:01:03.550
With that stated, let's actually
multiply and simplify

00:01:03.550 --> 00:01:06.660
this rational expression.

00:01:06.660 --> 00:01:08.610
So when we multiply, you just
multiply the numerator and

00:01:08.610 --> 00:01:14.190
multiply the denominator, so
you have 3x squared y times

00:01:14.190 --> 00:01:20.380
14a squared b in
the numerator.

00:01:20.380 --> 00:01:31.550
Then in the denominator, we have
2ab times 18xy squared.

00:01:31.550 --> 00:01:34.780
Let's see where we can
simplify this thing.

00:01:34.780 --> 00:01:39.220
We can divide the 14 by 2, and
the 2 by 2, and we get 14

00:01:39.220 --> 00:01:43.280
divided by 2 is 7, and
2 divided by 2 is 1.

00:01:43.280 --> 00:01:46.880
We could divide the 3 by 3 and
get 1, and divide the 18

00:01:46.880 --> 00:01:49.180
by 3, and get 6.

00:01:49.180 --> 00:01:50.890
Every time we divided the
numerator and the denominator

00:01:50.890 --> 00:01:54.640
by 2, now the numerator and the
denominator by 3, so we're

00:01:54.640 --> 00:01:56.650
not changing the expression.

00:01:56.650 --> 00:02:02.960
Then we can divide a squared
divided by a, so you're just

00:02:02.960 --> 00:02:04.940
left with an a in the numerator,
and a divided

00:02:04.940 --> 00:02:06.620
by a is just 1.

00:02:06.620 --> 00:02:07.810
You have a b over a b.

00:02:07.810 --> 00:02:10.060
Those guys cancel
each other out.

00:02:10.060 --> 00:02:14.020
You have an x squared divided by
an x, so x squared divided

00:02:14.020 --> 00:02:17.920
by x is x, and x divided by x
is just a 1, so this becomes

00:02:17.920 --> 00:02:20.100
an x over 1, or just an x.

00:02:20.100 --> 00:02:22.040
Finally, you have a y
over a y squared.

00:02:22.040 --> 00:02:22.220
if.

00:02:22.220 --> 00:02:23.945
You divide the numerator
by y, you get 1.

00:02:23.945 --> 00:02:27.990
If you divide the denominator by
y, you just get a y, and so

00:02:27.990 --> 00:02:29.430
what are we left with?

00:02:29.430 --> 00:02:32.410
We're left with in
the numerator,

00:02:32.410 --> 00:02:33.645
these 1's we can ignore.

00:02:33.645 --> 00:02:34.700
That doesn't really
change the number.

00:02:34.700 --> 00:02:40.200
We have a 7 times a times x.

00:02:40.200 --> 00:02:41.800
That's what we have
in the numerator.

00:02:41.800 --> 00:02:45.600
In the denominator,
we just have a 6y.

00:02:45.600 --> 00:02:53.350
And we have to add the
constraint that a, b, x, and y

00:02:53.350 --> 00:02:55.290
cannot equal 0.

00:02:55.290 --> 00:02:57.380
When you just look at this
expression right here, you're

00:02:57.380 --> 00:02:59.690
like, hey, what's
wrong with x?

00:02:59.690 --> 00:03:01.810
There isn't even any b here,
so it's a little bit of a

00:03:01.810 --> 00:03:07.340
bizarre statement, but you say,
hey, why can't x or a be

00:03:07.340 --> 00:03:09.200
equal to 0 over here?

00:03:09.200 --> 00:03:10.450
It doesn't make it undefined.

00:03:10.450 --> 00:03:12.790
But in order for these
to really be the same

00:03:12.790 --> 00:03:16.570
expressions, they have to
have the same domains.

00:03:16.570 --> 00:03:18.750
Or actually, if these were
functions that equal them, in

00:03:18.750 --> 00:03:22.875
order for that f of x to be
equal to this f of x right

00:03:22.875 --> 00:03:25.320
over here, you'd have to
constrain the domain in a

00:03:25.320 --> 00:03:27.040
similar way.

00:03:27.040 --> 00:03:29.220
This is a fundamentally
different expression if you

00:03:29.220 --> 00:03:31.280
allow x's and a's.

00:03:31.280 --> 00:03:33.070
In this one, you can't
allow x's and a's.

00:03:33.070 --> 00:03:35.510
For them to be really the same,
you have to put the same

00:03:35.510 --> 00:03:37.500
constraints on it.


Simplifying rational expressions 3

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.770
Simplify the rational
expression

00:00:02.770 --> 00:00:04.680
and state the domain.

00:00:04.680 --> 00:00:06.790
Let's see if we can start with
the domain part of the

00:00:06.790 --> 00:00:09.410
question, if we can start
with stating the domain.

00:00:09.410 --> 00:00:13.030
Now, the domain is the set of
all of the x values that you

00:00:13.030 --> 00:00:15.690
can legitimately input into
this if you view this as a

00:00:15.690 --> 00:00:19.210
function, if you said this is
f of x is equal to that.

00:00:19.210 --> 00:00:22.150
The domain is a set of all x
values that you could input

00:00:22.150 --> 00:00:24.210
into this function and
get something that is

00:00:24.210 --> 00:00:26.330
well-defined.

00:00:26.330 --> 00:00:29.300
The one x value that would make
this undefined is the x

00:00:29.300 --> 00:00:32.210
value that would make the
denominator equal 0-- the x

00:00:32.210 --> 00:00:34.690
value that would make
that equal 0.

00:00:34.690 --> 00:00:35.710
So when does that happen?

00:00:35.710 --> 00:00:37.850
Six minus x is equal to 0.

00:00:37.850 --> 00:00:40.860
Let's add x to both sides.

00:00:40.860 --> 00:00:48.950
We get 6 is equal to x, so the
domain of this function is

00:00:48.950 --> 00:01:02.200
equal to the set of all
real numbers except 6.

00:01:02.200 --> 00:01:05.570
So, x could be all real numbers
except 6, because if x

00:01:05.570 --> 00:01:09.220
is 6 then you're dividing
by 0, and then this

00:01:09.220 --> 00:01:11.290
expression is undefined.

00:01:11.290 --> 00:01:14.580
We've stated the domain, now
let's do the simplifying the

00:01:14.580 --> 00:01:15.510
rational expression.

00:01:15.510 --> 00:01:17.000
Let me rewrite it over here.

00:01:17.000 --> 00:01:23.260
We have x squared minus
36 over 6 minus x.

00:01:23.260 --> 00:01:26.120
Now, this might jump out at you
immediately, as it's that

00:01:26.120 --> 00:01:27.650
special type of binomial.

00:01:27.650 --> 00:01:30.830
It's of the form a squared minus
b squared, and we've

00:01:30.830 --> 00:01:32.310
seen this multiple times.

00:01:32.310 --> 00:01:37.160
This is equivalent to a plus
b times a minus b.

00:01:37.160 --> 00:01:41.390
And in this case, a
is x and b is 6.

00:01:41.390 --> 00:01:46.400
This top expression right here
can be factored as x plus 6

00:01:46.400 --> 00:01:52.120
times x minus 6, all of
that over 6 minus x.

00:01:52.120 --> 00:01:55.960
Now, at first you might say,
I have a x minus 6

00:01:55.960 --> 00:01:57.390
and a 6 minus x.

00:01:57.390 --> 00:02:01.660
Those aren't quite equal, but
what maybe will jump out at

00:02:01.660 --> 00:02:03.750
you is that these are the
negatives of each other.

00:02:03.750 --> 00:02:05.710
Try it out.

00:02:05.710 --> 00:02:11.130
Let's multiply by negative 1 and
then by negative 1 again.

00:02:11.130 --> 00:02:12.000
Think of it that way.

00:02:12.000 --> 00:02:17.110
If I multiply by negative 1
times negative 1, obviously,

00:02:17.110 --> 00:02:19.890
I'm just multiplying the
numerator by 1, so I'm not in

00:02:19.890 --> 00:02:21.780
any way changing
the numerator.

00:02:21.780 --> 00:02:25.390
What happens if we just multiply
the x minus 6 by that

00:02:25.390 --> 00:02:26.490
first negative 1?

00:02:26.490 --> 00:02:28.190
What happens to that
x minus 6?

00:02:28.190 --> 00:02:30.850
Let me rewrite the
whole expression.

00:02:30.850 --> 00:02:35.300
We have x plus 6, and I'm
going to distribute this

00:02:35.300 --> 00:02:36.740
negative 1.

00:02:36.740 --> 00:02:39.400
If I distribute the negative 1,
I have negative 1 times x

00:02:39.400 --> 00:02:41.050
is negative x.

00:02:41.050 --> 00:02:44.020
Negative 1 times negative
6 is plus 6.

00:02:44.020 --> 00:02:45.780
And then I have a negative
1 out here.

00:02:45.780 --> 00:02:50.540
I have a negative 1 times
negative 1, and all of that is

00:02:50.540 --> 00:02:52.960
over 6 minus x.

00:02:52.960 --> 00:02:54.920
Now, negative plus 6.

00:02:54.920 --> 00:02:58.600
This is the exact same thing
as 6 minus x if you just

00:02:58.600 --> 00:03:01.080
rearrange the two terms.
Negative x plus 6 is the same

00:03:01.080 --> 00:03:04.950
thing as 6 plus negative
x, or 6 minus x.

00:03:04.950 --> 00:03:06.820
Now you could cancel them out.

00:03:06.820 --> 00:03:11.650
6 minus x divided by 6 minus x,
and all you're left with is

00:03:11.650 --> 00:03:18.450
a negative 1-- I'll write it
out front-- times x plus 6.

00:03:18.450 --> 00:03:20.170
If you want, you can distribute
it and you would

00:03:20.170 --> 00:03:22.900
get negative x minus 6.

00:03:22.900 --> 00:03:25.290
That's the simplified
rational expression.

00:03:25.290 --> 00:03:27.300
In general, you don't have to
go through this exercise,

00:03:27.300 --> 00:03:29.120
multiplying by a negative
1 and a negative 1.

00:03:29.120 --> 00:03:32.490
But you should always be able
recognize that if you have a

00:03:32.490 --> 00:03:37.410
minus b over b minus a that that
is equal to negative 1.

00:03:37.410 --> 00:03:41.510
Or think of it this way: a
minus b is equal to the

00:03:41.510 --> 00:03:44.670
negative of b minus a.

00:03:44.670 --> 00:03:46.240
If you distribute this negative
sign, you get

00:03:46.240 --> 00:03:48.890
negative b plus a, which
is exactly what

00:03:48.890 --> 00:03:50.720
this is over here.

00:03:50.720 --> 00:03:52.450
We're all done.


Simplifying rational expressions 2

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.500
Simplify the rational
expression

00:00:02.500 --> 00:00:05.160
and state the domain.

00:00:05.160 --> 00:00:09.290
Once again, we have a trinomial
over a trinomial.

00:00:09.290 --> 00:00:10.930
To see if we can simplify
them, we need to

00:00:10.930 --> 00:00:11.920
factor both of them.

00:00:11.920 --> 00:00:14.430
That's also going to help us
figure out the domain.

00:00:14.430 --> 00:00:17.035
The domain is essentially
figuring out all of the valid

00:00:17.035 --> 00:00:20.270
x's that we can put into this
expression and not get

00:00:20.270 --> 00:00:21.530
something that's undefined.

00:00:21.530 --> 00:00:23.540
Let's factor the numerator
and the denominator.

00:00:23.540 --> 00:00:28.590
So let's start with the
numerator there, and since we

00:00:28.590 --> 00:00:34.200
have a 2 out front, factoring
by grouping will probably be

00:00:34.200 --> 00:00:36.680
the best way to go, so let's
just rewrite it here.

00:00:36.680 --> 00:00:38.190
I'm just working on the
numerator right now.

00:00:38.190 --> 00:00:42.290
2x squared plus 13x plus 20.

00:00:42.290 --> 00:00:46.270
We need to find two numbers,
a and b, that if I multiply

00:00:46.270 --> 00:00:49.955
them, a times b, needs to be
equal to-- let me write it

00:00:49.955 --> 00:00:51.455
over here on the right.

00:00:51.455 --> 00:00:57.320
a times b needs to be equal to
2 times 20, so it has to be

00:00:57.320 --> 00:00:59.220
equal to positive 40.

00:00:59.220 --> 00:01:04.110
And then a plus b has
to be equal to 13.

00:01:04.110 --> 00:01:05.990
The numbers that jump out
at me immediately

00:01:05.990 --> 00:01:08.340
are 5 and 8, right?

00:01:08.340 --> 00:01:09.430
5 times 8 is 40.

00:01:09.430 --> 00:01:11.660
5 plus 8 is 13.

00:01:11.660 --> 00:01:16.570
We can break this 13x into a
5x and an 8x, and so we can

00:01:16.570 --> 00:01:19.790
rewrite this as 2x squared.

00:01:19.790 --> 00:01:23.760
It'll break up the 13x into--
I'm going to write the 8x

00:01:23.760 --> 00:01:27.880
first. I'm going to
write 8x plus 5x.

00:01:27.880 --> 00:01:30.120
The reason why I wrote the 8x
first is because the 8 shares

00:01:30.120 --> 00:01:31.810
common factors with the
2, so maybe we can

00:01:31.810 --> 00:01:33.400
factor out a 2x here.

00:01:33.400 --> 00:01:34.850
It'll simplify it
a little bit.

00:01:34.850 --> 00:01:38.210
5 shares factors with the 20, so
let's see where this goes.

00:01:38.210 --> 00:01:41.450
We finally have a plus
20 here, and now

00:01:41.450 --> 00:01:42.220
we can group them.

00:01:42.220 --> 00:01:44.980
That's the whole point of
factoring by grouping.

00:01:44.980 --> 00:01:48.230
You group these first two
characters right here.

00:01:48.230 --> 00:01:53.320
Let's factor out a 2x, so this
would become 2x times-- well,

00:01:53.320 --> 00:01:56.400
2x squared divided by 2x
is just going to be x.

00:01:56.400 --> 00:02:00.235
8x divided by 2x is going
to be plus 4.

00:02:04.250 --> 00:02:06.840
Let's group these
two characters.

00:02:06.840 --> 00:02:08.919
And if we factor out a
5, what do we get?

00:02:08.919 --> 00:02:15.050
We get plus 5 times x plus 4.

00:02:15.050 --> 00:02:21.950
5x divided by 5 is x, 20
divided by 5 is 4.

00:02:21.950 --> 00:02:25.900
We have an x plus 4 in
both cases, so we

00:02:25.900 --> 00:02:27.590
can factor that out.

00:02:27.590 --> 00:02:30.850
We have x plus 4 times two
terms. We can undistribute it.

00:02:30.850 --> 00:02:38.430
This thing over here will be x
plus four times-- let me do it

00:02:38.430 --> 00:02:42.940
in that same color--
2x plus 5.

00:02:45.560 --> 00:02:47.480
And we've factored
this numerator

00:02:47.480 --> 00:02:48.485
expression right there.

00:02:48.485 --> 00:02:50.490
Now, let's do the same thing
with the denominator

00:02:50.490 --> 00:02:50.960
expression.

00:02:50.960 --> 00:02:53.160
I'll do that in a different--
I don't want

00:02:53.160 --> 00:02:54.010
to run out of colors.

00:02:54.010 --> 00:02:58.090
So the denominator is right over
here, let's do the same

00:02:58.090 --> 00:02:59.750
exercise with it.

00:02:59.750 --> 00:03:05.550
We have 2x squared
plus 17x plus 30.

00:03:05.550 --> 00:03:07.480
Let's look for an a and a b.

00:03:07.480 --> 00:03:11.220
When I multiply them, I get
2 times 30, which is 60.

00:03:11.220 --> 00:03:16.280
And an a plus a b, when
I add them, I get 17.

00:03:16.280 --> 00:03:19.230
Once again, 5 and
12 seem to work.

00:03:19.230 --> 00:03:21.040
So let's split this up.

00:03:21.040 --> 00:03:24.540
Let's split this up
into 2x squared.

00:03:24.540 --> 00:03:35.570
We're going to split up the 17x
into a 12x plus a 5x and

00:03:35.570 --> 00:03:36.950
that adds up to 17x.

00:03:36.950 --> 00:03:44.040
When you multiply 12 times 5,
you get 60, and then plus 30.

00:03:44.040 --> 00:03:51.620
Then on this first group right
here, we can factor out a 2x,

00:03:51.620 --> 00:03:57.460
so if you factor out a 2x, you
get 2x times x plus 6.

00:03:57.460 --> 00:04:02.500
In that second group, we can
factor out a 5, so you get

00:04:02.500 --> 00:04:07.750
plus 5 times x plus 6.

00:04:07.750 --> 00:04:12.880
Now, we can factor out an x plus
6, and we get we get x

00:04:12.880 --> 00:04:19.970
plus 6 times 2x plus 5.

00:04:23.600 --> 00:04:25.470
We've now factored the numerator
and the denominator.

00:04:25.470 --> 00:04:27.930
Let's rewrite both of these
expressions or write this

00:04:27.930 --> 00:04:30.370
entire rational expression with
the numerator and the

00:04:30.370 --> 00:04:31.770
denominator factored.

00:04:31.770 --> 00:04:36.210
The numerator is going to
be equal to x plus 4

00:04:36.210 --> 00:04:38.630
times 2x plus 5.

00:04:38.630 --> 00:04:40.220
We figured that out
right there.

00:04:40.220 --> 00:04:52.060
And then the denominator is
x plus 6 times 2x plus 5.

00:04:52.060 --> 00:04:54.205
It might already jump out at you
that you have 2x plus 5 in

00:04:54.205 --> 00:04:55.390
the numerator and the
denominator, and we

00:04:55.390 --> 00:04:56.570
can cancel them out.

00:04:56.570 --> 00:04:57.620
We will cancel them out.

00:04:57.620 --> 00:04:59.840
But before we do that, let's
work on the second part of

00:04:59.840 --> 00:05:00.860
this question.

00:05:00.860 --> 00:05:03.010
State the domain.

00:05:03.010 --> 00:05:06.680
What are the valid x values
that we could put in here?

00:05:06.680 --> 00:05:08.700
A more interesting question is
what are the x values that

00:05:08.700 --> 00:05:12.290
will make this rational
expression undefined?

00:05:12.290 --> 00:05:15.580
It's the x values that will make
the denominator equal to

00:05:15.580 --> 00:05:18.470
0, and when will the denominator
equal to 0?

00:05:18.470 --> 00:05:24.120
Well, either when x plus 6 is
equal to 0, or when 2x plus 5

00:05:24.120 --> 00:05:25.540
is equal to 0.

00:05:25.540 --> 00:05:26.830
We could just solve
for x here.

00:05:26.830 --> 00:05:29.530
Subtract 6 from both sides,
and you get x is equal to

00:05:29.530 --> 00:05:30.670
negative 6.

00:05:30.670 --> 00:05:33.510
If you subtract 5 from both
sides, you get 2x is equal to

00:05:33.510 --> 00:05:34.650
negative 5.

00:05:34.650 --> 00:05:36.337
Divide both sides by 2.

00:05:36.337 --> 00:05:40.520
You get x is equal
to negative 5/2.

00:05:40.520 --> 00:05:44.080
We could say the domain-- let
me write this over here.

00:05:44.080 --> 00:05:58.440
The domain is all real numbers
other than or except x is

00:05:58.440 --> 00:06:04.580
equal to negative 6 and x is
equal to negative 5/2.

00:06:04.580 --> 00:06:06.410
The reason why we have to
exclude those is those would

00:06:06.410 --> 00:06:08.880
make this denominator-- either
way you're right.

00:06:08.880 --> 00:06:11.150
It's going to go make the
denominator equal to 0, and it

00:06:11.150 --> 00:06:14.150
would make the entire rational
expression undefined.

00:06:14.150 --> 00:06:15.400
We've stated the domain.

00:06:15.400 --> 00:06:18.080
Now let's just simplify the
rational expression.

00:06:18.080 --> 00:06:20.490
We've already said that x cannot
be equal to negative

00:06:20.490 --> 00:06:23.310
5/2 or negative 6, so let's just
divide the numerator and

00:06:23.310 --> 00:06:24.990
the denominator by 2x plus 5.

00:06:24.990 --> 00:06:27.580
Or just looking at the 2x plus
5, we know that 2x plus 5

00:06:27.580 --> 00:06:32.630
won't be 0, because x won't be
equal to negative 5/2, and so

00:06:32.630 --> 00:06:34.480
we can cancel those out.

00:06:34.480 --> 00:06:38.380
The simplified rational
expression is just x plus 4

00:06:38.380 --> 00:06:41.020
over x plus 6.


Simplifying Rational Expressions 1 (old)

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:07.100
Given a rectangle with length a
squared plus 6a minus 27 and

00:00:07.100 --> 00:00:12.030
width a squared minus 9, write
the ratio of the width of the

00:00:12.030 --> 00:00:16.440
rectangle to its length as
a simplified rational

00:00:16.440 --> 00:00:17.470
expression.

00:00:17.470 --> 00:00:20.150
A rational expression is kind
of a fancy word essentially

00:00:20.150 --> 00:00:23.480
for a fraction or as a ratio.

00:00:23.480 --> 00:00:28.540
They want to us to express the
ratio of the length, so we'll

00:00:28.540 --> 00:00:34.390
put the length in the numerator,
to the width.

00:00:34.390 --> 00:00:38.285
In the problem, they tell us
that the length right here is

00:00:38.285 --> 00:00:48.870
a squared plus 6a minus 27 and
then the width they tell us

00:00:48.870 --> 00:00:52.670
right here is a squared
minus 9.

00:00:55.800 --> 00:00:59.280
So we've already expressed it as
a rational expression, but

00:00:59.280 --> 00:01:01.710
this isn't necessarily
a simplified rational

00:01:01.710 --> 00:01:02.680
expression.

00:01:02.680 --> 00:01:06.200
Whenever you have any rational
expression-- let me just go

00:01:06.200 --> 00:01:07.950
back to what you learned
a long time ago.

00:01:07.950 --> 00:01:12.520
If I had something like 2/6,
that's a rational expression,

00:01:12.520 --> 00:01:15.380
or that's also a fraction, but
this isn't in the most

00:01:15.380 --> 00:01:16.480
simplified form.

00:01:16.480 --> 00:01:18.420
What you could do is factor
both of these.

00:01:21.270 --> 00:01:23.310
If you do a prime factorization,
it's just 2,

00:01:23.310 --> 00:01:25.920
but 6 is 2 times 3.

00:01:25.920 --> 00:01:28.280
When you factor them, you see
that you have the same factor

00:01:28.280 --> 00:01:30.140
in the numerator and the
denominator, you can cancel

00:01:30.140 --> 00:01:33.370
them out, and you can
simplify it to 1/3.

00:01:33.370 --> 00:01:36.200
We're going to do the exact
same thing with these

00:01:36.200 --> 00:01:37.590
algebraic expressions.

00:01:37.590 --> 00:01:40.360
We can factor them and then see
if they have any of the

00:01:40.360 --> 00:01:42.720
same terms. If they have the
same terms in the numerator

00:01:42.720 --> 00:01:45.520
and the denominator, then we can
cancel them out, and then

00:01:45.520 --> 00:01:48.980
we'll just be left with
whatever's left over.

00:01:48.980 --> 00:01:51.650
Let's see if we can factor this
numerator right here.

00:01:51.650 --> 00:01:56.170
This is going to be equal to--
let me draw the line there.

00:01:56.170 --> 00:01:58.550
a squared plus 6a minus 27.

00:01:58.550 --> 00:02:02.340
Can we think of two numbers that
if we multiply them, we

00:02:02.340 --> 00:02:04.330
get negative 27?

00:02:04.330 --> 00:02:06.740
They're going to have to be of
different signs, because when

00:02:06.740 --> 00:02:08.350
you multiply them, you get a
negative number, and when you

00:02:08.350 --> 00:02:11.050
add them, you get positive 6.

00:02:11.050 --> 00:02:15.080
Let's see, 9 minus 3 is 6.

00:02:15.080 --> 00:02:22.420
9 times negative 3 is negative
27, so this top expression is

00:02:22.420 --> 00:02:27.790
a plus 9 times a minus 3.

00:02:27.790 --> 00:02:29.280
We factored it there.

00:02:29.280 --> 00:02:31.840
What does this bottom expression
factor into?

00:02:31.840 --> 00:02:35.560
This is that standard form, if
you remember, a squared minus

00:02:35.560 --> 00:02:38.785
b squared, where in this
case, b would be 3.

00:02:38.785 --> 00:02:40.330
b squared would be 9.

00:02:40.330 --> 00:02:45.380
We know that this is equal to
a plus b times a minus b.

00:02:45.380 --> 00:02:47.980
In this situation, we know this
is a special form that

00:02:47.980 --> 00:02:55.930
would be factored into a
plus 3 times a minus 3.

00:02:55.930 --> 00:02:59.360
We see there is a term that is
common to both the numerator

00:02:59.360 --> 00:03:01.740
and the denominator:
a minus 3.

00:03:01.740 --> 00:03:04.300
You have a minus 3 in the
numerator and you have an a

00:03:04.300 --> 00:03:06.850
minus 3 in the denominator.

00:03:06.850 --> 00:03:11.100
So we can divide the numerator
and the denominator by a minus

00:03:11.100 --> 00:03:12.530
3, or that they would
cancel out.

00:03:12.530 --> 00:03:16.630
a minus 3 divided by a minus
3, assuming that a is not

00:03:16.630 --> 00:03:18.850
equal to 3, because then it
would become undefined, but if

00:03:18.850 --> 00:03:22.360
we assume a does not equal to 3,
something divided by itself

00:03:22.360 --> 00:03:24.990
will just be 1.

00:03:24.990 --> 00:03:32.910
Then we'll just be left with
a plus 9 over a plus 3, and

00:03:32.910 --> 00:03:36.310
that's about as simplified as
we can get for this rational

00:03:36.310 --> 00:03:37.560
expression.


Non-linear systems of equations 2

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.510
Solve the system of equations
by substitution.

00:00:03.510 --> 00:00:06.300
Check your solution by graphing
the equation.

00:00:06.300 --> 00:00:07.500
Let's do it by substitution.

00:00:07.500 --> 00:00:10.890
We know that y is equal to this
thing over here, and we

00:00:10.890 --> 00:00:14.270
also know that y is equal
to this thing over here.

00:00:14.270 --> 00:00:17.270
So they're going to intersect
when they both equal the same

00:00:17.270 --> 00:00:20.410
y, or when this thing is equal
to this thing, or when

00:00:20.410 --> 00:00:29.540
negative x squared is equal to
2x squared plus 3x minus 6.

00:00:29.540 --> 00:00:32.915
The x value, which these two are
equal, is going to be the

00:00:32.915 --> 00:00:35.610
x values where these y's are
equal, so that's going to be

00:00:35.610 --> 00:00:37.480
their point of intersection.

00:00:37.480 --> 00:00:39.900
That will be an x and y
pair that satisfies

00:00:39.900 --> 00:00:42.070
both of these equations.

00:00:42.070 --> 00:00:43.600
Let's solve for x here.

00:00:43.600 --> 00:00:46.270
A good starting point-- let's
just add x squared to both

00:00:46.270 --> 00:00:55.990
sides, and we end up with
0 is equal to 3x squared

00:00:55.990 --> 00:01:00.010
plus 3x minus 6.

00:01:00.010 --> 00:01:02.520
Then, just to simplify this on
the right hand side, we could

00:01:02.520 --> 00:01:06.030
divide both sides of the
equation by 3-- we can divide

00:01:06.030 --> 00:01:11.150
everything by 3-- and we're
left with 0 is equal to x

00:01:11.150 --> 00:01:16.500
squared plus x minus 2.

00:01:16.500 --> 00:01:18.810
We could do the quadratic
equation, or complete the

00:01:18.810 --> 00:01:22.840
square, or all sorts of crazy
things, but this is actually

00:01:22.840 --> 00:01:27.460
very factorable-- 0 is equal
to two numbers, which are 2

00:01:27.460 --> 00:01:28.590
and negative 1.

00:01:28.590 --> 00:01:30.270
When you multiply them, you get
negative 2, and when you

00:01:30.270 --> 00:01:32.130
add them, you get positive 1.

00:01:32.130 --> 00:01:37.500
So this is x plus 2
times x minus 1.

00:01:37.500 --> 00:01:42.430
That tells us that x plus 2 is
equal to 0, or x minus 1 could

00:01:42.430 --> 00:01:45.420
be equal to 0.

00:01:45.420 --> 00:01:47.100
Subtract 2 from both
sides of this.

00:01:47.100 --> 00:01:49.210
You could get x is equal
to negative 2.

00:01:49.210 --> 00:01:53.890
Or add 1 to both sides of
this, x is equal to 1.

00:01:53.890 --> 00:01:58.290
These are our two solutions-- x
is equal to negative 2 or x

00:01:58.290 --> 00:02:00.350
is equal to 1.

00:02:00.350 --> 00:02:01.540
Let's verify it.

00:02:01.540 --> 00:02:04.750
When I put x is equal to
negative 2-- let's do it over

00:02:04.750 --> 00:02:07.670
here-- what do I get?

00:02:07.670 --> 00:02:10.020
Negative 2 squared
is positive 4.

00:02:10.020 --> 00:02:13.160
And then you put a negative
there, so it's negative 4--

00:02:13.160 --> 00:02:18.520
the point here is negative
2, and then negative 4.

00:02:18.520 --> 00:02:20.540
That's what happens when you
put negative 2 there.

00:02:20.540 --> 00:02:22.060
And then what happens
when I put 1?

00:02:22.060 --> 00:02:24.370
1 squared is 1, and then if you
put a negative there, it's

00:02:24.370 --> 00:02:25.110
negative 1.

00:02:25.110 --> 00:02:26.630
So it's 1, negative 1.

00:02:26.630 --> 00:02:29.200
And so these are both points on
this equation right here,

00:02:29.200 --> 00:02:30.600
on this function.

00:02:30.600 --> 00:02:32.610
If you look at this one over
here, if you put negative 2

00:02:32.610 --> 00:02:37.080
over here-- 2 times negative
2 squared.

00:02:37.080 --> 00:02:38.660
Negative 2 squared
is positive 4.

00:02:38.660 --> 00:02:41.140
2 times positive 4 is 8.

00:02:41.140 --> 00:02:44.535
3 times negative 2 is negative
6, so 8 minus 6.

00:02:47.470 --> 00:02:52.010
8 minus 6 is 2, and 2 minus
6 is equal to negative 4.

00:02:52.010 --> 00:02:57.910
The point negative 2, negative
4 is on this function.

00:02:57.910 --> 00:02:59.460
They both share that
point in common, so

00:02:59.460 --> 00:03:00.980
they intersect there.

00:03:00.980 --> 00:03:07.400
If I put 1 into this, I
get 2 plus 3 minus 6.

00:03:07.400 --> 00:03:12.280
2 plus 3 is 5, minus 6, when x
is 1, which is negative 1.

00:03:12.280 --> 00:03:16.600
The point 1, negative 1 is
also on this top graph.

00:03:16.600 --> 00:03:19.670
We can plot these points:
negative 2 comma negative 4.

00:03:19.670 --> 00:03:23.745
Negative 2-- 1, 2, 3,
4-- is right there.

00:03:23.745 --> 00:03:26.010
That's going to be a point
of intersection.

00:03:26.010 --> 00:03:27.770
Then we have the point
1, negative 1.

00:03:27.770 --> 00:03:33.330
1, negative 1 is also going
to be a point of

00:03:33.330 --> 00:03:35.160
intersection there.

00:03:35.160 --> 00:03:37.550
Let's graph this second one and
just verify-- they say,

00:03:37.550 --> 00:03:41.530
check your solution by graphing
the equation.

00:03:41.530 --> 00:03:42.530
Let's graph the first one.

00:03:42.530 --> 00:03:44.770
This is pretty easy to
graph-- y is equal

00:03:44.770 --> 00:03:46.430
to negative x squared.

00:03:46.430 --> 00:03:49.140
It's going to intersect
the point 0, 0.

00:03:49.140 --> 00:03:50.910
It's going to be a
downward slope,

00:03:50.910 --> 00:03:52.730
downward opening parabola.

00:03:57.390 --> 00:04:00.390
When x is positive or negative
1, y is going to be negative

00:04:00.390 --> 00:04:03.660
1, because you square them and
then you take the negative.

00:04:03.660 --> 00:04:05.820
When x is positive or negative
2, y is going

00:04:05.820 --> 00:04:08.030
to be negative 4.

00:04:08.030 --> 00:04:09.690
So the graph will look
something like this--

00:04:09.690 --> 00:04:11.720
4, 5, 6, 7, 8, 9.

00:04:11.720 --> 00:04:16.800
The graph will look something
like that.

00:04:16.800 --> 00:04:19.700
That is that equation
right here.

00:04:19.700 --> 00:04:25.100
This is going to be an upward
opening parabola, and a quick

00:04:25.100 --> 00:04:28.920
way-- we could do all of that
completing the square, but if

00:04:28.920 --> 00:04:32.250
you think about it, let's just
apply the quadratic equation

00:04:32.250 --> 00:04:32.820
right here.

00:04:32.820 --> 00:04:35.140
I want to show you something,
a quick way of where the

00:04:35.140 --> 00:04:37.120
vertex formula comes
from when you apply

00:04:37.120 --> 00:04:39.190
the quadratic formula.

00:04:39.190 --> 00:04:41.900
If you take this, and you wanted
to find its zeros, you

00:04:41.900 --> 00:04:46.380
would say x is equal to negative
b-- which is negative

00:04:46.380 --> 00:04:52.250
3-- plus or minus the square
root of b squared-- which is

00:04:52.250 --> 00:04:55.530
9-- minus 4ac.

00:04:55.530 --> 00:04:59.000
Minus 4 times 2 times
negative 6.

00:05:01.510 --> 00:05:07.780
All of that over 2 times
a, or 2 times 2.

00:05:10.470 --> 00:05:13.260
Now, we can evaluate this and
figure out the zeros of this,

00:05:13.260 --> 00:05:15.550
but the real point why I wanted
to show you this is

00:05:15.550 --> 00:05:17.780
because we always have
two solutions.

00:05:17.780 --> 00:05:19.570
Let me actually write
down the quadratic

00:05:19.570 --> 00:05:21.220
formula here for you.

00:05:21.220 --> 00:05:24.110
The quadratic formula is
negative b plus or minus the

00:05:24.110 --> 00:05:30.150
square root of b squared
minus 4ac over 2a.

00:05:30.150 --> 00:05:34.050
You always end up-- or if this
is a positive number-- with

00:05:34.050 --> 00:05:37.360
two solutions, and they're
equal distant.

00:05:37.360 --> 00:05:41.990
They're this far away, this far
over 2a away from negative

00:05:41.990 --> 00:05:45.620
b over 2a-- we could write this
as negative b over 2a

00:05:45.620 --> 00:05:48.380
plus or minus the square
root of b squared

00:05:48.380 --> 00:05:52.210
minus 4ac over 2a.

00:05:52.210 --> 00:05:55.430
You have at most two solutions
that are equal distant from

00:05:55.430 --> 00:05:57.380
this x value right here.

00:05:57.380 --> 00:06:00.520
We've seen in multiple videos:
what is that point that is

00:06:00.520 --> 00:06:02.310
equidistant from the
two solutions,

00:06:02.310 --> 00:06:04.400
that is right in between?

00:06:04.400 --> 00:06:07.170
That's going to be the line of
symmetry, or the x value of

00:06:07.170 --> 00:06:08.300
the vertex.

00:06:08.300 --> 00:06:11.320
This is where the x value of the
vertex formula comes from:

00:06:11.320 --> 00:06:16.720
x val of vertex is negative
b over 2a.

00:06:16.720 --> 00:06:19.500
If we wanted to find the vertex,
the x value of this

00:06:19.500 --> 00:06:21.800
guy right here, we just take
negative b-- which is negative

00:06:21.800 --> 00:06:27.200
3-- over 2 times a, 2
times 2, which is 4.

00:06:27.200 --> 00:06:29.770
So, x is equal to negative
3/4 is the

00:06:29.770 --> 00:06:31.480
vertex of this parabola.

00:06:31.480 --> 00:06:35.130
And when x is equal to negative
3/4, what is y?

00:06:35.130 --> 00:06:37.990
We could actually-- that's a
little bit more complicated

00:06:37.990 --> 00:06:41.090
right here, but I'll
just go through it.

00:06:41.090 --> 00:06:51.310
It's going to be 2 times
9 over 16 minus 9

00:06:51.310 --> 00:06:54.300
over 4 minus 6.

00:06:54.300 --> 00:06:57.540
Let me actually get the
calculator out for this one.

00:06:57.540 --> 00:06:59.850
It's going to be 18 over 60.

00:06:59.850 --> 00:07:01.240
Let me just get the calculator
out, because it will be

00:07:01.240 --> 00:07:03.130
simpler, and I don't want
to waste your time doing

00:07:03.130 --> 00:07:04.380
arithmetic.

00:07:06.020 --> 00:07:23.530
It's going to be 18 divided by
16 minus 9 divided by 4 minus

00:07:23.530 --> 00:07:29.300
6 is equal to negative 7.125.

00:07:29.300 --> 00:07:33.110
This is equal to
negative 7.125.

00:07:33.110 --> 00:07:36.220
So the vertex occurs when x is
negative 3/4-- when x is right

00:07:36.220 --> 00:07:42.190
there-- and y is negative 7.

00:07:42.190 --> 00:07:50.170
This is essentially 7 1/8, so
1, 2, 3, 4, 5, 6, 7.125.

00:07:50.170 --> 00:07:52.980
So it's a little over 7.

00:07:52.980 --> 00:07:58.240
The vertex of this top graph
is right over there.

00:07:58.240 --> 00:07:59.630
It's symmetric around
the vertex.

00:08:02.250 --> 00:08:05.610
That is the line of symmetry,
and so this top graph is going

00:08:05.610 --> 00:08:06.920
to look something like this.

00:08:11.150 --> 00:08:13.100
Just like that and we're done.

00:08:13.100 --> 00:08:16.160
We've found our two points of
intersection, right over there

00:08:16.160 --> 00:08:17.870
and right over there, and
when you graph it it

00:08:17.870 --> 00:08:20.490
looks pretty good.


Non-linear systems of equations 3

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:03.350
Solve the system of equations
using any method.

00:00:03.350 --> 00:00:06.930
We have y is equal to 2 times
the quantity x minus 4

00:00:06.930 --> 00:00:08.700
squared plus 3.

00:00:08.700 --> 00:00:10.960
We also have y is equal
to negative x squared

00:00:10.960 --> 00:00:13.460
plus 2 x minus 2.

00:00:13.460 --> 00:00:16.470
The solution-- it might be one,
it might be none, or it

00:00:16.470 --> 00:00:22.580
might be two solutions-- to this
system occurs for the x

00:00:22.580 --> 00:00:24.640
values that generate
the same y values.

00:00:24.640 --> 00:00:26.570
There's the same x and
y that satisfy

00:00:26.570 --> 00:00:28.870
both of these equations.

00:00:28.870 --> 00:00:31.910
In order to find the x values,
they need to equal the same y

00:00:31.910 --> 00:00:34.030
values, so this y has
to be that y value.

00:00:34.030 --> 00:00:37.060
So the solution is going to
occur when this guy right

00:00:37.060 --> 00:00:45.080
here-- negative x squared plus
2x minus 2 is equal to that

00:00:45.080 --> 00:00:49.780
guy up there, or equal
to 2 times x minus 4

00:00:49.780 --> 00:00:52.690
squared plus 3.

00:00:52.690 --> 00:00:55.120
Now let's just try
to solve for x.

00:00:55.120 --> 00:00:59.980
The left hand side-- we're going
to have to multiply this

00:00:59.980 --> 00:01:03.040
out, so let's do that first.
It's negative x squared plus

00:01:03.040 --> 00:01:05.530
2x minus 2 is equal to.

00:01:05.530 --> 00:01:10.110
And on the right hand side, 2
times x minus 4 squared is x

00:01:10.110 --> 00:01:14.455
squared minus 8x
plus 16 plus 3.

00:01:14.455 --> 00:01:17.980
This is going to be equal
to 2x squared-- I'm just

00:01:17.980 --> 00:01:27.680
distributing the 2-- minus 16x
plus 32 plus 3, which is equal

00:01:27.680 --> 00:01:34.960
to 2x squared minus
16x plus 35.

00:01:34.960 --> 00:01:36.930
That's, of course, going to be
equal to this thing on the

00:01:36.930 --> 00:01:42.050
left hand side, negative x
squared plus 2x minus 2.

00:01:46.510 --> 00:01:49.040
Let's just get rid of this whole
thing from the left hand

00:01:49.040 --> 00:01:53.600
side all at once by adding
x squared to both sides.

00:01:53.600 --> 00:01:54.720
We can all do it in one step.

00:01:54.720 --> 00:01:56.340
We're going to add x squared
to both sides.

00:01:56.340 --> 00:02:01.040
Let's subtract 2x from
both sides, and let's

00:02:01.040 --> 00:02:02.450
add 2 to both sides.

00:02:07.000 --> 00:02:09.310
On the left hand side, those
cancel out, those cancel out,

00:02:09.310 --> 00:02:10.095
those cancel out.

00:02:10.095 --> 00:02:15.430
You're left with 0 is equal to
2x squared plus x squared is

00:02:15.430 --> 00:02:17.390
3x squared.

00:02:17.390 --> 00:02:22.940
Negative 16x minus 2x is
negative 18x, and then

00:02:22.940 --> 00:02:26.500
35 plus 2 is 37.

00:02:26.500 --> 00:02:29.060
So we just have a plain vanilla
quadratic equation

00:02:29.060 --> 00:02:30.220
right here.

00:02:30.220 --> 00:02:32.510
We might as well apply the
quadratic formula here to try

00:02:32.510 --> 00:02:33.920
to solve it.

00:02:33.920 --> 00:02:39.000
Our solutions are going to be
x is equal to negative b.

00:02:39.000 --> 00:02:42.860
Well, b is negative 18, so
negative b is positive 18.

00:02:42.860 --> 00:02:49.960
It's 18 plus or minus the square
root of 18 squared

00:02:49.960 --> 00:02:59.990
minus 4 times 3 times
c-- times 37.

00:02:59.990 --> 00:03:05.290
All of that is over 2 times
a-- 2 times 3, which is 6.

00:03:05.290 --> 00:03:07.400
Let's think about what
this is going to be.

00:03:07.400 --> 00:03:12.680
Over here, we have 18 plus or
minus the square root of--

00:03:12.680 --> 00:03:14.220
let's just use a calculator.

00:03:14.220 --> 00:03:19.860
I could multiply it out but I
think-- we have 18 squared

00:03:19.860 --> 00:03:28.170
minus 4 times 3 times 37,
which is negative 120.

00:03:28.170 --> 00:03:31.760
It's 18 plus or minus the square
root of negative 120.

00:03:31.760 --> 00:03:32.820
You might have even been
able to figure out

00:03:32.820 --> 00:03:33.770
that this is negative.

00:03:33.770 --> 00:03:35.030
4 times 3 is 12.

00:03:35.030 --> 00:03:39.460
12 times 37 is going to be
a bigger number than 18.

00:03:39.460 --> 00:03:41.850
Although it's not 100% obvious,
but you might be able

00:03:41.850 --> 00:03:43.170
to just get the intuition
there.

00:03:43.170 --> 00:03:45.290
We definitely end up with a
negative number under the

00:03:45.290 --> 00:03:46.440
radical here.

00:03:46.440 --> 00:03:49.950
Now, if we're dealing with
real numbers, there is no

00:03:49.950 --> 00:03:53.000
square root of negative 120.

00:03:53.000 --> 00:03:56.850
So there is no solution to
this quadratic equation.

00:03:56.850 --> 00:04:00.030
There is no solution.

00:04:00.030 --> 00:04:01.500
If we wanted to, we could
have just looked at the

00:04:01.500 --> 00:04:02.100
discriminant.

00:04:02.100 --> 00:04:05.660
The discriminant is this part--
b squared minus 4ac.

00:04:05.660 --> 00:04:08.960
We see the discriminant is
negative, there's no solution,

00:04:08.960 --> 00:04:12.350
which means that these two
guys-- these two equations--

00:04:12.350 --> 00:04:13.410
never intersect.

00:04:13.410 --> 00:04:20.310
There is no solution
to the system.

00:04:20.310 --> 00:04:23.060
There are no x values that
when you put into both of

00:04:23.060 --> 00:04:26.860
these equations give you
the exact same y value.

00:04:26.860 --> 00:04:30.460
Let's think a little bit about
why that happened.

00:04:30.460 --> 00:04:33.280
This one is already in kind
of our y-intercept form.

00:04:33.280 --> 00:04:36.070
It's an upward opening
parabola, so it looks

00:04:36.070 --> 00:04:37.240
something like this.

00:04:37.240 --> 00:04:39.370
I'll do my best to draw
it-- just a quick and

00:04:39.370 --> 00:04:40.990
dirty version of it.

00:04:40.990 --> 00:04:44.670
Let me draw my axes in
a neutral color.

00:04:44.670 --> 00:04:48.930
Let's say that this right here
is my y-axis, that right there

00:04:48.930 --> 00:04:51.000
is my x-axis.

00:04:51.000 --> 00:04:52.130
x and y.

00:04:52.130 --> 00:04:54.540
This vertex-- it's in the vertex
form-- occurs when x is

00:04:54.540 --> 00:04:56.595
equal to 4 and y
is equal to 3.

00:04:56.595 --> 00:04:59.490
So x is equal to 4 and
y is equal to 3.

00:04:59.490 --> 00:05:01.010
It's an upward opening
parabola.

00:05:01.010 --> 00:05:03.210
We have a positive coefficient
out here.

00:05:03.210 --> 00:05:05.240
So this will look something
like this.

00:05:08.460 --> 00:05:11.990
I don't know the exact thing,
but that's close enough.

00:05:11.990 --> 00:05:13.960
Now, what will this
thing look like?

00:05:13.960 --> 00:05:16.690
It's a downward opening parabola
and we can actually

00:05:16.690 --> 00:05:19.910
put this in vertex form.

00:05:19.910 --> 00:05:22.880
Let me put the second equation
in vertex form,

00:05:22.880 --> 00:05:23.925
just so we have it.

00:05:23.925 --> 00:05:25.010
So we have a good sense.

00:05:25.010 --> 00:05:27.730
So, y is equal to-- we could
factor in a negative 1--

00:05:27.730 --> 00:05:32.670
negative x squared
minus 2x plus 2.

00:05:36.200 --> 00:05:41.210
Actually, let me put the plus
2 further out-- plus 2, all

00:05:41.210 --> 00:05:43.080
the way up out there.

00:05:43.080 --> 00:05:46.390
Then we could say, half of
negative 2 is negative 1.

00:05:46.390 --> 00:05:49.100
You square it, so you
have a plus 1 and

00:05:49.100 --> 00:05:51.390
then a minus 1 there.

00:05:51.390 --> 00:05:56.170
This part right over here, we
can rewrite as x minus 1

00:05:56.170 --> 00:06:02.130
squared, so it becomes negative
x minus 1 squared.

00:06:02.130 --> 00:06:03.715
Let me just do it one
step at a time.

00:06:03.715 --> 00:06:06.100
I don't want to skip steps.

00:06:06.100 --> 00:06:09.820
Negative x minus 1 squared
minus 1 plus 2.

00:06:09.820 --> 00:06:12.250
So that's plus 1 out here.

00:06:12.250 --> 00:06:16.060
Or if we want to distribute
the negative, we get y is

00:06:16.060 --> 00:06:21.440
equal to negative x minus
1 squared minus 1.

00:06:21.440 --> 00:06:25.030
Here the vertex occurs at x is
equal to 1, y is equal to

00:06:25.030 --> 00:06:26.280
negative 1.

00:06:28.750 --> 00:06:29.840
The vertex is there,
and this is a

00:06:29.840 --> 00:06:31.350
downward opening parabola.

00:06:31.350 --> 00:06:34.820
We have a negative coefficient
out here on the second degree

00:06:34.820 --> 00:06:37.310
term, so it's going to look
something like this.

00:06:41.840 --> 00:06:44.640
So as you see, they
don't intersect.

00:06:44.640 --> 00:06:47.030
This vertex is above it
and it opens upward.

00:06:47.030 --> 00:06:48.320
This is its minimum point.

00:06:48.320 --> 00:06:50.610
And it's above this guy's
maximum point.

00:06:50.610 --> 00:06:53.790
So they will never intersect,
so there is no solution to

00:06:53.790 --> 00:06:55.710
this system of equations.


Applying Quadratic Functions 1

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:04.370
A suspension bridge has two
towers that rise 500 feet

00:00:04.370 --> 00:00:07.390
above the road and are connected
by cables that hang

00:00:07.390 --> 00:00:11.410
in the shape of a parabola that
can be represented by y

00:00:11.410 --> 00:00:19.990
is equal to 1/8,960 times x
minus 2,100 squared plus 8,

00:00:19.990 --> 00:00:23.200
where x and y are measured in
feet and x is the distance

00:00:23.200 --> 00:00:26.450
from the left tower in the
direction of the right tower.

00:00:26.450 --> 00:00:28.980
What is the distance between
the two towers?

00:00:28.980 --> 00:00:31.330
Let's draw ourselves
a diagram.

00:00:31.330 --> 00:00:37.130
If we draw the leftmost
tower-- I'm

00:00:37.130 --> 00:00:38.640
going to do it in orange.

00:00:38.640 --> 00:00:41.960
So we have the left
tower right here.

00:00:41.960 --> 00:00:45.510
That tower is at x
is equal to 0.

00:00:45.510 --> 00:00:49.490
We'll call that x is equal to 0,
or maybe just call that the

00:00:49.490 --> 00:00:51.480
0 point right there.

00:00:51.480 --> 00:00:53.940
Then x is just measuring the
distance from that leftmost

00:00:53.940 --> 00:00:57.600
tower to the rightmost tower, so
the rightmost tower in the

00:00:57.600 --> 00:01:00.060
direction of the right tower.

00:01:00.060 --> 00:01:02.145
This right here, maybe
the right tower

00:01:02.145 --> 00:01:05.099
is right over here.

00:01:05.099 --> 00:01:07.680
They're both 500 feet high.

00:01:07.680 --> 00:01:10.510
The right tower is right over
there, and we don't

00:01:10.510 --> 00:01:11.240
know where it is.

00:01:11.240 --> 00:01:11.930
That's we have to figure out.

00:01:11.930 --> 00:01:13.880
We have to figure out the
distance between the two, but

00:01:13.880 --> 00:01:16.650
we just need to know that x is
measuring this distance.

00:01:16.650 --> 00:01:19.810
It's measuring the distance
from the left tower in the

00:01:19.810 --> 00:01:22.050
direction of the right tower, so
it's just traditional, just

00:01:22.050 --> 00:01:23.540
like our coordinate plane.

00:01:23.540 --> 00:01:25.020
That is what x is measuring.

00:01:25.020 --> 00:01:29.560
If you say x is equal to a
million feet, you're going to

00:01:29.560 --> 00:01:32.190
get something way beyond
that second tower.

00:01:32.190 --> 00:01:33.910
If you say x is one foot, you're
going to get something

00:01:33.910 --> 00:01:36.840
in the direction of the second
tower from the left tower.

00:01:36.840 --> 00:01:39.840
So that's x, and then y, we can
assume is the height of

00:01:39.840 --> 00:01:42.680
the towers, or is just the
height in general that we're

00:01:42.680 --> 00:01:43.830
dealing with.

00:01:43.830 --> 00:01:45.860
This is the y-axis, so we're
just dealing in the first

00:01:45.860 --> 00:01:49.470
quadrant of the coordinate
plane, and that is y.

00:01:49.470 --> 00:01:52.220
They're telling us that this
parabola, the shape of the

00:01:52.220 --> 00:01:54.350
parabola, can be represented--
or even better, they're saying

00:01:54.350 --> 00:01:57.320
that the cable that connects
these two is in

00:01:57.320 --> 00:01:58.420
the shape of a parabola.

00:01:58.420 --> 00:02:01.690
Let me draw that parabola
right there.

00:02:01.690 --> 00:02:03.550
They're the same height, so
the parabola will look

00:02:03.550 --> 00:02:04.800
something like this.

00:02:09.289 --> 00:02:12.840
What we can do to figure out,
or one possible technique--

00:02:12.840 --> 00:02:15.530
let me draw a better
parabola there.

00:02:15.530 --> 00:02:18.290
That's even worse.

00:02:18.290 --> 00:02:20.200
That's about as good
as I can do.

00:02:20.200 --> 00:02:23.320
One technique for finding the
distance between the two is if

00:02:23.320 --> 00:02:27.040
we could figure out the
minimum point of this

00:02:27.040 --> 00:02:31.920
parabola, which is also going
to be the vertex of that

00:02:31.920 --> 00:02:33.970
parabola, that's going
to be halfway

00:02:33.970 --> 00:02:35.940
between the two towers.

00:02:35.940 --> 00:02:38.090
If we can figure out this
x-coordinate, we can say

00:02:38.090 --> 00:02:40.450
that's going to be halfway
between the two towers and

00:02:40.450 --> 00:02:42.170
then just multiply by two.

00:02:42.170 --> 00:02:44.960
Let's see if we can figure out
this minimum point, because

00:02:44.960 --> 00:02:47.450
the parabola is symmetric around
it, and these are both

00:02:47.450 --> 00:02:48.830
at the same height.

00:02:48.830 --> 00:02:50.530
Let's see if we can figure
out that minimum point.

00:02:50.530 --> 00:02:57.920
Let me rewrite this equation: We
have y is equal to 1/8,960

00:02:57.920 --> 00:03:03.070
times x minus 2,100
squared plus 8.

00:03:03.070 --> 00:03:06.280
How do we find the minimum
point here?

00:03:06.280 --> 00:03:09.460
This whole expression that's
squared, this whole expression

00:03:09.460 --> 00:03:12.180
is always going to be positive,
or it's actually

00:03:12.180 --> 00:03:13.770
always going to be
non-negative.

00:03:13.770 --> 00:03:17.010
It could be equal to 0, and so
this whole thing is always

00:03:17.010 --> 00:03:19.920
going to be greater than
or equal to 0.

00:03:19.920 --> 00:03:22.520
This whole thing I'm squaring
off in orange is always going

00:03:22.520 --> 00:03:24.440
to be greater than
or equal to 0.

00:03:24.440 --> 00:03:27.540
So what is the lowest value
that y can take on?

00:03:27.540 --> 00:03:31.200
The lowest valued that y can
take on will happen when this

00:03:31.200 --> 00:03:37.930
expression is equal to 0, and
this expression equals 0 only

00:03:37.930 --> 00:03:41.670
when x minus 2,100 squared
is equal to 0.

00:03:41.670 --> 00:03:44.820
That's only going to be equal
to 0 when x minus 2,100 is

00:03:44.820 --> 00:03:50.290
equal to 0, so x-- let me write
it this way-- equals 0,

00:03:50.290 --> 00:03:53.650
and I'm just talking about
this orange part-- when x

00:03:53.650 --> 00:03:57.650
minus 2,100 is equal to 0, or
if you add 2,100 to both

00:03:57.650 --> 00:04:01.890
sides, or x is equal to 2,100.

00:04:01.890 --> 00:04:05.180
When x is equal to 2,100, just
out of curiosity, what is y?

00:04:05.180 --> 00:04:09.350
If x is 2,100, this thing is 0
times 0, and y is just going

00:04:09.350 --> 00:04:10.900
to be equal to that 8 there.

00:04:10.900 --> 00:04:13.460
So this coordinate right here
is the coordinate x is equal

00:04:13.460 --> 00:04:18.630
to 2,100, y is equal to 8, and
it's going to be halfway

00:04:18.630 --> 00:04:21.070
between our two towers.

00:04:21.070 --> 00:04:23.630
If this is at 2,100, how
far is this tower away?

00:04:23.630 --> 00:04:25.350
Well, it's going to
be twice 2,100.

00:04:25.350 --> 00:04:27.920
It's going to be 4,200,
and everything we're

00:04:27.920 --> 00:04:28.720
dealing with is feet.

00:04:28.720 --> 00:04:32.630
It's going to be 4,200 feet away
from that first tower.

00:04:32.630 --> 00:04:38.160
So the distance between the
two towers is 4,200 feet.


Applying Quadratic Functions 2

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:07.110
A cliff diver dives off of a
cliff 85 feet above the water.

00:00:07.110 --> 00:00:10.850
If the diver's initial vertical
velocity is negative

00:00:10.850 --> 00:00:14.480
5 feet per second-- so he's
going downward, so they tell

00:00:14.480 --> 00:00:17.700
us right here-- 5 feet per
second downward, and the

00:00:17.700 --> 00:00:21.680
acceleration of gravity is
negative 32 feet per second

00:00:21.680 --> 00:00:25.660
squared, or 32 feet per second
downward-- so that means after

00:00:25.660 --> 00:00:30.110
every second, you're going 32
feet per second faster in the

00:00:30.110 --> 00:00:33.620
downward direction-- how long is
the diver in the air before

00:00:33.620 --> 00:00:36.170
entering the water?

00:00:36.170 --> 00:00:39.400
So let's just set a variable t
to be what we want to solve

00:00:39.400 --> 00:00:43.640
for, which is how long is
the diver in the air?

00:00:43.640 --> 00:00:45.165
Let me just write
time in the air.

00:00:51.230 --> 00:00:55.490
And we have the initial velocity
of the diver right

00:00:55.490 --> 00:00:58.330
here, the initial vertical
velocity, and that's what we

00:00:58.330 --> 00:00:58.860
care about.

00:00:58.860 --> 00:01:00.710
We just care about what's going
on in the up and down

00:01:00.710 --> 00:01:02.390
direction, the vertical
direction.

00:01:02.390 --> 00:01:06.415
So our initial velocity is
negative 5 feet per second.

00:01:10.410 --> 00:01:13.660
Now, what's going to be his
velocity right when he hits

00:01:13.660 --> 00:01:16.910
the water, right when
he enters the water?

00:01:16.910 --> 00:01:18.393
What's going to be his
final velocity?

00:01:22.930 --> 00:01:25.580
He's going to start at negative
5 feet per second, so

00:01:25.580 --> 00:01:30.780
it's going to be negative 5 feet
per second, that's his

00:01:30.780 --> 00:01:31.970
initial velocity.

00:01:31.970 --> 00:01:36.000
And every second that goes by,
he's going to be going

00:01:36.000 --> 00:01:40.680
negative 32 feet per second
faster after every second.

00:01:40.680 --> 00:01:43.970
He's accelerating downward
at that rate.

00:01:43.970 --> 00:01:46.780
So if you multiply the
acceleration times time,

00:01:46.780 --> 00:01:48.540
that's how much his velocity
is increased.

00:01:48.540 --> 00:01:59.960
So it's going to be his initial
velocity minus 32 feet

00:01:59.960 --> 00:02:02.940
per second squared.

00:02:02.940 --> 00:02:07.660
That's the acceleration times
time-- times the time that

00:02:07.660 --> 00:02:09.440
he's in the air, times this t.

00:02:09.440 --> 00:02:12.660
Obviously, this will be in
seconds, so we have 1 second

00:02:12.660 --> 00:02:14.580
over seconds squared.

00:02:14.580 --> 00:02:16.290
You'll just have a second in
the denominator, so you'll

00:02:16.290 --> 00:02:18.210
have feet per second and
feet per second.

00:02:18.210 --> 00:02:20.260
The units all work out.

00:02:20.260 --> 00:02:21.310
But hopefully this
makes sense.

00:02:21.310 --> 00:02:24.880
He's starting off at moving
downward at 5 feet per second.

00:02:24.880 --> 00:02:27.330
The negative is telling you the
direction is downward, so

00:02:27.330 --> 00:02:31.330
he's moving downward at
5 feet per second.

00:02:31.330 --> 00:02:34.070
So after 1 second,
he'll be going 37

00:02:34.070 --> 00:02:35.520
feet per second downward.

00:02:35.520 --> 00:02:39.040
After 2 seconds, he'll be
going-- what, 32 times 2 is

00:02:39.040 --> 00:02:43.430
64, plus 9-- 69 feet per
second downwards.

00:02:43.430 --> 00:02:45.270
So this is his final velocity.

00:02:45.270 --> 00:02:46.880
So what's his average
velocity?

00:02:46.880 --> 00:02:49.320
If his initial velocity is
negative 5 feet per second,

00:02:49.320 --> 00:02:51.410
his final velocity is this.

00:02:51.410 --> 00:02:53.360
It's a linear relation.

00:02:53.360 --> 00:02:58.270
At every second that goes by,
it's linearly increasing, or

00:02:58.270 --> 00:03:00.650
decreasing, if you want to view
it as a negative number.

00:03:00.650 --> 00:03:02.810
So the average velocity is just
going to be the average

00:03:02.810 --> 00:03:06.030
of these two values.

00:03:06.030 --> 00:03:07.910
We're really just learning a lot
of physics here, but this

00:03:07.910 --> 00:03:10.500
will set us up with a nice
quadratic equation that we can

00:03:10.500 --> 00:03:13.000
use the quadratic formula for.

00:03:13.000 --> 00:03:17.230
So our average velocity is going
to be the average of

00:03:17.230 --> 00:03:18.360
those two velocities.

00:03:18.360 --> 00:03:20.780
So it's going to be-- and I'll
skip the units right now--

00:03:20.780 --> 00:03:29.760
it's going to be negative 5
minus 5, minus 32t, all of

00:03:29.760 --> 00:03:30.890
that over 2.

00:03:30.890 --> 00:03:32.550
I'm just averaging
these two things.

00:03:32.550 --> 00:03:33.845
That's my initial velocity.

00:03:33.845 --> 00:03:34.985
This is my final velocity.

00:03:34.985 --> 00:03:37.080
I'm just taking the average
of the two.

00:03:37.080 --> 00:03:39.110
So this is going to
be equal to what?

00:03:39.110 --> 00:03:45.470
This is negative 10 minus 32t
over 2, which is equal to

00:03:45.470 --> 00:03:49.380
negative 5 minus 16t.

00:03:49.380 --> 00:03:51.960
I'm just dividing
everything by 2.

00:03:51.960 --> 00:03:54.100
So this is my average
velocity.

00:03:54.100 --> 00:03:57.450
Now, we can just apply our
simple distance is equal to

00:03:57.450 --> 00:03:58.700
rate times time.

00:04:01.760 --> 00:04:03.180
Now, what's the distance
that he's

00:04:03.180 --> 00:04:04.940
traveling in this situation?

00:04:04.940 --> 00:04:09.140
Well, he's diving off of a cliff
85 feet above the water,

00:04:09.140 --> 00:04:10.510
so the distance that
he's going to be

00:04:10.510 --> 00:04:17.019
traveling is 85 feet.

00:04:17.019 --> 00:04:20.519
He starts 85 feet above the
water, but the distance he'll

00:04:20.519 --> 00:04:23.800
travel, he's going to
go 85 feet down.

00:04:23.800 --> 00:04:28.790
So the distance he's going to
travel is negative 85 feet.

00:04:28.790 --> 00:04:31.310
We use negative for down,
we use positive for up.

00:04:31.310 --> 00:04:32.790
Everything in this problem is

00:04:32.790 --> 00:04:34.500
happening in the down direction.

00:04:34.500 --> 00:04:38.040
So negative 85 feet is
going to be equal

00:04:38.040 --> 00:04:40.250
to the average velocity.

00:04:40.250 --> 00:04:41.820
But we just figured out
the average velocity.

00:04:41.820 --> 00:04:43.780
It's this thing right here.

00:04:43.780 --> 00:04:45.020
That's our average velocity.

00:04:45.020 --> 00:04:48.170
So it's going to be-- let me do
it in another color; let me

00:04:48.170 --> 00:04:52.630
do it in this blue-- our
average velocity is

00:04:52.630 --> 00:04:55.880
negative 5 minus 16t.

00:04:55.880 --> 00:04:59.410
That's our average velocity,
velocity average.

00:04:59.410 --> 00:05:02.650
And then we want to multiply
that times time.

00:05:02.650 --> 00:05:06.940
This is our velocity, our
rate-- I'm mixing up the

00:05:06.940 --> 00:05:10.050
colors-- this is our rate, and
then we want to multiply it by

00:05:10.050 --> 00:05:15.240
times, so rate times time is
equal to distance travelled.

00:05:15.240 --> 00:05:17.450
Let's solve for t.

00:05:17.450 --> 00:05:20.160
And we'll skip the units for
now, just because that'll kind

00:05:20.160 --> 00:05:22.500
of get in the way of
maybe the learning.

00:05:22.500 --> 00:05:26.010
So we have negative 85 is equal
to-- when we distribute

00:05:26.010 --> 00:05:32.370
this t, we get negative
5t minus 16t squared.

00:05:32.370 --> 00:05:35.810
And now we can add 85 to both
sides of this equation.

00:05:35.810 --> 00:05:38.500
Let's add 85 to both sides.

00:05:38.500 --> 00:05:45.520
And we are left with
0 is equal to--

00:05:45.520 --> 00:05:46.790
let's rearrange this.

00:05:46.790 --> 00:05:54.210
So we'll a negative 16t squared
minus 5t, plus 85.

00:05:54.210 --> 00:05:57.990
So this, essentially,
is the equation.

00:05:57.990 --> 00:06:01.340
If we solve for t, we'll know
how long he's been in the air,

00:06:01.340 --> 00:06:04.040
because we've used all of
the other constraints.

00:06:04.040 --> 00:06:06.340
So this is a straight up
quadratic equation, so we

00:06:06.340 --> 00:06:08.810
could use the quadratic
formula here.

00:06:08.810 --> 00:06:12.320
So it's going to
be negative b.

00:06:12.320 --> 00:06:13.300
Now what's b?

00:06:13.300 --> 00:06:18.400
b here is negative 5, so
negative b, which is negative

00:06:18.400 --> 00:06:23.250
negative 5, plus or minus the
square root of b squared.

00:06:23.250 --> 00:06:29.500
Negative 5 squared is 25 minus
4, times a, which is negative

00:06:29.500 --> 00:06:35.730
16, times c, which is 85.

00:06:35.730 --> 00:06:41.070
All of that over 2 times a. a
is negative 16, so 2 times

00:06:41.070 --> 00:06:43.110
that is negative 32.

00:06:43.110 --> 00:06:47.470
So the time is equal to positive
5, right, negative

00:06:47.470 --> 00:06:52.720
negative 5 is positive 5, plus
or minus the square root of--

00:06:52.720 --> 00:06:55.250
we probably want to take out
our calculator for this.

00:06:57.860 --> 00:07:00.560
Let's see, let's
evaluate this.

00:07:00.560 --> 00:07:02.640
So we have a negative and a
negative, so those two are

00:07:02.640 --> 00:07:04.380
going to cancel-- let
put my calculator

00:07:04.380 --> 00:07:06.550
aside right for a second.

00:07:06.550 --> 00:07:09.310
So we have negative out here,
and then we have a negative,

00:07:09.310 --> 00:07:10.930
so these two are going
to cancel out.

00:07:10.930 --> 00:07:14.590
So it's going to be 25 plus
4, times 16, times 85.

00:07:14.590 --> 00:07:17.000
So let's figure out
what that is.

00:07:17.000 --> 00:07:24.450
So if we have 25 plus-- I'll put
some parentheses here-- 4

00:07:24.450 --> 00:07:33.330
times 16, times 85 is equal
to-- so we got 25 plus 4,

00:07:33.330 --> 00:07:38.630
times 16, times 85--
is equal to 5,465.

00:07:38.630 --> 00:07:40.120
And we could take the
square root of this.

00:07:40.120 --> 00:07:47.440
So let's just take the square
root of it, and we have 73.9.

00:07:47.440 --> 00:07:50.400
So let's just stick with 73.9.

00:07:50.400 --> 00:07:53.860
So this is going to be equal
to-- let me delete this right

00:07:53.860 --> 00:07:57.790
here-- this is going to be equal
to, we already figured

00:07:57.790 --> 00:08:01.380
out what the square root is, we
just evaluated it-- let me

00:08:01.380 --> 00:08:04.870
delete that right there-- this
is going to be equal to plus

00:08:04.870 --> 00:08:10.730
or minus 73.9, all of that
over negative 32.

00:08:10.730 --> 00:08:15.670
Now, we can add 73.9, and we
can subtract 73.9, but what

00:08:15.670 --> 00:08:17.850
happens if we subtract 73.9?

00:08:17.850 --> 00:08:20.020
Well, actually, let's think
about this carefully.

00:08:20.020 --> 00:08:20.960
Let's do both of these.

00:08:20.960 --> 00:08:22.010
Let's just go through them.

00:08:22.010 --> 00:08:25.740
So the two combinations, so the
two possibilities for t,

00:08:25.740 --> 00:08:32.730
is we have 5 plus 73.9, which
is 78.9, all of that over

00:08:32.730 --> 00:08:34.020
negative 32.

00:08:34.020 --> 00:08:35.659
That's one possibility for t.

00:08:35.659 --> 00:08:39.420
The other one is 5 minus 73.9.

00:08:39.420 --> 00:08:43.530
And let's see, what's
73.9 minus 5?

00:08:43.530 --> 00:08:48.560
It is negative 68.9.

00:08:48.560 --> 00:08:54.360
73 minus 5 is 68, swap the
negatives, so it's negative

00:08:54.360 --> 00:08:58.680
68.9 over negative 32.

00:08:58.680 --> 00:09:00.610
In this case, the negatives
cancel out.

00:09:00.610 --> 00:09:04.040
So which of these-- can we
use both of these times?

00:09:04.040 --> 00:09:06.500
Well, this right here is going
to be a negative number, and

00:09:06.500 --> 00:09:07.520
we don't want a negative time.

00:09:07.520 --> 00:09:09.280
It's definitely a positive
amount of time

00:09:09.280 --> 00:09:09.900
that he's in the air.

00:09:09.900 --> 00:09:10.620
[UNINTELLIGIBLE]

00:09:10.620 --> 00:09:12.360
negative time, are we
going back in time?

00:09:12.360 --> 00:09:13.100
Who knows?

00:09:13.100 --> 00:09:14.490
So we can't even use that.

00:09:14.490 --> 00:09:16.406
So the time in the air is going
to be this, it's going

00:09:16.406 --> 00:09:22.690
to be 68.9 divided by 32, so
let's figure out what that is.

00:09:22.690 --> 00:09:32.380
So it's 68.9 divided by 32
is equal to 2.15 seconds.

00:09:32.380 --> 00:09:40.240
So his time in the air is going
to be 2.15 seconds.

00:09:40.240 --> 00:09:41.820
And we're done.