Quadratic Regression

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:01.810
I have some data here.

00:00:01.810 --> 00:00:04.570
It says a golf ball is
hit down a fairway.

00:00:04.570 --> 00:00:07.120
The following table shows the
height of the ball with

00:00:07.120 --> 00:00:08.610
respect to time.

00:00:08.610 --> 00:00:12.200
The ball is hit at an angle of
70 degrees with the horizontal

00:00:12.200 --> 00:00:14.620
with a speed of 40 meters
per second.

00:00:14.620 --> 00:00:17.680
And then they give us a bunch
of, essentially data samples.

00:00:17.680 --> 00:00:19.590
At time 0, we're at 0 meters.

00:00:19.590 --> 00:00:23.300
At time 0.5 seconds, we're
at 17.2 meters.

00:00:23.300 --> 00:00:25.440
So they give us a bunch
of data points.

00:00:25.440 --> 00:00:29.390
And what I want to do is use
these data, or use these data

00:00:29.390 --> 00:00:34.250
points, to essentially find a
quadratic function that fits

00:00:34.250 --> 00:00:37.270
it best. Or I guess to view it
the other way, I want to model

00:00:37.270 --> 00:00:39.630
this data or I want to model
this phenomenon, I want to

00:00:39.630 --> 00:00:41.810
model the golf ball being hit
down the fairway, or the

00:00:41.810 --> 00:00:45.190
height of the golf ball using
a quadratic equation, or

00:00:45.190 --> 00:00:46.175
quadratic function.

00:00:46.175 --> 00:00:49.040
And we're going to do it using
a graphing calculator.

00:00:49.040 --> 00:00:50.830
And actually, you know, if you
do a little bit of physics,

00:00:50.830 --> 00:00:53.250
you can figure out what
the actual equation

00:00:53.250 --> 00:00:54.330
would be in a vacuum.

00:00:54.330 --> 00:00:58.370
But this is actual data taken
from a golf ball hit in air.

00:00:58.370 --> 00:01:00.310
And that changes things a bunch,
it makes the physics

00:01:00.310 --> 00:01:01.320
really complicated.

00:01:01.320 --> 00:01:03.900
So we'll actually look at the
measurements and then try to

00:01:03.900 --> 00:01:05.695
figure out what the actual
function is.

00:01:05.695 --> 00:01:07.320
And actually, the calculator
is going to do most of the

00:01:07.320 --> 00:01:08.160
work for us.

00:01:08.160 --> 00:01:09.490
I won't go into the details
of what the

00:01:09.490 --> 00:01:11.090
calculators algorithm is.

00:01:11.090 --> 00:01:12.810
I just want to show
you how to do it.

00:01:12.810 --> 00:01:16.210
It's safe to say that this is
going to be well approximated

00:01:16.210 --> 00:01:18.910
by some type of quadratic
function or a parabola.

00:01:18.910 --> 00:01:21.970
So let's try to fit
a function to it.

00:01:21.970 --> 00:01:24.540
So what we want to do is we
want to hit stat on our

00:01:24.540 --> 00:01:25.900
calculator.

00:01:25.900 --> 00:01:27.850
You hit stat, then
you hit edit.

00:01:27.850 --> 00:01:31.010
That's what we use to input
our information.

00:01:31.010 --> 00:01:32.350
We can click enter twice.

00:01:32.350 --> 00:01:34.600
This just says these are the two
variable names for where

00:01:34.600 --> 00:01:36.680
we input our data.

00:01:36.680 --> 00:01:38.400
I'm just hitting enter twice.

00:01:38.400 --> 00:01:40.450
And now I can start
entering the data.

00:01:40.450 --> 00:01:44.140
And I'll use my keyboard
to do this.

00:01:44.140 --> 00:01:46.560
So the first data point is 0.

00:01:46.560 --> 00:01:46.960
Enter.

00:01:46.960 --> 00:01:48.350
The x is 0, y is 0.

00:01:48.350 --> 00:01:49.320
Press enter again.

00:01:49.320 --> 00:01:54.540
The next data point,
0.5 and 17.2.

00:01:54.540 --> 00:01:59.330
I'm just looking at this right
here, 0.5 and 17.2.

00:01:59.330 --> 00:02:00.900
Next data point.

00:02:00.900 --> 00:02:04.110
We have x is 1.5.

00:02:04.110 --> 00:02:08.000
When x is 1.5, y is 42.9.

00:02:08.000 --> 00:02:11.640
When x is 2, y is 51.6.

00:02:11.640 --> 00:02:15.940
When x is 2.5, y is 57.7.

00:02:15.940 --> 00:02:20.720
When x is 3, y is 61.2.

00:02:20.720 --> 00:02:23.250
I just keep pressing
enter every time.

00:02:23.250 --> 00:02:28.660
When x is 3.5, y is 62.3
meters in the air.

00:02:28.660 --> 00:02:34.630
When x is 4, y is 61
meters in the air.

00:02:34.630 --> 00:02:40.060
And when x is 4.5, y is 57.2.

00:02:40.060 --> 00:02:43.250
So I've entered all
of the data.

00:02:43.250 --> 00:02:44.790
And then the next thing I
want to do, I'm going

00:02:44.790 --> 00:02:45.660
to plot all of it.

00:02:45.660 --> 00:02:48.440
But before I plot all of this
data, I want to make sure that

00:02:48.440 --> 00:02:50.970
I have the proper range
on my display.

00:02:50.970 --> 00:02:54.730
So I'm going to go to
graph, hit range.

00:02:54.730 --> 00:02:55.700
And see.

00:02:55.700 --> 00:02:57.780
My x minimum, I want
it to be 0.

00:02:57.780 --> 00:02:59.610
That's the lowest value here.

00:02:59.610 --> 00:03:03.090
And my x maximum, I could
set it actually,

00:03:03.090 --> 00:03:05.950
a little bit lower.

00:03:05.950 --> 00:03:07.710
Well, I set my x maximum,
that's fine.

00:03:07.710 --> 00:03:09.190
I'll go up to 7 seconds.

00:03:09.190 --> 00:03:11.460
I'll make the scale, that's
the measurements on the

00:03:11.460 --> 00:03:12.500
x-axis, as 1.

00:03:12.500 --> 00:03:14.380
y minimum is 0.

00:03:14.380 --> 00:03:16.350
y maximum, 70 looks
pretty good.

00:03:16.350 --> 00:03:18.280
It seems like it'll cover all of
these numbers, and some of

00:03:18.280 --> 00:03:19.550
these get pretty close.

00:03:19.550 --> 00:03:21.370
They're in the 60's, so
70's pretty good.

00:03:21.370 --> 00:03:23.220
And I'll make the y scale 5.

00:03:23.220 --> 00:03:24.870
I don't have to make any changes
here, but you could

00:03:24.870 --> 00:03:26.390
make changes if your calculator

00:03:26.390 --> 00:03:28.510
isn't set to this already.

00:03:28.510 --> 00:03:31.000
And so we can go back to stat.

00:03:31.000 --> 00:03:34.480
And then what we want to do is
have the calculator calculate

00:03:34.480 --> 00:03:38.420
a quadratic function that
matches these points as well

00:03:38.420 --> 00:03:39.620
as possible.

00:03:39.620 --> 00:03:41.530
So we'll go to calc.

00:03:41.530 --> 00:03:43.890
I'm assuming that means
for calculate.

00:03:43.890 --> 00:03:45.940
This is saying, where
is the data?

00:03:45.940 --> 00:03:48.400
It's the same variables we used
to input the data, so

00:03:48.400 --> 00:03:49.940
I'll just click enter.

00:03:49.940 --> 00:03:51.400
And enter twice.

00:03:51.400 --> 00:03:52.970
And now here are the
different types of

00:03:52.970 --> 00:03:54.330
regressions we can do.

00:03:54.330 --> 00:03:55.530
We can do a linear regression.

00:03:55.530 --> 00:03:57.710
We can do an exponential
regression, power regression.

00:03:57.710 --> 00:04:01.650
You click more and you'll see
a power 2 regression.

00:04:01.650 --> 00:04:04.370
This means a second
degree regression.

00:04:04.370 --> 00:04:06.870
So here, we can literally
click-- we

00:04:06.870 --> 00:04:08.260
just select this option.

00:04:08.260 --> 00:04:10.670
That means a quadratic or this
would be a third degree

00:04:10.670 --> 00:04:12.010
regression, a fourth
degree regression.

00:04:12.010 --> 00:04:15.400
So we're just going to do
a quadratic regression.

00:04:15.400 --> 00:04:19.279
And it figured out the
coefficients of the quadratic

00:04:19.279 --> 00:04:21.670
that best matches this data.

00:04:21.670 --> 00:04:22.600
So I'm going to write it down.

00:04:22.600 --> 00:04:25.080
So it's negative 5.2.

00:04:25.080 --> 00:04:26.490
It's negative 5.2.

00:04:26.490 --> 00:04:27.190
Scroll over a bit.

00:04:27.190 --> 00:04:29.890
35.99999.

00:04:29.890 --> 00:04:33.560
And then 0.292514.

00:04:33.560 --> 00:04:37.950
So let me write these
down over here.

00:04:37.950 --> 00:04:39.980
So let me take it off
the screen so I can

00:04:39.980 --> 00:04:41.900
keep looking at it.

00:04:41.900 --> 00:04:45.750
And so we have-- it's telling
us, the calculator has told

00:04:45.750 --> 00:04:52.700
us, that if the quadratic is
equal to ax squared plus bx

00:04:52.700 --> 00:04:56.010
plus c, it's telling us that the
first coefficient, a, is

00:04:56.010 --> 00:05:01.270
negative 5.20128.

00:05:01.270 --> 00:05:02.990
I'll probably round that
off a little bit.

00:05:02.990 --> 00:05:09.980
It's telling us b is equal to
35.9-- let me scroll over my

00:05:09.980 --> 00:05:15.532
calculator a bit-- 993.

00:05:15.532 --> 00:05:17.590
I'll probably round these.

00:05:17.590 --> 00:05:20.060
9934.

00:05:20.060 --> 00:05:23.070
And it's telling my c values
that it calculates for the

00:05:23.070 --> 00:05:29.170
best fit quadratic is 0.2925.

00:05:29.170 --> 00:05:31.540
Which is interesting.

00:05:31.540 --> 00:05:33.670
Before I did this problem, I
actually figured out what this

00:05:33.670 --> 00:05:35.030
quadratic would be
in a vacuum.

00:05:35.030 --> 00:05:37.510
I just used kind of some of
the physics principles.

00:05:37.510 --> 00:05:39.760
And then you have this
number in a vacuum.

00:05:39.760 --> 00:05:41.920
If you didn't have air
resistance, this number would

00:05:41.920 --> 00:05:44.300
be negative 4.9.

00:05:44.300 --> 00:05:47.410
This number I think turned out
to be something like 37.

00:05:47.410 --> 00:05:50.120
And this number turned out to--
well, this was 0, because

00:05:50.120 --> 00:05:51.720
you start at the ground.

00:05:51.720 --> 00:05:52.560
But this is interesting.

00:05:52.560 --> 00:05:55.410
This is actual data and actual
data is always going to be a

00:05:55.410 --> 00:05:58.030
little bit different, or
sometimes a lot different than

00:05:58.030 --> 00:06:00.990
the theoretical ideal, if you
had no air resistance.

00:06:00.990 --> 00:06:03.010
So that's what we're
getting right here.

00:06:03.010 --> 00:06:04.860
And so we can now use this.

00:06:04.860 --> 00:06:07.570
This is telling me, my graphing
calculator's telling

00:06:07.570 --> 00:06:10.970
me, that the quadratic I should
use-- you know, y is a

00:06:10.970 --> 00:06:16.980
function of x-- is negative
5.2x squared

00:06:16.980 --> 00:06:25.260
plus 35.99x plus 0.29.

00:06:25.260 --> 00:06:26.700
I rounded them off
a little bit.

00:06:26.700 --> 00:06:28.840
So now we can use this
information, or maybe I

00:06:28.840 --> 00:06:29.670
should say of t.

00:06:29.670 --> 00:06:32.570
I could have made this-- well,
x is equal to time.

00:06:32.570 --> 00:06:34.076
I could have made it
t instead, but I

00:06:34.076 --> 00:06:35.140
think you get the idea.

00:06:35.140 --> 00:06:38.150
And now we can use this
information to figure out what

00:06:38.150 --> 00:06:40.900
the ball was probably doing
at other points in time.

00:06:40.900 --> 00:06:43.330
So, for example, if we wanted
to know what was

00:06:43.330 --> 00:06:47.870
happening at time 5.

00:06:47.870 --> 00:06:52.280
What's y of, let's say
5.2 seconds into it?

00:06:52.280 --> 00:06:54.140
So what happens a little
bit further off?

00:06:54.140 --> 00:06:58.400
We're extrapolating, we're going
beyond what the data

00:06:58.400 --> 00:06:59.370
that we have right here.

00:06:59.370 --> 00:07:01.470
So y of 5.2.

00:07:01.470 --> 00:07:02.320
We can do it two ways.

00:07:02.320 --> 00:07:04.110
We can look at it graphically.

00:07:04.110 --> 00:07:06.270
So now that we've calculated
this regression, here are the

00:07:06.270 --> 00:07:07.530
coefficients again.

00:07:07.530 --> 00:07:09.450
As you see here, I'm just
scrolling to the left.

00:07:09.450 --> 00:07:10.300
It gave me the coefficients.

00:07:10.300 --> 00:07:17.100
Negative 5.2, 35.99, and then
it gave me the 0.2925.

00:07:17.100 --> 00:07:19.580
So now that it calculated, we
could actually plot all the

00:07:19.580 --> 00:07:22.970
points and draw the
regression.

00:07:22.970 --> 00:07:25.330
I want to pick this draw
option up here.

00:07:25.330 --> 00:07:29.090
So I can either exit and
then I pick draw.

00:07:29.090 --> 00:07:31.570
And first I'll just
draw the data.

00:07:31.570 --> 00:07:32.430
So that's scattered.

00:07:32.430 --> 00:07:34.610
It'll make a scattered plot
of all of the data.

00:07:34.610 --> 00:07:35.520
I select it there.

00:07:35.520 --> 00:07:39.730
And notice, it drew the data
points as they were given.

00:07:39.730 --> 00:07:40.750
That's the data points
right there.

00:07:40.750 --> 00:07:43.340
5.2 is going to be out
here some place.

00:07:43.340 --> 00:07:45.790
And now I can draw the
actual regression.

00:07:45.790 --> 00:07:48.270
This is draw the regression
using these

00:07:48.270 --> 00:07:49.850
coefficients right here.

00:07:49.850 --> 00:07:52.070
So draw regression.

00:07:52.070 --> 00:07:54.880
And you see it actually
fits the data very,

00:07:54.880 --> 00:07:56.960
very, very, very well.

00:07:56.960 --> 00:08:00.330
So let's use this function that
seems to fit the data

00:08:00.330 --> 00:08:04.170
very, very, very well to
actually come up with how high

00:08:04.170 --> 00:08:07.050
was the ball at 5.2 seconds?

00:08:07.050 --> 00:08:09.600
So let's exit from here.

00:08:09.600 --> 00:08:10.670
And let's just calculate it.

00:08:10.670 --> 00:08:23.490
It is negative 5.2 times
5.2 squared.

00:08:23.490 --> 00:08:26.450
Just by coincidence, I picked a
time that equals that first

00:08:26.450 --> 00:08:26.610
coefficient.

00:08:26.610 --> 00:08:28.670
Therefore, it's the positive
version of it.

00:08:28.670 --> 00:08:36.650
So 5.2 squared plus 35.99 times
5.2-- remember, we're

00:08:36.650 --> 00:08:38.900
dealing with when time is equal
to 5.2, or where x is

00:08:38.900 --> 00:08:43.659
equal to 5.2-- plus 0.29.

00:08:43.659 --> 00:08:46.510
So we're just going to evaluate
this function at time

00:08:46.510 --> 00:08:49.600
or at x is equal to 5.2.

00:08:49.600 --> 00:08:54.870
And we get 46.83 feet.

00:08:54.870 --> 00:09:01.580
So it's equal to 46.83, or
I should say 0.83 meters.

00:09:01.580 --> 00:09:02.370
So I hope you enjoyed that.

00:09:02.370 --> 00:09:04.620
I just wanted to show you a
quick thing of how a graphing

00:09:04.620 --> 00:09:05.580
calculator is useful.

00:09:05.580 --> 00:09:08.160
Maybe in the future, I'll make
a more advanced video on the

00:09:08.160 --> 00:09:12.580
actual algorithm or the actual
process that the calculator

00:09:12.580 --> 00:09:16.160
used to figure out
this best curve.


Identifying Exponential Models

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:03.110
Two videos ago, I told you that
when you're dealing with

00:00:03.110 --> 00:00:07.280
an exponential function, that
when you increase x by a

00:00:07.280 --> 00:00:11.720
constant amount that the y
values will increase by a

00:00:11.720 --> 00:00:12.840
constant factor.

00:00:12.840 --> 00:00:16.580
When we increased x from
0 to 1, we increased y

00:00:16.580 --> 00:00:18.300
by a factor of 1.5.

00:00:18.300 --> 00:00:21.650
We multiplied 1.5 times
120 to get 180.

00:00:21.650 --> 00:00:25.470
When we increased x by 1 again,
once again, y increased

00:00:25.470 --> 00:00:27.610
by a factor of 1.5.

00:00:27.610 --> 00:00:30.320
We multiplied 1.5 times 180.

00:00:30.320 --> 00:00:32.540
What I want to do in this video
is to kind of quickly

00:00:32.540 --> 00:00:35.980
show you just going from the
basic definition of an

00:00:35.980 --> 00:00:39.070
exponential function, why
this is the case.

00:00:39.070 --> 00:00:42.300
Why this works for exponential
functions.

00:00:42.300 --> 00:00:46.140
So let's just start with a
general exponential, y is

00:00:46.140 --> 00:00:49.120
equal to A times r to the x.

00:00:49.120 --> 00:00:52.050
And let's look at some x
values and then their

00:00:52.050 --> 00:00:53.350
corresponding y values.

00:00:56.270 --> 00:01:01.990
These are my x values, those
are my y values over there.

00:01:01.990 --> 00:01:07.100
Now let's say that x is equal
to, let me just pick an

00:01:07.100 --> 00:01:07.850
arbitrary number.

00:01:07.850 --> 00:01:09.980
Let's say that x
is equal to b.

00:01:09.980 --> 00:01:12.170
I haven't used that
variable yet.

00:01:12.170 --> 00:01:14.030
And what's y going
to be equal to?

00:01:14.030 --> 00:01:19.840
It's going to be equal to A
times r to the bth power.

00:01:19.840 --> 00:01:22.190
Now, let's increase x by 1.

00:01:22.190 --> 00:01:24.510
This would also work if you
increased x by really, just a

00:01:24.510 --> 00:01:25.400
constant factor.

00:01:25.400 --> 00:01:27.470
But let's increase it by 1.

00:01:27.470 --> 00:01:31.380
So we're now going to look at
x is equal to b plus 1.

00:01:31.380 --> 00:01:35.240
Then y is going to be equal
to A times r to

00:01:35.240 --> 00:01:37.560
the b plus 1 power.

00:01:37.560 --> 00:01:38.780
Let's do it one more time.

00:01:38.780 --> 00:01:40.570
Let's increase x by 1.

00:01:40.570 --> 00:01:43.460
You get b plus 2.

00:01:43.460 --> 00:01:45.640
And now Ax-- sorry.

00:01:45.640 --> 00:01:52.460
Now y would be A times r
to the b plus 2 power.

00:01:52.460 --> 00:01:55.320
Now let's see what happens when
you take the ratio of

00:01:55.320 --> 00:01:59.270
each successive y terms that
correspond to the x terms when

00:01:59.270 --> 00:02:01.060
you increase by a
constant factor.

00:02:01.060 --> 00:02:04.100
And notice, every time we
increased x by exactly 1.

00:02:04.100 --> 00:02:05.830
If this was 0, this
was 1, this is 2.

00:02:05.830 --> 00:02:08.360
If this is negative 5, this
would be negative 4, this be

00:02:08.360 --> 00:02:09.759
negative 3.

00:02:09.759 --> 00:02:12.090
So let's see what happens
with our y.

00:02:12.090 --> 00:02:17.000
So over here, if we look at the
factor that y changed by,

00:02:17.000 --> 00:02:26.070
we could just divide Ar to the
b plus 1 by Ar to the b and

00:02:26.070 --> 00:02:27.220
what do we get?

00:02:27.220 --> 00:02:29.460
Well you might already recognize
you could use

00:02:29.460 --> 00:02:31.690
straight up exponent
rules right here.

00:02:31.690 --> 00:02:36.270
The A's cancel out and then this
is going to be equal to r

00:02:36.270 --> 00:02:41.870
to the b plus 1 minus b.

00:02:41.870 --> 00:02:42.810
Same base.

00:02:42.810 --> 00:02:44.000
You're taking the quotient.

00:02:44.000 --> 00:02:45.830
You subtract exponents.

00:02:45.830 --> 00:02:48.200
So this is just going
to be equal to r.

00:02:51.280 --> 00:02:55.820
Let's see what happens when
we go from here to here.

00:02:55.820 --> 00:03:01.180
Once again, take the y value
when x is equal to b plus 2,

00:03:01.180 --> 00:03:03.940
so Ar to the b plus 2.

00:03:03.940 --> 00:03:06.590
Divide it by the previous
y value.

00:03:06.590 --> 00:03:11.450
Divide it by Ar to
the b plus 1.

00:03:11.450 --> 00:03:20.200
These cancel out and we're left
with r to the b plus 2

00:03:20.200 --> 00:03:28.370
minus b plus 1, which is equal
to r to the b plus 2

00:03:28.370 --> 00:03:30.540
minus b minus 1.

00:03:30.540 --> 00:03:31.580
These cancel out.

00:03:31.580 --> 00:03:32.790
2 minus 1.

00:03:32.790 --> 00:03:36.130
Once again, you're
just left with r.

00:03:36.130 --> 00:03:39.870
So hopefully that satisfies any
curiosity you might have

00:03:39.870 --> 00:03:45.060
had of why you're able to use
that as the way to evaluate

00:03:45.060 --> 00:03:46.830
whether this data
was exponential.

00:03:46.830 --> 00:03:49.050
Why we're able to just say,
hey, let's take this term

00:03:49.050 --> 00:03:52.790
divided by the previous term
and see what number we get.

00:03:52.790 --> 00:03:55.140
This term divided by that term
and if we get the same number

00:03:55.140 --> 00:03:58.760
every time, we're dealing with
an exponential function.


Identifying Quadratic Models

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:06.610
In the last video, I told you
that for a quadratic function,

00:00:06.610 --> 00:00:10.660
that the change in the change
of y is constant.

00:00:10.660 --> 00:00:13.060
What I mean by that-- this is
what we did in the last video.

00:00:13.060 --> 00:00:15.240
This was a quadratic,
this was data

00:00:15.240 --> 00:00:17.140
from a quadratic function.

00:00:17.140 --> 00:00:21.670
That when x increased from
1 to 2, y increased by 2.

00:00:21.670 --> 00:00:25.270
Then when we increased
x from 2 to 3, y

00:00:25.270 --> 00:00:26.220
didn't increase at all.

00:00:26.220 --> 00:00:27.690
It increased by 0.

00:00:27.690 --> 00:00:31.040
So the change in y was
clearly not constant.

00:00:31.040 --> 00:00:34.420
And then we went from 3 to 4,
y actually decreased by

00:00:34.420 --> 00:00:35.270
negative 2.

00:00:35.270 --> 00:00:39.110
So the change in y was not
constant, but the change in

00:00:39.110 --> 00:00:41.630
the changes of y-- we had a
change in y of 2, then a

00:00:41.630 --> 00:00:45.620
change in y of 0-- so our change
in the change of y was

00:00:45.620 --> 00:00:46.870
negative 2.

00:00:46.870 --> 00:00:49.450
Then we went from a
0 to a negative 2.

00:00:49.450 --> 00:00:52.910
So the change in the change
of y was negative 2.

00:00:52.910 --> 00:00:55.660
What I want to do in this video
is show you why that

00:00:55.660 --> 00:00:58.350
works for a quadratic
function.

00:00:58.350 --> 00:01:00.630
And just to make the math
simple, I'm going to pick the

00:01:00.630 --> 00:01:02.320
simplest of quadratic
functions.

00:01:02.320 --> 00:01:05.389
And if you have some free time
on your hands, you might try

00:01:05.389 --> 00:01:07.600
it out for the more
general case.

00:01:07.600 --> 00:01:10.560
But I'm going to show you
that it works for y

00:01:10.560 --> 00:01:13.050
is equal to x squared.

00:01:13.050 --> 00:01:14.950
And you could try it later
with ax squared

00:01:14.950 --> 00:01:16.360
plus bx plus c.

00:01:16.360 --> 00:01:19.080
And I'm going to increment
x by 1.

00:01:19.080 --> 00:01:21.410
This'll work as long
as you increase x

00:01:21.410 --> 00:01:22.930
by a constant amount.

00:01:22.930 --> 00:01:25.350
But 1 makes the math a
little bit simpler.

00:01:25.350 --> 00:01:27.380
So let's pick some x's.

00:01:30.810 --> 00:01:32.165
I'm going to keep things
a little bit general.

00:01:32.165 --> 00:01:35.710
So it's going to be very
abstract and algebraicy for a

00:01:35.710 --> 00:01:36.130
little bit.

00:01:36.130 --> 00:01:37.860
But I think you might
find it satisfying.

00:01:37.860 --> 00:01:40.690
Then we'll figure out
our y's given the x.

00:01:40.690 --> 00:01:43.330
So let's say that x is x1.

00:01:43.330 --> 00:01:45.600
So when I write this little
subscript 1 here, that means

00:01:45.600 --> 00:01:49.070
I'm picking a particular x, a
particular example of the

00:01:49.070 --> 00:01:53.680
variable x, a particular x in
the domain of this function.

00:01:53.680 --> 00:01:56.500
So in that situation, what's
y going to be equal to?

00:01:56.500 --> 00:02:00.060
Well, y is just going to
be equal to x1 squared.

00:02:00.060 --> 00:02:03.900
Let me do that in orange, keep
the colors consistent.

00:02:03.900 --> 00:02:05.550
x1 squared.

00:02:05.550 --> 00:02:08.550
Now let's say we increment
x by 1.

00:02:08.550 --> 00:02:11.590
So our next one is going
to be x1 plus 1.

00:02:11.590 --> 00:02:14.030
I'm just going 1 above
that number there.

00:02:14.030 --> 00:02:15.990
If this was 3, this
is going to be 4.

00:02:15.990 --> 00:02:18.880
If this was 0, this
is going to be 1.

00:02:18.880 --> 00:02:24.240
Then this is going to be x1 plus
1 squared, which is equal

00:02:24.240 --> 00:02:30.930
to x1 squared plus 2x1 plus 1.

00:02:30.930 --> 00:02:34.090
All right, let's increase
this by 1.

00:02:34.090 --> 00:02:37.320
So then we get to x1 plus 2.

00:02:37.320 --> 00:02:39.600
If this is 0, this
is 1, this is 2.

00:02:39.600 --> 00:02:42.730
If this is 10, this
is 11, this is 12.

00:02:42.730 --> 00:02:48.570
So then y, when we have this
x value, will be x1 plus 2

00:02:48.570 --> 00:02:56.205
squared, which is equal to x1
squared plus 4x1 plus 4.

00:02:56.205 --> 00:02:59.680
Let's do two more actually.

00:02:59.680 --> 00:03:00.830
Now let's increase
by another 1.

00:03:00.830 --> 00:03:02.680
I think you see what
I'm doing.

00:03:02.680 --> 00:03:05.000
x1 plus 3.

00:03:05.000 --> 00:03:08.250
The y value when x is x1
plus 3 is going to

00:03:08.250 --> 00:03:12.260
be x1 plus 3 squared.

00:03:12.260 --> 00:03:20.230
Which is equal to x1 squared
plus 6x1, plus 9.

00:03:20.230 --> 00:03:22.170
Let's do one more.

00:03:22.170 --> 00:03:26.550
So when x is going to be-- let's
increment it by 1 more--

00:03:26.550 --> 00:03:33.970
x1 plus 4, y is going to be x1
plus 4 squared, which is equal

00:03:33.970 --> 00:03:41.560
to x1 squared plus
8x1, plus 16.

00:03:41.560 --> 00:03:45.680
Now, given this, let's figure
out the changes in y.

00:03:45.680 --> 00:03:49.120
So let me write here
the change in y.

00:03:49.120 --> 00:03:52.510
The change in y as we increment
x by 1 each time.

00:03:52.510 --> 00:03:56.230
So here the change in y, we'll
take this point or this y

00:03:56.230 --> 00:03:58.010
value minus this y value.

00:03:58.010 --> 00:04:01.720
If we take this y value minus
this y value, so from here to

00:04:01.720 --> 00:04:09.110
here, our change in y is going
to be x1 squared plus 2x1,

00:04:09.110 --> 00:04:14.312
plus 1, minus x1 squared.

00:04:14.312 --> 00:04:19.899
So these cancel out and we
just get 2x1 plus 1.

00:04:19.899 --> 00:04:22.970
Then when we go from this
point to this point, our

00:04:22.970 --> 00:04:31.670
change in why is going to be x1
squared plus 4x1, plus 4,

00:04:31.670 --> 00:04:37.960
minus x1 squared, minus
2x1, minus 1.

00:04:37.960 --> 00:04:40.050
So what is this going
to be equal to?

00:04:40.050 --> 00:04:45.630
This is going to be equal to--
these guys cancel out-- 4x1

00:04:45.630 --> 00:04:48.480
minus 2x1 is 2x1.

00:04:48.480 --> 00:04:50.790
And then 4 minus 1 is 3.

00:04:50.790 --> 00:04:53.540
So plus 3.

00:04:53.540 --> 00:04:55.330
Now let's do the next one.

00:04:55.330 --> 00:04:59.740
When I go from this y value to
that y value, change in y is

00:04:59.740 --> 00:05:08.460
going to be x1 squared plus
6x1, plus 9, minus

00:05:08.460 --> 00:05:09.910
this guy over here.

00:05:09.910 --> 00:05:14.560
x1 squared plus 4x1, plus 4.

00:05:14.560 --> 00:05:16.860
These cancel out.

00:05:16.860 --> 00:05:19.250
Sorry, I have to distribute
the negative sign.

00:05:19.250 --> 00:05:20.670
I'm subtracting this
whole thing.

00:05:20.670 --> 00:05:23.950
Put a minus there and a minus
negative sign there.

00:05:23.950 --> 00:05:29.070
6x1 minus 4x1, that is 2x1.

00:05:29.070 --> 00:05:30.980
And then I have 9 minus 4.

00:05:30.980 --> 00:05:32.450
So plus 5.

00:05:32.450 --> 00:05:33.780
So this is the change in y.

00:05:33.780 --> 00:05:36.240
The first change in y was that,
the second change in y

00:05:36.240 --> 00:05:38.260
is that, third change
in y is that.

00:05:38.260 --> 00:05:40.450
Let's do this one over here.

00:05:40.450 --> 00:05:46.820
So our change in y is x1 squared
plus 8x1, plus 16,

00:05:46.820 --> 00:05:50.360
minus this value over here.

00:05:50.360 --> 00:05:54.710
Minus x1 squared minus
6x1, minus 9.

00:05:54.710 --> 00:05:56.980
I'm distributing
the minus sign.

00:05:56.980 --> 00:05:58.440
That cancels out.

00:05:58.440 --> 00:06:02.410
8x1 minus 6x1 is 2x1.

00:06:02.410 --> 00:06:07.820
16 minus 9 is equal to 7.

00:06:07.820 --> 00:06:09.560
So plus 7.

00:06:09.560 --> 00:06:13.200
So our change in y's from here
to here is 2x plus 1, right

00:06:13.200 --> 00:06:15.360
there-- 2x1 plus 1.

00:06:15.360 --> 00:06:17.680
Then it's 2x1 plus 3.

00:06:17.680 --> 00:06:19.720
Then it's 2x1 plus 5.

00:06:19.720 --> 00:06:22.110
And then it's 2x1 plus 7.

00:06:22.110 --> 00:06:24.320
So what is the change
in the change of y?

00:06:24.320 --> 00:06:27.170
Notice, these are not
the same values.

00:06:27.170 --> 00:06:28.550
They're increasing.

00:06:28.550 --> 00:06:30.430
And so we're kind of
jumping the gun.

00:06:30.430 --> 00:06:37.200
What is the change in delta y?

00:06:37.200 --> 00:06:46.080
So if we go from 2x plus 1 to
2x plus 3, well, if you

00:06:46.080 --> 00:06:48.430
subtract that from that, the
2x's are going to cancel out.

00:06:48.430 --> 00:06:50.610
3 minus 1 is 2.

00:06:50.610 --> 00:06:55.290
If you go from 2x plus 3 to 2x
plus 5, once again, the 2x's

00:06:55.290 --> 00:06:55.840
cancel out.

00:06:55.840 --> 00:06:58.460
We've just increased by 2.

00:06:58.460 --> 00:07:04.530
If you go from 2x1 plus 5 to 2x1
plus 7, once again, you're

00:07:04.530 --> 00:07:06.750
increasing by 2.

00:07:06.750 --> 00:07:09.940
So for at least the situation
of y is equal to x squared,

00:07:09.940 --> 00:07:13.410
hopefully it satisfies you that
no matter which x1 you

00:07:13.410 --> 00:07:16.140
pick, as long as you keep
increasing by 1-- and it's

00:07:16.140 --> 00:07:18.920
actually true if you increase
by any constant number-- the

00:07:18.920 --> 00:07:22.150
change in your y values
will not be constant.

00:07:22.150 --> 00:07:25.950
Notice this is not that,
which is not that,

00:07:25.950 --> 00:07:27.230
which is not that.

00:07:27.230 --> 00:07:30.950
But the change in the change in
y's-- the change in y's are

00:07:30.950 --> 00:07:34.080
increasing in this example
at a constant rate.

00:07:34.080 --> 00:07:36.980
So this was completely analogous
to what we did here.

00:07:36.980 --> 00:07:38.720
And if you have some time on
your hands, you can actually

00:07:38.720 --> 00:07:41.250
prove this or show it or kind
of prove, you know, get

00:07:41.250 --> 00:07:44.470
comfort that this will work for
the more general case of y

00:07:44.470 --> 00:07:47.840
is equal to ax squared or ax
squared plus bx plus c.

00:07:47.840 --> 00:07:51.340
Or it will work for ax squared
plus bx plus c, if we

00:07:51.340 --> 00:07:54.290
increment just by any arbitrary
constant here.

00:07:54.290 --> 00:07:56.990
So you could put a k, 2k, 3k.

00:07:56.990 --> 00:07:58.480
The algebra gets a little
bit hairier, that's

00:07:58.480 --> 00:07:59.460
why I didn't do it.

00:07:59.460 --> 00:08:01.085
But it should still work.

00:08:01.085 --> 00:08:04.010
So hopefully you found that
kind of satisfying.


Linear, Quadratic, and Exponential Models

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:02.870
In this video, we're going
to play a bit of a game.

00:00:02.870 --> 00:00:06.210
I have three different data
sets here, and we need to

00:00:06.210 --> 00:00:10.000
figure out which of these data
sets come from either a

00:00:10.000 --> 00:00:18.893
linear, quadratic, or
exponential function.

00:00:22.800 --> 00:00:25.840
And just as a reminder, linear
function is of the form, y is

00:00:25.840 --> 00:00:28.110
equal to mx plus b.

00:00:28.110 --> 00:00:33.510
A quadratic is of the form, y
is equal to ax squared plus

00:00:33.510 --> 00:00:36.020
bx, plus c.

00:00:36.020 --> 00:00:40.570
And an exponential is of the
form, y is equal to some

00:00:40.570 --> 00:00:43.210
constant-- let's call
it a-- times some

00:00:43.210 --> 00:00:46.230
number to the x power.

00:00:46.230 --> 00:00:48.450
Now, the way to tell them
apart-- and I'm going to

00:00:48.450 --> 00:00:49.170
actually show you.

00:00:49.170 --> 00:00:52.330
It'll probably be intuitive for
you for a linear function,

00:00:52.330 --> 00:00:53.890
and maybe even for
an exponential.

00:00:53.890 --> 00:00:56.760
The quadratic, the technique I'm
going to use might not be

00:00:56.760 --> 00:00:59.570
intuitive, and I'll actually
prove it to you in the next

00:00:59.570 --> 00:01:02.160
video, or at least show you an
example in the general case of

00:01:02.160 --> 00:01:03.630
why it makes sense.

00:01:03.630 --> 00:01:05.820
But the way to tell a linear
function, is that when you

00:01:05.820 --> 00:01:09.810
increment x by a constant
amount, y will always change

00:01:09.810 --> 00:01:11.040
by a constant amount.

00:01:11.040 --> 00:01:14.690
So when you always increase x by
1, y should always increase

00:01:14.690 --> 00:01:16.570
by a certain amount.

00:01:16.570 --> 00:01:22.220
So just using that as our first
guideline, let's see

00:01:22.220 --> 00:01:27.600
which of these represent
a linear function.

00:01:27.600 --> 00:01:29.970
So in all of these examples, as
we go from one data point

00:01:29.970 --> 00:01:31.955
to the next, we're increasing
x by 1.

00:01:31.955 --> 00:01:33.880
And it's very important that
you pay attention to that,

00:01:33.880 --> 00:01:36.760
because if you weren't, you
would have to adjust things a

00:01:36.760 --> 00:01:37.090
little bit.

00:01:37.090 --> 00:01:40.920
But in all of these, we're
always increasing x by 1.

00:01:40.920 --> 00:01:42.900
So let's look at our
change in y's.

00:01:42.900 --> 00:01:44.940
When we go from the first data
point to the second data

00:01:44.940 --> 00:01:47.750
point, our change in y is 60.

00:01:47.750 --> 00:01:49.400
When we go to the second
to the third, our

00:01:49.400 --> 00:01:51.680
change in y is 90.

00:01:51.680 --> 00:01:54.630
So that immediately tells us
we're not dealing with a

00:01:54.630 --> 00:01:55.590
linear function.

00:01:55.590 --> 00:01:59.580
When our change in x is 1, our
change in y here is 60.

00:01:59.580 --> 00:02:04.230
Then it's 90, and as you go here
it's like a 135, so this

00:02:04.230 --> 00:02:07.090
one is clearly not linear.

00:02:07.090 --> 00:02:09.030
Let's look at this
one over here.

00:02:09.030 --> 00:02:12.600
Our first change in y
is 2, then our next

00:02:12.600 --> 00:02:14.230
change in y is 0.

00:02:14.230 --> 00:02:15.700
Once again, not linear.

00:02:15.700 --> 00:02:20.370
As we increment x by a constant
amount, our y does

00:02:20.370 --> 00:02:22.780
not increase or decrease
at a constant amount.

00:02:22.780 --> 00:02:25.740
So once again, this
one is not linear.

00:02:25.740 --> 00:02:26.650
Now let's look over here.

00:02:26.650 --> 00:02:30.850
Once again, we're increasing
x by 1 on every data point.

00:02:30.850 --> 00:02:32.740
And when we to go from this
first data point to the second

00:02:32.740 --> 00:02:35.990
data point, y decreases by 3.

00:02:35.990 --> 00:02:38.380
Then we go from the second to
the third, once again, y

00:02:38.380 --> 00:02:39.980
decreases by 3.

00:02:39.980 --> 00:02:42.760
Third to the fourth,
y decreases by 3.

00:02:42.760 --> 00:02:46.350
1 to negative 2, once again,
decreases by 3.

00:02:46.350 --> 00:02:50.170
And we just keep decreasing by
3, so whenever we change x by

00:02:50.170 --> 00:02:55.980
1, when we increase x by 1,
y goes down by negative 3.

00:02:55.980 --> 00:02:57.855
So we actually have
a constant slope.

00:02:57.855 --> 00:03:02.180
We can say our change in y over
change in x is equal to

00:03:02.180 --> 00:03:06.250
when x goes up by 1,
y goes down by 3.

00:03:06.250 --> 00:03:08.110
This is a definition of slope.

00:03:08.110 --> 00:03:10.620
This is the m in this
equation right here.

00:03:10.620 --> 00:03:14.315
Our changes are constant, so
this one right here is linear.

00:03:18.140 --> 00:03:20.620
Now quadratic, there's an
interesting, we could call it

00:03:20.620 --> 00:03:22.230
a trick right now, because
I haven't really

00:03:22.230 --> 00:03:23.810
showed you why it works.

00:03:23.810 --> 00:03:30.680
But in a quadratic, your changes
in y are not going to

00:03:30.680 --> 00:03:32.850
be constant, as was the
case in the linear.

00:03:32.850 --> 00:03:36.600
But the change in the changes
of y will be constant.

00:03:36.600 --> 00:03:37.380
Let me write that down.

00:03:37.380 --> 00:03:39.320
It probably makes no sense to
you right now, but when you

00:03:39.320 --> 00:03:41.830
see an example it'll
be kind of fun.

00:03:41.830 --> 00:03:52.315
Change in change
in y constant.

00:03:55.120 --> 00:03:56.770
Let me show you what I mean.

00:03:56.770 --> 00:03:59.080
So over here, let's look at
this first one over here.

00:03:59.080 --> 00:04:02.300
So we went from 120 to 180, that
was an increase of 60.

00:04:02.300 --> 00:04:05.410
180 to 270, that was
an increase of 90.

00:04:05.410 --> 00:04:12.230
Then we went from 270 to 405,
which is an increase of 135.

00:04:12.230 --> 00:04:14.960
Now, when I talk about the
change in the change of y,

00:04:14.960 --> 00:04:17.620
these are the change in y's,
what are the change in the

00:04:17.620 --> 00:04:18.779
change of y's?

00:04:18.779 --> 00:04:22.210
Well, we had a change of 60,
then we had a change of 90, so

00:04:22.210 --> 00:04:27.050
that was a change in the
change of y of 30, or a

00:04:27.050 --> 00:04:29.460
difference of the difference,
you could view them that way

00:04:29.460 --> 00:04:31.460
if you like.

00:04:31.460 --> 00:04:33.210
Then we had a change of 90, and
then we had a change of a

00:04:33.210 --> 00:04:39.180
135, so the change in the
change, let's see, that is 45.

00:04:39.180 --> 00:04:42.030
So here, the change in the
change in y, or the difference

00:04:42.030 --> 00:04:44.290
of the difference
is not constant.

00:04:44.290 --> 00:04:48.760
We grew by 60, and then we
grew by 90, which was

00:04:48.760 --> 00:04:49.870
30 more than 60.

00:04:49.870 --> 00:04:52.780
Then we grew by 135, which
is 45 more than 90.

00:04:52.780 --> 00:04:54.630
So this is not constant
over here.

00:04:54.630 --> 00:04:58.240
So this tells us this is not
a quadratic function.

00:04:58.240 --> 00:05:00.680
Now let's look at
this over here.

00:05:00.680 --> 00:05:03.580
The change in y-- let's
write that down.

00:05:03.580 --> 00:05:04.910
This clearly was not linear.

00:05:04.910 --> 00:05:09.480
We increased by 2 here,
increased by 0 here, increased

00:05:09.480 --> 00:05:13.670
by negative 2 here, increased by
negative 4 here, increased

00:05:13.670 --> 00:05:15.600
by negative 6 there.

00:05:15.600 --> 00:05:17.930
The change in y's are definitely
not constant.

00:05:17.930 --> 00:05:20.590
But let's look at the change
in the change of y's.

00:05:20.590 --> 00:05:24.350
So we increased by 2, then we
increased by 0, so this was a

00:05:24.350 --> 00:05:27.560
negative 2 in the change in
the change of y, or the

00:05:27.560 --> 00:05:29.270
difference of the difference.

00:05:29.270 --> 00:05:33.020
Then when you go from 0 to
negative 2, we changed by 0,

00:05:33.020 --> 00:05:36.485
then we changed by negative 2,
that is a change in the change

00:05:36.485 --> 00:05:38.050
of negative 2.

00:05:38.050 --> 00:05:40.490
Then you go from negative 2 to
negative 4, the change in the

00:05:40.490 --> 00:05:42.220
change of negative 2.

00:05:42.220 --> 00:05:43.730
This one is starting
to look quadratic.

00:05:43.730 --> 00:05:47.120
We changed by negative 4 than we
changed by negative 6, once

00:05:47.120 --> 00:05:50.610
again, a change in the
change of negative 2.

00:05:50.610 --> 00:05:56.580
So this data right here fits our
requirement for quadratic,

00:05:56.580 --> 00:05:59.490
and I'm just kind of giving it
to you as a trick right now.

00:05:59.490 --> 00:06:00.560
In the next video I'll
actually show

00:06:00.560 --> 00:06:01.230
you why this works.

00:06:01.230 --> 00:06:03.470
I'll show you an example, just
so you can get the gut feeling

00:06:03.470 --> 00:06:04.890
of why it should work.

00:06:04.890 --> 00:06:09.300
So this one right here
is quadratic.

00:06:09.300 --> 00:06:11.930
So you might say, hey, Sal has
given us three examples, one

00:06:11.930 --> 00:06:13.400
of them is going to be linear,
one of them's going to be

00:06:13.400 --> 00:06:16.100
quadratic, one's going to be
exponential, this one's

00:06:16.100 --> 00:06:17.810
probably going to
be exponential.

00:06:17.810 --> 00:06:19.535
And you are probably right.

00:06:19.535 --> 00:06:22.690
But let me give you the way
that we can figure out if

00:06:22.690 --> 00:06:24.890
something is exponential.

00:06:24.890 --> 00:06:29.580
So when x increases by 1 in an
exponential, you should always

00:06:29.580 --> 00:06:32.790
have a constant multiplying
factor.

00:06:32.790 --> 00:06:35.520
So when you go from one term to
the next term, they should

00:06:35.520 --> 00:06:39.210
be a similar ratio from
one to the next.

00:06:39.210 --> 00:06:41.530
So instead of just subtracting--
in this case,

00:06:41.530 --> 00:06:45.040
120 from 180, and 180 from
270-- to test for an

00:06:45.040 --> 00:06:49.220
exponential function, what you
want to do is check whether

00:06:49.220 --> 00:06:52.340
the ratio of when you increment
x by a constant

00:06:52.340 --> 00:06:55.550
amount, whether the ratio
of the y's are the same.

00:06:55.550 --> 00:07:02.760
So when we increased x by 1, y
went from 120 to 180, so it

00:07:02.760 --> 00:07:06.760
increased by 18 over
12 is 1.5.

00:07:06.760 --> 00:07:08.970
So we went from this
term to this term.

00:07:08.970 --> 00:07:10.755
Let me clear all of the
stuff out here.

00:07:16.180 --> 00:07:20.690
So when we went from here to
here, this was times 1.5.

00:07:20.690 --> 00:07:24.310
And then when we go from 180 to
270, once again, that looks

00:07:24.310 --> 00:07:26.300
like we're going times
1.5, right?

00:07:26.300 --> 00:07:30.070
1/2 of 180 is 90, add that to
270, so once again, we're

00:07:30.070 --> 00:07:32.490
increasing by 1.5.

00:07:32.490 --> 00:07:37.360
270 to 405, 1/2 of 270 is 135,
that's how much we're

00:07:37.360 --> 00:07:38.120
increasing by.

00:07:38.120 --> 00:07:40.580
So once again, we're
increasing by 1.5.

00:07:40.580 --> 00:07:44.390
So every step of this way, we
are multiplying the previous

00:07:44.390 --> 00:07:46.520
term by 1.5 to get the next.

00:07:46.520 --> 00:07:53.200
And that actually tells us that
this right here is a 1.5.

00:07:53.200 --> 00:07:56.820
But our point here isn't to
actually figure out the

00:07:56.820 --> 00:07:58.990
formulas, the point here
is to classify them.

00:07:58.990 --> 00:08:03.080
So now we know definitively that
this right here, at least

00:08:03.080 --> 00:08:09.950
given the data we have, is
an exponential function.

00:08:09.950 --> 00:08:14.800
Anyway, hopefully you
found that fun.


Khan Academy Vision and Social Return

WEBVTT
Kind: captions
Language: en

00:00:00.945 --> 00:00:05.884
Voiceover: What I want to go
over in this video is what Khan
Academy is in its current form

00:00:05.884 --> 00:00:10.450
and where I see it going,
what I hope it will grow into.

00:00:10.450 --> 00:00:17.872
Most of you are probably already
familiar with the video library
that's available at Khanacademy.org.

00:00:17.872 --> 00:00:25.552
There already about 1,400 videos
covering everything from basic
mathematics to college level calculus,

00:00:25.552 --> 00:00:29.604
biology, chemistry,
differential equations,

00:00:29.604 --> 00:00:32.252
there's even a piece on economics,

00:00:32.252 --> 00:00:35.197
but I really see it growing
well beyond the video library

00:00:35.197 --> 00:00:37.678
and I've already started
working on the software piece.

00:00:37.678 --> 00:00:41.126
This right here, this
map structure you see,

00:00:41.126 --> 00:00:43.526
this is the Khan Academy knowledge map,

00:00:43.526 --> 00:00:48.434
(writing) Knowledge, knowledge map.

00:00:48.434 --> 00:00:58.076
and you can access it at
Khanexercises.appspot.com

00:01:00.760 --> 00:01:02.399
You just need a Google
ID and you can log in,

00:01:02.399 --> 00:01:06.199
it's completely free and
hopefully it always will be.

00:01:06.199 --> 00:01:11.011
If you click on the knowledge
map tab you'll see something
like this for yourself.

00:01:11.011 --> 00:01:16.618
This shows you, essentially, the
current curriculum of the software.

00:01:16.618 --> 00:01:19.897
I want to be very clear, the
videos, there's 1,400 videos,

00:01:19.897 --> 00:01:22.243
far more extensive than
the software piece.

00:01:22.243 --> 00:01:25.575
The software piece right now, because
it frankly is more labor intensive,

00:01:25.575 --> 00:01:29.806
or since I've been the only labor, maybe I
should say since it's more Sal intensive,

00:01:29.806 --> 00:01:34.921
than the video piece there's only
about 70 software modules right now

00:01:34.921 --> 00:01:39.794
and this covers from basic
arithmetic through about Algebra II.

00:01:39.794 --> 00:01:43.254
The idea here is to start
everyone at the most basic level,

00:01:43.254 --> 00:01:44.718
a level that anyone can start at.

00:01:44.718 --> 00:01:48.817
This module right here is 1 + 1 = 2 or

00:01:48.817 --> 00:01:53.662
2 + 3 = 5 and as they get
streaks of 10 in a row right

00:01:53.662 --> 00:01:58.920
it keeps moving them up, at their own
pace, to more and more advanced topics.

00:01:58.920 --> 00:02:01.208
Over here we have level
4 linear equations,

00:02:01.208 --> 00:02:05.122
systems of equations, multiplying
expressions, so on and so forth.

00:02:05.122 --> 00:02:06.916
Just to get an idea of
what these look like

00:02:06.916 --> 00:02:09.718
and if you log in you can just click on
the modules and start working on them,

00:02:09.718 --> 00:02:12.966
but this is a screen shots
of one of the modules.

00:02:12.966 --> 00:02:16.324
This is a screen shot
of multiplying decimals.

00:02:16.324 --> 00:02:20.199
The paradigm that I think this
type of software introduces,

00:02:20.199 --> 00:02:21.925
although it might seem very subtle,

00:02:21.925 --> 00:02:25.832
I think it's quite transformational to
how we traditionally teach and learn.

00:02:25.832 --> 00:02:28.755
The traditional model
is you have a lecture,

00:02:28.755 --> 00:02:31.920
you have a lecture that normally occurs

00:02:31.920 --> 00:02:35.197
while everyone's in a room
together at some scheduled time.

00:02:35.197 --> 00:02:39.322
Then you have practice in the form of
homework that normally goes on at home,

00:02:39.322 --> 00:02:42.832
usually on your own and if you
get stuck there's not much help

00:02:42.832 --> 00:02:44.257
and you get very little feedback.

00:02:44.257 --> 00:02:46.798
The book might have every
other problems answer,

00:02:46.798 --> 00:02:49.677
but you have no idea why
you got it right or wrong.

00:02:49.677 --> 00:02:53.952
Then you have some type of
assessment, some type of quiz.

00:02:53.952 --> 00:03:00.250
You maybe get 80 or 90% right
on that quiz, you are labeled
a B student or an A student

00:03:00.250 --> 00:03:02.803
and then you move on to the next topic.

00:03:02.803 --> 00:03:04.453
(writing) Next topic.

00:03:04.453 --> 00:03:10.774
And because of this structure, and
it's no individuals fault, it's more
mainly a bi-product of the system,

00:03:10.774 --> 00:03:15.747
everyone has to learn at the
same pace, the lecture is 1
to 30 people at their pace.

00:03:15.747 --> 00:03:19.353
It doesn't matter if half the class
is lost or half the class is bored,

00:03:19.353 --> 00:03:20.932
that's what everyone's getting.

00:03:20.932 --> 00:03:24.128
No matter how good of a teacher you
are you have to cater to everyone,

00:03:24.128 --> 00:03:28.295
so some percentage will be lost
and some percentage will be bored.

00:03:28.295 --> 00:03:32.453
Then no mater whether you
get 10% wrong or 20% wrong

00:03:32.453 --> 00:03:37.337
or 30% wrong as long as you are
"passing" you move on to the next topic.

00:03:37.337 --> 00:03:42.333
Anyone familiar with math or science knows
that if you have some core weakenesses,

00:03:42.333 --> 00:03:46.694
if there's 10% of exponent
problems you didn't know how to do,

00:03:46.694 --> 00:03:49.937
later on when you move
to more advanced topics

00:03:49.937 --> 00:03:53.654
that's going to be a major
hole in your foundation,

00:03:53.654 --> 00:03:58.067
it's going to make it almost impossible
to master more advanced topics.

00:03:58.067 --> 00:03:59.828
So, this is the traditional model.

00:03:59.828 --> 00:04:03.487
What this software is advocating
or maybe what I am advocating,

00:04:03.487 --> 00:04:09.255
is a model where all of this
is happening simultaneously in
a very data rich environment.

00:04:09.255 --> 00:04:13.152
You start off with instructions,
that's the whole point of lectures.

00:04:13.152 --> 00:04:17.248
When you do it within this
framework you're getting your
instruction from the videos.

00:04:17.248 --> 00:04:19.664
(writing) You're getting your
instruction from the videos.

00:04:19.664 --> 00:04:24.284
Let's say you're seeing the
multiplying decimals module for
the first time in your life,

00:04:24.284 --> 00:04:28.273
so you're right here, right
here on the knowledge map.

00:04:28.273 --> 00:04:31.448
You've just gotten 10 in a row
right on level 4 multiplication,

00:04:31.448 --> 00:04:34.984
so it is now suggesting that you
work on multiplying decimals,

00:04:34.984 --> 00:04:37.682
this will be orange in your knowledge map.

00:04:37.682 --> 00:04:41.490
So you're seeing this for the first
time, you have no idea how to do it,

00:04:41.490 --> 00:04:45.154
you can click on watch video
and this window will pop up

00:04:45.154 --> 00:04:49.737
and you can watch one of, well in this
case, the multiplying decimals video

00:04:49.737 --> 00:04:53.515
from Khanacademy.org it
will be pumped directly in.

00:04:53.515 --> 00:04:57.383
Once you watch it you can start
doing some practice problems.

00:04:57.383 --> 00:05:00.276
Once again the paradigm
here is very different.

00:05:00.276 --> 00:05:03.955
If you don't know how to do this
particular problem you can click hint

00:05:03.955 --> 00:05:08.287
and it will give you the exact
steps, step by step for this problem.

00:05:08.287 --> 00:05:11.923
So, when the person, or the viewer first
went here they just saw the problem

00:05:11.923 --> 00:05:15.581
and all of this highlighting
and all of that, that happened
after they started clicking hint.

00:05:15.581 --> 00:05:21.920
I encourage you to experiment on it,
with it on Khanexercises.appspot.com.

00:05:21.920 --> 00:05:29.939
Once you start getting good
at it, you don't move on
after only getting 90% or 80%

00:05:29.939 --> 00:05:33.523
and it labels you as a B+
student or as an A- student.

00:05:33.523 --> 00:05:37.861
This says I want you to keep doing
these multiplying decimal problems

00:05:37.861 --> 00:05:42.542
and I, the software, will keep generating
multiplying decimal problems for you,

00:05:42.542 --> 00:05:46.799
even if it takes you a thousand
problems until you get 10 in a row.

00:05:46.799 --> 00:05:49.619
(writing) 10 in a row.

00:05:49.619 --> 00:05:53.784
The paradigm or the idea
that I'm trying to advocate

00:05:53.784 --> 00:05:58.248
is for you not to move on until you
have complete proficiency in a concept.

00:05:58.248 --> 00:06:00.422
Right here on this streak
as you get problems right,

00:06:00.422 --> 00:06:03.607
you can try it out, the stars
fill up until you get 10 in a row.

00:06:03.607 --> 00:06:05.067
Once you get 10 in a row it will say,

00:06:05.067 --> 00:06:10.644
"Hey you know what you're doing, we think
you should move on to the next concept."

00:06:10.644 --> 00:06:13.648
The other area where this is kind
of ... this is the user experience,

00:06:13.648 --> 00:06:18.588
but behind the scenes here we have a
super, super data rich environment.

00:06:18.588 --> 00:06:20.884
Just to give you an idea
of the potential here ...

00:06:20.884 --> 00:06:23.660
Everything you're seeing,
this is already built,

00:06:23.660 --> 00:06:26.076
but I think this can all be
taken to even another level.

00:06:26.076 --> 00:06:29.192
This right here is data
I get from YouTube.

00:06:29.192 --> 00:06:33.248
(writing) That right there
is data I get from YouTube.

00:06:33.248 --> 00:06:36.248
This really is, on some level,
the educational holy grail.

00:06:36.248 --> 00:06:41.320
This is the actual data on
the Khan Academy balancing
chemical equations videos.

00:06:41.320 --> 00:06:43.191
This is the video here,

00:06:43.191 --> 00:06:48.516
but right here YouTube shows me the
attention span as someone watches videos.

00:06:48.516 --> 00:06:52.830
(writing) This is attention,
this is attention.

00:06:52.830 --> 00:06:57.530
They're measuring it against the
average YouTube video of this length.

00:06:57.530 --> 00:07:03.405
Let's say this is 7 minutes,
so at 7 minutes into the
video I'm doing pretty well,

00:07:03.405 --> 00:07:04.963
I'm still above the average.

00:07:04.963 --> 00:07:08.725
There's always a natural drop off rate,
but my drop off rate is doing better

00:07:08.725 --> 00:07:13.878
and actually it improves relative
to the average as we go further
and further through the video.

00:07:13.878 --> 00:07:15.323
So you can imagine the potential here.

00:07:15.323 --> 00:07:19.498
You can keep recording videos
and see which ones have the
best audience attention.

00:07:19.498 --> 00:07:25.497
I've seen some of my videos where
this graphic looks more like this.

00:07:25.497 --> 00:07:27.195
(writing) Where the graphic
looks more like this.

00:07:27.195 --> 00:07:31.357
I can go back and say gee, something must
have happened right there in the video,

00:07:31.357 --> 00:07:35.025
at maybe minute 8, that
caused people to drop off,

00:07:35.025 --> 00:07:37.535
maybe I said the word
orthogonality without defining it

00:07:37.535 --> 00:07:40.993
or I skipped a step in an equation
and I can either annotate that video

00:07:40.993 --> 00:07:44.436
or I could re-record the video so that
the new version does something like ...

00:07:44.436 --> 00:07:46.410
(writing) does something like that.

00:07:46.410 --> 00:07:51.225
Just off of the viewer data, off of
YouTube, you can do tremendous things,

00:07:51.225 --> 00:07:53.688
things that were never
possible in education before.

00:07:53.688 --> 00:07:57.612
When you couple that, when you actually
couple that with the exercises,

00:07:57.612 --> 00:08:01.636
when you actually know when
they watched the video, in
the context of what exercise,

00:08:01.636 --> 00:08:05.263
how they did on the exercises before
and after watching the videos,

00:08:05.263 --> 00:08:10.737
how many times they clicked on
hint, then all of the sudden you
can do far more powerful things.

00:08:10.737 --> 00:08:14.660
This right here is actual
data collected for a teacher.

00:08:14.660 --> 00:08:18.594
This is a daily spreadsheet
that is automatically generated

00:08:18.594 --> 00:08:21.883
for a teacher using Khan
Academy at a summer camp.

00:08:21.883 --> 00:08:23.927
These are all of the students.

00:08:23.927 --> 00:08:25.732
I've blurred them out for their privacy.

00:08:25.732 --> 00:08:29.595
Each column here is one of the
exercises in the Khan Academy

00:08:29.595 --> 00:08:35.254
and green means that the student has
already gotten 10 on a row in that module.

00:08:35.254 --> 00:08:37.244
Purple means that they're
working on it right now,

00:08:37.244 --> 00:08:39.291
but it seems like everything is good.

00:08:39.291 --> 00:08:42.740
Red means that they're working on it,
but they've done an awful lot of problems

00:08:42.740 --> 00:08:46.271
without having gotten 10 in a row
yet, so this might be a problem area.

00:08:46.271 --> 00:08:52.024
The model that, I'm at least advocating
is, everyone works at their own pace,

00:08:52.024 --> 00:08:56.531
at home and in the
classroom using the software

00:08:56.531 --> 00:08:59.461
and the teacher gets these daily
reports, and it tells them,

00:08:59.461 --> 00:09:03.096
look teacher everyone is doing
fine, everyone is progressing,

00:09:03.096 --> 00:09:08.998
except for these 2 kids right here
are stuck on level 2 division.

00:09:08.998 --> 00:09:15.399
Why don't you take them aside,
have a very focused 30 minute,
1 to 2 sessions with them

00:09:15.399 --> 00:09:20.671
kind of a tutorial with them and
then move on to the next 2 students.

00:09:20.671 --> 00:09:24.000
There's a couple of students who are
having trouble with level 4 subtraction,

00:09:24.000 --> 00:09:28.375
so that way the student is
always being directly catered too

00:09:28.375 --> 00:09:34.276
and the teachers time is always
spent directly, directly hitting
pain points for the student.

00:09:34.276 --> 00:09:40.947
Just to highlight the power, at least in
my mind, of this asynchronous learning.

00:09:40.947 --> 00:09:45.896
This is the actual data of that
same cohort of students as they
work through the summer camps.

00:09:45.896 --> 00:09:49.363
Just to understand what
this chart is showing,

00:09:49.363 --> 00:09:54.666
this vertical axis right here is
the modules that they completed

00:09:54.666 --> 00:09:59.065
and the horizontal axis right
here is days on the summer camp.

00:09:59.065 --> 00:10:01.653
The black line is the class mean.

00:10:01.653 --> 00:10:10.476
This is saying 5 days into the camp the
average student had completed 20 modules.

00:10:10.476 --> 00:10:15.115
These green lines above and below
that are standard deviations
below and above the mean.

00:10:15.115 --> 00:10:19.846
I've plotted some, what I consider
interesting students trajectories here.

00:10:19.846 --> 00:10:22.778
I could plot all of the students here,
but it would be a very messy graph.

00:10:22.778 --> 00:10:25.752
What I want to show you in
particular is one student right here.

00:10:25.752 --> 00:10:30.612
On day 5 if you were to do an assessment
using a traditional learning model,

00:10:30.612 --> 00:10:33.595
if you were to do an
assessment on day 5 right here,

00:10:33.595 --> 00:10:39.255
you would say that this
student, this student in purple,

00:10:39.255 --> 00:10:42.327
that she is more than one
standard deviation below the mean.

00:10:42.327 --> 00:10:44.853
You would say that maybe she
needs some type of remediation.

00:10:44.853 --> 00:10:49.338
She is a C student or a D student,
maybe she shouldn't be in this class,

00:10:49.338 --> 00:10:53.255
but when you do it with an asynchronous
learning model you allow that student

00:10:53.255 --> 00:10:56.077
to work at their own pace, you don't
have to slow down the rest of the class

00:10:56.077 --> 00:10:58.444
for him or her, and this was a her.

00:10:58.444 --> 00:11:03.379
It turns out that she just needed a
little bit extra time on negative numbers.

00:11:03.379 --> 00:11:06.050
This was a cohort of rising 8th graders.

00:11:06.050 --> 00:11:08.770
Everyone else, it was a little
bit of a review for her,

00:11:08.770 --> 00:11:11.836
she needed extra time on negative numbers
and we were collecting data on everything.

00:11:11.836 --> 00:11:16.418
Each of these columns
represent a problem she did.

00:11:16.418 --> 00:11:19.865
This right here, this first column
was the very first problem she did.

00:11:19.865 --> 00:11:24.652
She got it wrong, that's why it's
red and it took her 33 or 35 seconds.

00:11:24.652 --> 00:11:26.615
The next one she got right,
maybe she guessed it,

00:11:26.615 --> 00:11:28.783
or maybe it was 2 positive
numbers being added

00:11:28.783 --> 00:11:32.779
and then the next two she got
wrong, then she got one right,
maybe she guessed, who knows.

00:11:32.779 --> 00:11:35.056
Then this right here,
actually, I don't plot it,

00:11:35.056 --> 00:11:39.264
but this is actually where she watched
the video, that's why it took her so long.

00:11:39.264 --> 00:11:43.650
Then after that she started to
get some problems right, that's
what the blue bars represent.

00:11:43.650 --> 00:11:46.484
Not only did she get some ... she
got a few wrong here or there,

00:11:46.484 --> 00:11:49.916
but at the same time, the time it took
her to do problems also went down,

00:11:49.916 --> 00:11:51.854
you see the height of
these bars are going down.

00:11:51.854 --> 00:11:54.950
This is actually a distribution
of the times it took her.

00:11:54.950 --> 00:12:00.093
As you can see, you can actually
get granular date of what
peoples' actual pain points are,

00:12:00.093 --> 00:12:04.052
what remediation they need
without slowing anyone else down

00:12:04.052 --> 00:12:07.551
and then directly tackle that and
you get all of the data on it.

00:12:07.551 --> 00:12:12.608
At the same time, once she was able to
get through this, this one pain point,

00:12:12.608 --> 00:12:15.253
not only was she not below the average,

00:12:15.253 --> 00:12:18.452
but she was able to rocket
forward after she got through this

00:12:18.452 --> 00:12:21.608
and she actually, at the end of ... this
was actually only a month and a half,

00:12:21.608 --> 00:12:24.598
she finished at the 2nd or
3rd best student of the class.

00:12:24.598 --> 00:12:28.335
Just to give an idea, these
judgements that we place on
students are so arbitrary,

00:12:28.335 --> 00:12:31.089
back here you might have
called her a remedial student,

00:12:31.089 --> 00:12:36.536
out here you would say wow she
just rocketed straight through
level 3 linear equations,

00:12:36.536 --> 00:12:41.447
which are equations of this
form right here, which is
really above her grade level

00:12:41.447 --> 00:12:43.785
and would say, hey she's
maybe a gifted student.

00:12:43.785 --> 00:12:46.933
When you let every student work at
their own pace, at their own time,

00:12:46.933 --> 00:12:50.859
not pushing people forward before
they're ready or holding people back,

00:12:50.859 --> 00:12:56.350
you really do see amazing
results out of students.

00:12:56.350 --> 00:13:00.516
Now, everything I've talked about so
far, everything I've talked about so far,

00:13:00.516 --> 00:13:02.580
we've talked about the video library.

00:13:02.580 --> 00:13:07.458
(writing) we've talked about the video
library, we've talked about the exercises.

00:13:07.458 --> 00:13:13.281
(writing) we've talked about
the exercises and how that
could be, the combination here,

00:13:13.281 --> 00:13:17.202
could be transformational for a classroom,
everyone working at their own pace

00:13:17.202 --> 00:13:19.519
and even if the software isn't being used,

00:13:19.519 --> 00:13:21.342
there have actually been a few
teachers who've already emailed me,

00:13:21.342 --> 00:13:25.677
that they've flipped the model, instead
of doing lectures in the classroom

00:13:25.677 --> 00:13:28.650
and homework at home, they're now
doing homework in the classroom

00:13:28.650 --> 00:13:32.361
and using the Khan Academy videos
at home which makes a lot of sense

00:13:32.361 --> 00:13:35.563
because you do the videos at your
own time, you can pause, repeat,

00:13:35.563 --> 00:13:40.404
look up stuff, you can review videos
at your own pace, at your own time,

00:13:40.404 --> 00:13:45.311
as you need them and then when you're
actually doing problems in the classroom

00:13:45.311 --> 00:13:49.195
when you're surrounded by your peers
and the teacher who can help you along

00:13:49.195 --> 00:13:53.330
and really kind of give you assistance
and feedback when you need it.

00:13:53.330 --> 00:13:55.859
That's already happening,
even before the software,

00:13:55.859 --> 00:13:59.323
but I think when you have the
software it becomes even more rich,

00:13:59.323 --> 00:14:02.779
especially with the data collection,
but there is a situation of,

00:14:02.779 --> 00:14:05.987
what if one of these
students were getting stuck.

00:14:05.987 --> 00:14:10.904
Let's say they're not working in
the framework of a classroom that
has already adopted this paradigm,

00:14:10.904 --> 00:14:17.907
what do they do, at least my view
point of the answer, that you
have peer-to-peer instruction.

00:14:17.907 --> 00:14:26.256
(writing) peer-to-peer, peer-to-peer
instruction, peer-to-peer instruction.

00:14:26.256 --> 00:14:29.799
So you've watched the videos, we
know you've watched the video,

00:14:29.799 --> 00:14:31.706
the software knows
you've watched the video,

00:14:31.706 --> 00:14:36.588
you've done the exercises, we
know that you've done so many that
you're having a pain point here.

00:14:36.588 --> 00:14:42.172
What we can automatically do is
set up a peer-to-peer instruction

00:14:42.172 --> 00:14:46.343
with some other teacher, student,
tutor, it could be in the same city,

00:14:46.343 --> 00:14:49.601
it could be on the other side of the
planet and we could do it virtually;

00:14:49.601 --> 00:14:53.263
the same way that I started off with
my cousins where you have a pen tablet,

00:14:53.263 --> 00:14:58.720
you're doing this on a computer
but you have some type of
internet telephone connection

00:14:58.720 --> 00:15:02.716
whether it's Skype or Google
voice or whatever else and
you're using a shared whiteboard,

00:15:02.716 --> 00:15:06.956
kind of something like what I'm writing on
right now, but it's live and in real time.

00:15:06.956 --> 00:15:08.953
You could actually even record.

00:15:08.953 --> 00:15:11.869
(writing) You could
record the interaction.

00:15:11.869 --> 00:15:14.052
One, that's valuable from
a safety point of view,

00:15:14.052 --> 00:15:18.457
but it's also valuable to have this
huge library of other interactions that

00:15:18.457 --> 00:15:20.750
maybe if I can't find a tutor immediately,

00:15:20.750 --> 00:15:24.259
I can dig through the recordings of
other people who've had the same problem

00:15:24.259 --> 00:15:26.259
and I can get my question answered.

00:15:26.259 --> 00:15:32.331
The power here is not only are you able
to leverage all of the student body

00:15:32.331 --> 00:15:35.054
of this virtual academy
to teach each other,

00:15:35.054 --> 00:15:40.744
but you also are doing it in
a very data rich environment
because you know what happens.

00:15:40.744 --> 00:15:45.889
Each of these, in isolation, I
think, is already a huge value add,

00:15:45.889 --> 00:15:49.553
but when you do them together it
becomes a super-duper value add

00:15:49.553 --> 00:15:52.430
because not only do you have
the peer-to-peer instruction,

00:15:52.430 --> 00:15:58.221
you use the exercises to
figure out exactly what type of
peer-to-peer instruction you needed,

00:15:58.221 --> 00:16:03.085
you know who to pair someone with because
you know maybe Sal is having trouble

00:16:03.085 --> 00:16:07.603
with level 1 exponents, Bill, we already
know has mastered level 1 exponents.

00:16:07.603 --> 00:16:11.652
So, let's set up that interaction and
then you have data on before and after.

00:16:11.652 --> 00:16:14.518
(writing) Before and after.

00:16:14.518 --> 00:16:19.806
You will even be able to assess
how good was that interaction.

00:16:19.806 --> 00:16:24.744
You can have a qualitative rating
from both the tutor and the tutee,

00:16:24.744 --> 00:16:26.621
I guess you could say or from the student.

00:16:26.621 --> 00:16:31.261
You can also have data on what happened
before and after that interaction

00:16:31.261 --> 00:16:35.320
and is it statistically significant
compared to other similar interactions.

00:16:35.320 --> 00:16:41.504
When you couple this, I really
think you are approaching
the holy grain of education.

00:16:41.504 --> 00:16:47.518
This right here is the virtual school
that I see Khan Academy growing into.

00:16:47.518 --> 00:16:51.993
Obviously these first two pieces, this
piece has been built to a large degree.

00:16:51.993 --> 00:16:56.325
I intend to continue to build it and
continue to add, really all topics.

00:16:56.325 --> 00:17:01.050
This is already starting to be
built, but I envision, right
now it goes through Algebra II,

00:17:01.050 --> 00:17:02.822
if you look at the knowledge map.

00:17:02.822 --> 00:17:08.201
I completely see more and more
of these, you can see physics
problems budding off of this,

00:17:08.201 --> 00:17:12.173
you can see calculus budding off
of this, you can see chemistry
budding off at certain points.

00:17:12.173 --> 00:17:15.969
You can see genetics problems budding off
of this, probability, you can imagine.

00:17:15.969 --> 00:17:21.814
Right now there are 70 modules,
I imagine a world where you
can have 500 or 5,000 modules.

00:17:21.814 --> 00:17:26.886
That piece can obviously built
up, you could have other types of
exercises that are complimentary,

00:17:26.886 --> 00:17:29.143
maybe emersive type exercises.

00:17:29.143 --> 00:17:31.993
Then the part that I think has a lot
of value that has yet to be built

00:17:31.993 --> 00:17:38.135
is really the peer-to-peer piece,
but all of the technology already
exists today, I think, to do it.

00:17:38.135 --> 00:17:42.546
Just to give, I think, a hint of
where I think the social value is here

00:17:42.546 --> 00:17:45.732
and I'm going to do a quick
back of the envelope calculation

00:17:45.732 --> 00:17:51.931
because I think there is a huge
amount of ... it seems appealing
to a large percentage of people

00:17:51.931 --> 00:17:58.045
but just to get an idea of how
scalable, how much leverage this
type of a concept could have,

00:17:58.045 --> 00:17:59.631
or how much social value it can create.

00:17:59.631 --> 00:18:02.953
Let's just talk about the videos alone,
let's not even talk about the exercises

00:18:02.953 --> 00:18:07.582
or the peer-to-peer interactions and let's
just talk about the videos only in English

00:18:07.582 --> 00:18:10.442
although they are already
being translated into Spanish

00:18:10.442 --> 00:18:13.286
and Tamil, and there's a group in Poland.

00:18:13.286 --> 00:18:16.112
They could very easily be translated
to every major language in the world.

00:18:16.112 --> 00:18:20.906
Let's just talk about English,
let's say we have about 500 million,

00:18:20.906 --> 00:18:25.645
(writing) There's about 500 million
English speakers world wide.

00:18:25.645 --> 00:18:29.186
That includes, I think there's
about 350 million native speakers

00:18:29.186 --> 00:18:33.846
and 150 million people who are
speaking English as a second language.

00:18:33.846 --> 00:18:36.684
Let's just say, this is a back
of the envelope calculation.

00:18:36.684 --> 00:18:42.278
Let's just say that 10% could
potentially use these type of videos,

00:18:42.278 --> 00:18:44.581
that they are at some point in
their life where they want to learn.

00:18:44.581 --> 00:18:47.401
Maybe they're teachers who want
a new way to teach something

00:18:47.401 --> 00:18:49.985
or there are people going back to college,

00:18:49.985 --> 00:18:55.986
or they are your traditional
K-12 or college student,

00:18:55.986 --> 00:18:59.583
so 10% of the population
could stand to benefit.

00:18:59.583 --> 00:19:02.239
This is, you could kind of
view this as the market,

00:19:02.239 --> 00:19:06.595
but I speak of this in kind of a social
enterprise point of view, as the market,

00:19:06.595 --> 00:19:08.769
people who could benefit from it.

00:19:08.769 --> 00:19:12.982
We're already talking about 50 million, I
actually think this number is much larger

00:19:12.982 --> 00:19:17.278
because you should include parents
and teachers and really anyone
who wants to learn anything.

00:19:17.278 --> 00:19:22.054
Let's just say 10% of the population, so
you have a 50 million addressable market,

00:19:22.054 --> 00:19:26.984
and let's say that for whatever
reason only 10% adopt the videos,

00:19:26.984 --> 00:19:30.900
I'm only talking about the videos
here, so let's just cut that down.

00:19:30.900 --> 00:19:36.319
We're talking about 10% of
the 10% addressable market of
the English speaking world,

00:19:36.319 --> 00:19:42.515
which is really only 10% of
the world, I think making very
conservative assumptions here.

00:19:42.515 --> 00:19:50.873
Sp, 10% penetration into the market
that, for whatever reason, it's
free and it's easily available

00:19:50.873 --> 00:19:55.205
and probably the most
extensive resource available,

00:19:55.205 --> 00:19:58.048
let's say just for
assumption only 10% adopt it,

00:19:58.048 --> 00:20:03.723
we're talking about 5 million
learners on a regular basis using it.

00:20:03.723 --> 00:20:07.077
Let's say that they each
watch a video roughly a week.

00:20:07.077 --> 00:20:15.756
We're talking about times say
50 video a year per person,

00:20:15.756 --> 00:20:17.851
they're just watching one video a week.

00:20:17.851 --> 00:20:21.716
The data I'm seeing right
now is people are on average
watching a video every 3 days.

00:20:21.716 --> 00:20:25.248
Actually, some of them are
watching 2, 3 videos a day,

00:20:25.248 --> 00:20:26.950
but if we average all of
the unique viewers out,

00:20:26.950 --> 00:20:30.268
it looks like they're watching a video
once every 2 or 3 days, so once a week,

00:20:30.268 --> 00:20:32.550
I think, is a very
conservative assumption.

00:20:32.550 --> 00:20:34.524
Remember, we're only talking
about the videos right now,

00:20:34.524 --> 00:20:37.528
we're not talking about even the
exercises or the peer-to-peer
instruction or all of that.

00:20:37.528 --> 00:20:43.423
If you multiply that out, you
have 250 million video views.

00:20:43.423 --> 00:20:47.120
(writing) views per year.

00:20:47.120 --> 00:20:52.190
Khan Academy isn't there right
now because it doesn't have quite
that, really that mind share yet.

00:20:52.190 --> 00:20:54.954
I don't think most people are
even aware that it even exists,

00:20:54.954 --> 00:20:57.482
but I'm talking about the reality
where people do know it exists.

00:20:57.482 --> 00:21:01.743
You have 250 million video views a year,
we're talking about only the videos,

00:21:01.743 --> 00:21:04.261
only in English, we're making
some conservative assumptions.

00:21:04.261 --> 00:21:08.567
Let's say that a video view, the
social value of a video view.

00:21:08.567 --> 00:21:15.937
(writing) Let's say the social
value, so the social value.

00:21:15.937 --> 00:21:17.889
Let's say, I don't know, it's $1.

00:21:17.889 --> 00:21:19.531
I'm just making an assumption here.

00:21:19.531 --> 00:21:21.941
My personal opinion, I think
it's far more than that.

00:21:21.941 --> 00:21:26.466
Private tutoring in any of these topics
can cost anywhere from $10 to $50

00:21:26.466 --> 00:21:29.224
and this, I think, is frankly
better than private tutoring.

00:21:29.224 --> 00:21:32.690
My cousin's have even told me that
I'm better on YouTube than I am live

00:21:32.690 --> 00:21:36.394
because it's a less stressful environment,
they can pause, they can repeat,

00:21:36.394 --> 00:21:40.826
but let's say that the social value is
$1 which I think is very conservative.

00:21:40.826 --> 00:21:47.094
Then the total social return,
that just the videos, just the
videos using these assumptions

00:21:47.094 --> 00:21:49.122
would produce, just in English ...

00:21:49.122 --> 00:21:56.252
(writing) The social return in
this scenario would be $1 times
250 million video views a year,

00:21:56.252 --> 00:22:03.183
which would be $250 million,
$250 million annually, annually.

00:22:03.183 --> 00:22:07.118
I'm not doing some type of
present value where I'm looking
at all of the future value

00:22:07.118 --> 00:22:12.685
created over the next 25 or 50
years, this is on an annual basis.

00:22:12.685 --> 00:22:14.854
This is using these
conservative estimates.

00:22:14.854 --> 00:22:19.001
If you had translated to
the top 10 languages you can
multiply this number by 10.

00:22:19.001 --> 00:22:24.573
If you make this 10% into
20% you can multiple this by
2, it will be 500 million.

00:22:24.573 --> 00:22:28.799
If you present value even this
number you're going to get
something in the billions,

00:22:28.799 --> 00:22:31.912
depending on how you present value and
your discount rate and all of that.

00:22:31.912 --> 00:22:36.620
You're going to get a number that's in
excess of multiple billions of dollars.

00:22:36.620 --> 00:22:43.320
This is all, in my mind, for a
very, very, very small relative
investment to support all of this.

00:22:43.320 --> 00:22:46.123
Remember, the social return is
calculated just from the video views.

00:22:46.123 --> 00:22:48.989
If you actually think about
it replacing actual education,

00:22:48.989 --> 00:22:53.130
you compare it to what's actually spent on
a per pupil basis in the developed world,

00:22:53.130 --> 00:22:57.395
a $10,000 per pupil then the
numbers become really ridiculous,

00:22:57.395 --> 00:23:02.294
but if you just look at the
$250 million all of this, that's
the return just on the videos,

00:23:02.294 --> 00:23:07.660
but all of this could be built,
I think, by a team of 5-10
highly motivated engineers.

00:23:07.660 --> 00:23:16.772
(writing) 5-10, built and
maintained, motivated, 5-10
engineers including myself.

00:23:16.772 --> 00:23:20.858
So, you're really talking about
an operating budget depending
on the scale you do it at

00:23:20.858 --> 00:23:24.192
on the order of a million dollars a year.

00:23:24.192 --> 00:23:29.234
The videos, obviously I can keep
going and I would pay myself
far less than a million dollars,

00:23:29.234 --> 00:23:32.195
so just there you're
getting a huge social value,

00:23:32.195 --> 00:23:36.134
but even here if you're
looking at an operating budget
of a million dollars a year

00:23:36.134 --> 00:23:43.965
you really are talking about something
that has the impact to transform
education on a world wide basis

00:23:43.965 --> 00:23:49.208
and have a social return well,
well in excess of what I think
of this as kind of the base case.

00:23:49.208 --> 00:23:50.708
Even if this was the best case,

00:23:50.708 --> 00:23:57.441
a million dollar a year
getting a $250 million a year
payback is pretty tremendous.

00:23:57.441 --> 00:23:59.937
The last thing I want to highlight
is that this is all evergreen,

00:23:59.937 --> 00:24:05.167
this is all evergreen, these
videos and these modules they don't
have to be changed every year.

00:24:05.167 --> 00:24:08.267
Math and Science does
not change that fast.

00:24:08.267 --> 00:24:11.983
Once some of this, once most
of this content is built out,

00:24:11.983 --> 00:24:17.191
if I, God forbid, got hit by
a bus tomorrow, this social
return will still exist.

00:24:17.191 --> 00:24:20.733
It's not like as soon as these engineers
disappear that all of this disappears.

00:24:20.733 --> 00:24:23.378
As long as this is maintained
by some skeleton crew,

00:24:23.378 --> 00:24:28.929
you will still continue to generate
this much in social return.

00:24:28.929 --> 00:24:31.444
Anyway, I want to leave you there
and hopefully that gives you a sense

00:24:31.444 --> 00:24:35.637
of what I see Khan Academy, what
it is today, what I see it becoming

00:24:35.637 --> 00:24:39.637
and, I think, a sense of the real
value that this can have for the world.


How to use the quadratic formula

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:04.610
In this video, I'm going to
expose you to what is maybe

00:00:04.610 --> 00:00:07.860
one of at least the top five
most useful formulas in

00:00:07.860 --> 00:00:08.550
mathematics.

00:00:08.550 --> 00:00:11.460
And if you've seen many of my
videos, you know that I'm not

00:00:11.460 --> 00:00:13.660
a big fan of memorizing
things.

00:00:13.660 --> 00:00:17.400
But I will recommend you
memorize it with the caveat

00:00:17.400 --> 00:00:20.776
that you also remember how to
prove it, because I don't want

00:00:20.776 --> 00:00:22.940
you to just remember
things and not know

00:00:22.940 --> 00:00:23.870
where they came from.

00:00:23.870 --> 00:00:25.970
But with that said, let me
show you what I'm talking

00:00:25.970 --> 00:00:32.330
about: it's the quadratic
formula.

00:00:32.330 --> 00:00:37.620
And as you might guess, it is to
solve for the roots, or the

00:00:37.620 --> 00:00:40.900
zeroes of quadratic equations.

00:00:40.900 --> 00:00:43.450
So let's speak in very general
terms and I'll

00:00:43.450 --> 00:00:44.850
show you some examples.

00:00:44.850 --> 00:00:48.360
So let's say I have an equation
of the form ax

00:00:48.360 --> 00:00:54.070
squared plus bx plus
c is equal to 0.

00:00:54.070 --> 00:00:55.620
You should recognize this.

00:00:55.620 --> 00:01:00.450
This is a quadratic equation
where a, b and c are-- Well, a

00:01:00.450 --> 00:01:02.730
is the coefficient on the x
squared term or the second

00:01:02.730 --> 00:01:05.540
degree term, b is the
coefficient on the x term and

00:01:05.540 --> 00:01:08.270
then c, is, you could imagine,
the coefficient on the x to

00:01:08.270 --> 00:01:10.550
the zero term, or it's
the constant term.

00:01:10.550 --> 00:01:13.550
Now, given that you have a
general quadratic equation

00:01:13.550 --> 00:01:16.700
like this, the quadratic formula
tells us that the

00:01:16.700 --> 00:01:24.100
solutions to this equation are
x is equal to negative b plus

00:01:24.100 --> 00:01:32.020
or minus the square root of
b squared minus 4ac, all

00:01:32.020 --> 00:01:34.530
of that over 2a.

00:01:34.530 --> 00:01:38.490
And I know it seems crazy and
convoluted and hard for you to

00:01:38.490 --> 00:01:41.160
memorize right now, but as you
get a lot more practice you'll

00:01:41.160 --> 00:01:44.170
see that it actually is a pretty
reasonable formula to

00:01:44.170 --> 00:01:46.300
stick in your brain someplace.

00:01:46.300 --> 00:01:48.570
And you might say, gee, this is
a wacky formula, where did

00:01:48.570 --> 00:01:49.340
it come from?

00:01:49.340 --> 00:01:50.660
And in the next video I'm
going to show you

00:01:50.660 --> 00:01:51.260
where it came from.

00:01:51.260 --> 00:01:54.920
But I want you to get used to
using it first. But it really

00:01:54.920 --> 00:01:57.670
just came from completing
the square on this

00:01:57.670 --> 00:01:58.750
equation right there.

00:01:58.750 --> 00:02:00.880
If you complete the square here,
you're actually going to

00:02:00.880 --> 00:02:04.250
get this solution and that
is the quadratic

00:02:04.250 --> 00:02:06.390
formula, right there.

00:02:06.390 --> 00:02:09.949
So let's apply it to some
problems. Let's start off with

00:02:09.949 --> 00:02:12.680
something that we could have
factored just to verify that

00:02:12.680 --> 00:02:14.120
it's giving us the
same answer.

00:02:14.120 --> 00:02:19.660
So let's say we have x
squared plus 4x minus

00:02:19.660 --> 00:02:22.890
21 is equal to 0.

00:02:22.890 --> 00:02:30.230
So in this situation-- let me
do that in a different color

00:02:30.230 --> 00:02:32.370
--a is equal to 1, right?

00:02:32.370 --> 00:02:34.415
The coefficient on the
x squared term is 1.

00:02:37.790 --> 00:02:40.910
b is equal to 4, the coefficient
on the x-term.

00:02:40.910 --> 00:02:44.240
And then c is equal
to negative 21,

00:02:44.240 --> 00:02:45.470
the constant term.

00:02:45.470 --> 00:02:49.860
And let's just plug it in the
formula, so what do we get?

00:02:49.860 --> 00:02:54.470
We get x, this tells us that
x is going to be equal to

00:02:54.470 --> 00:02:56.200
negative b.

00:02:56.200 --> 00:03:02.346
Negative b is negative 4-- I put
the negative sign in front

00:03:02.346 --> 00:03:05.660
of that --negative b
plus or minus the

00:03:05.660 --> 00:03:08.770
square root of b squared.

00:03:08.770 --> 00:03:11.230
b squared is 16, right?

00:03:11.230 --> 00:03:20.170
4 squared is 16, minus 4 times
a, which is 1, times c, which

00:03:20.170 --> 00:03:21.870
is negative 21.

00:03:21.870 --> 00:03:24.750
So we can put a 21 out there
and that negative sign will

00:03:24.750 --> 00:03:28.790
cancel out just like that with
that-- Since this is the first

00:03:28.790 --> 00:03:31.400
time we're doing it, let me
not skip too many steps.

00:03:31.400 --> 00:03:34.800
So negative 21, just so you
can see how it fit in, and

00:03:34.800 --> 00:03:39.330
then all of that over 2a.

00:03:39.330 --> 00:03:42.760
a is 1, so all of that over 2.

00:03:42.760 --> 00:03:45.460
So what does this simplify, or
hopefully it simplifies?

00:03:45.460 --> 00:03:50.120
So we get x is equal to negative
4 plus or minus the

00:03:50.120 --> 00:03:53.110
square root of-- Let's see we
have a negative times a

00:03:53.110 --> 00:03:55.890
negative, that's going to
give us a positive.

00:03:55.890 --> 00:04:00.470
And we had 16 plus, let's see
this is 6, 4 times 1 is 4

00:04:00.470 --> 00:04:03.160
times 21 is 84.

00:04:03.160 --> 00:04:06.380
16 plus 84 is 100.

00:04:06.380 --> 00:04:07.810
That's nice.

00:04:07.810 --> 00:04:09.180
That's a nice perfect square.

00:04:09.180 --> 00:04:13.380
All of that over 2, and so this
is going to be equal to

00:04:13.380 --> 00:04:19.160
negative 4 plus or
minus 10 over 2.

00:04:19.160 --> 00:04:23.060
We could just divide both of
these terms by 2 right now.

00:04:23.060 --> 00:04:27.840
So this is equal to negative 4
divided by 2 is negative 2

00:04:27.840 --> 00:04:32.000
plus or minus 10 divided
by 2 is 5.

00:04:32.000 --> 00:04:38.490
So that tells us that x could be
equal to negative 2 plus 5,

00:04:38.490 --> 00:04:44.930
which is 3, or x could be equal
to negative 2 minus 5,

00:04:44.930 --> 00:04:47.240
which is negative 7.

00:04:47.240 --> 00:04:49.420
So the quadratic formula
seems to have given us

00:04:49.420 --> 00:04:50.500
an answer for this.

00:04:50.500 --> 00:04:53.240
You can verify just by
substituting back in that

00:04:53.240 --> 00:04:55.840
these do work, or you could even
just try to factor this

00:04:55.840 --> 00:04:56.610
right here.

00:04:56.610 --> 00:04:59.110
You say what two numbers when
you take their product, you

00:04:59.110 --> 00:05:02.230
get negative 21 and when you
take their sum you get

00:05:02.230 --> 00:05:04.200
positive 4?

00:05:04.200 --> 00:05:08.740
So you'd get x plus 7
times x minus 3 is

00:05:08.740 --> 00:05:10.690
equal to negative 21.

00:05:10.690 --> 00:05:14.970
Notice 7 times negative 3 is
negative 21, 7 minus 3 is

00:05:14.970 --> 00:05:16.160
positive 4.

00:05:16.160 --> 00:05:20.850
You would get x plus-- sorry
it's not negative --21 is

00:05:20.850 --> 00:05:22.590
equal to 0.

00:05:22.590 --> 00:05:23.890
There should be a 0 there.

00:05:23.890 --> 00:05:29.380
So you get x plus 7 is equal
to 0, or x minus

00:05:29.380 --> 00:05:30.810
3 is equal to 0.

00:05:30.810 --> 00:05:36.850
X could be equal to negative
7 or x could be equal to 3.

00:05:36.850 --> 00:05:39.390
So it definitely gives us the
same answer as factoring, so

00:05:39.390 --> 00:05:42.700
you might say, hey why bother
with this crazy mess?

00:05:42.700 --> 00:05:45.140
And the reason we want to bother
with this crazy mess is

00:05:45.140 --> 00:05:48.150
it'll also work for problems
that are hard to factor.

00:05:48.150 --> 00:05:50.210
And let's do a couple of
those, let's do some

00:05:50.210 --> 00:05:53.080
hard-to-factor problems
right now.

00:05:53.080 --> 00:05:57.410
So let's scroll down to get
some fresh real estate.

00:05:57.410 --> 00:06:01.150
Let's rewrite the formula again,
just in case we haven't

00:06:01.150 --> 00:06:04.580
had it memorized yet. x is going
to be equal to negative

00:06:04.580 --> 00:06:11.180
b plus or minus the square root
of b squared minus 4ac,

00:06:11.180 --> 00:06:14.480
all of that over 2a.

00:06:14.480 --> 00:06:17.540
I'll supply this to
another problem.

00:06:17.540 --> 00:06:30.320
Let's say we have the equation
3x squared plus 6x is equal to

00:06:30.320 --> 00:06:31.720
negative 10.

00:06:31.720 --> 00:06:33.610
Well, the first thing we want
to do is get it in the form

00:06:33.610 --> 00:06:35.980
where all of our terms or on the
left-hand side, so let's

00:06:35.980 --> 00:06:38.450
add 10 to both sides
of this equation.

00:06:38.450 --> 00:06:46.680
We get 3x squared plus the
6x plus 10 is equal to 0.

00:06:46.680 --> 00:06:49.930
And now we can use a
quadratic formula.

00:06:49.930 --> 00:06:51.320
So let's apply it here.

00:06:51.320 --> 00:06:53.050
So a is equal to 3.

00:06:53.050 --> 00:06:59.900
That is a, this is b and
this right here is c.

00:06:59.900 --> 00:07:01.870
So the quadratic formula
tells us the

00:07:01.870 --> 00:07:04.060
solutions to this equation.

00:07:04.060 --> 00:07:09.570
The roots of this quadratic
function, I guess

00:07:09.570 --> 00:07:10.570
we could call it.

00:07:10.570 --> 00:07:13.300
x is going to be equal
to negative b.

00:07:13.300 --> 00:07:17.920
b is 6, so negative 6
plus or minus the

00:07:17.920 --> 00:07:21.350
square root of b squared.

00:07:21.350 --> 00:07:29.270
b is 6, so we get 6 squared
minus 4 times a, which is 3

00:07:29.270 --> 00:07:32.630
times c, which is 10.

00:07:32.630 --> 00:07:37.050
Let's stretch out the radical
little bit, all of that over 2

00:07:37.050 --> 00:07:42.190
times a, 2 times 3.

00:07:42.190 --> 00:07:46.380
So we get x is equal to negative
6 plus or minus the

00:07:46.380 --> 00:07:57.320
square root of 36 minus-- this
is interesting --minus 4 times

00:07:57.320 --> 00:07:59.090
3 times 10.

00:07:59.090 --> 00:08:05.690
So this is minus-- 4
times 3 times 10.

00:08:05.690 --> 00:08:11.500
So this is minus 120.

00:08:11.500 --> 00:08:15.010
All of that over 6.

00:08:15.010 --> 00:08:17.750
So this is interesting, you
might already realize why it's

00:08:17.750 --> 00:08:18.220
interesting.

00:08:18.220 --> 00:08:19.970
What is this going
to simplify to?

00:08:19.970 --> 00:08:23.610
36 minus 120 is what?

00:08:23.610 --> 00:08:30.630
That's 84.

00:08:30.630 --> 00:08:33.860
We make this into a 10,
this will become an

00:08:33.860 --> 00:08:36.159
11, this is a 4.

00:08:36.159 --> 00:08:39.590
It is 84, so this is going to be
equal to negative 6 plus or

00:08:39.590 --> 00:08:42.799
minus the square root of-- But
not positive 84, that's if

00:08:42.799 --> 00:08:44.640
it's 120 minus 36.

00:08:44.640 --> 00:08:46.920
We have 36 minus 120.

00:08:46.920 --> 00:08:53.630
It's going to be negative
84 all of that 6.

00:08:53.630 --> 00:08:55.140
So you might say, gee,
this is crazy.

00:08:55.140 --> 00:08:56.930
What a this silly quadratic
formula you're

00:08:56.930 --> 00:08:58.000
introducing me to, Sal?

00:08:58.000 --> 00:08:58.790
It's worthless.

00:08:58.790 --> 00:09:03.020
It just gives me a square root
of a negative number.

00:09:03.020 --> 00:09:04.380
It's not giving me an answer.

00:09:04.380 --> 00:09:06.750
And the reason why it's not
giving you an answer, at least

00:09:06.750 --> 00:09:11.450
an answer that you might want,
is because this will have no

00:09:11.450 --> 00:09:12.700
real solutions.

00:09:17.910 --> 00:09:20.260
In the future, we're going to
introduce something called an

00:09:20.260 --> 00:09:23.310
imaginary number, which is a
square root of a negative

00:09:23.310 --> 00:09:27.720
number, and then we can actually
express this in terms

00:09:27.720 --> 00:09:28.810
of those numbers.

00:09:28.810 --> 00:09:31.540
So this actually does have
solutions, but they involve

00:09:31.540 --> 00:09:32.660
imaginary numbers.

00:09:32.660 --> 00:09:34.690
So this actually has no real
solutions, we're taking the

00:09:34.690 --> 00:09:37.760
square root of a negative
number.

00:09:37.760 --> 00:09:41.280
So the b squared with the b
squared minus 4ac, if this

00:09:41.280 --> 00:09:45.330
term right here is negative,
then you're not going to have

00:09:45.330 --> 00:09:47.290
any real solutions.

00:09:47.290 --> 00:09:48.610
And let's verify that
for ourselves.

00:09:48.610 --> 00:09:51.240
Let's get our graphic calculator
out and let's graph

00:09:51.240 --> 00:09:54.600
this equation right here.

00:09:54.600 --> 00:09:57.710
So, let's get the graphs that y
is equal to-- that's what I

00:09:57.710 --> 00:10:10.510
had there before --3x squared
plus 6x plus 10.

00:10:10.510 --> 00:10:11.920
So that's the equation and we're
going to see where it

00:10:11.920 --> 00:10:13.210
intersects the x-axis.

00:10:13.210 --> 00:10:15.370
Where does it equal 0?

00:10:15.370 --> 00:10:17.150
So let me graph it.

00:10:20.510 --> 00:10:23.700
Notice, this thing just comes
down and then goes back up.

00:10:23.700 --> 00:10:26.880
Its vertex is sitting here
above the x-axis and it's

00:10:26.880 --> 00:10:27.710
upward-opening.

00:10:27.710 --> 00:10:30.290
It never intersects
the x-axis.

00:10:30.290 --> 00:10:33.262
So at no point will this
expression, will this

00:10:33.262 --> 00:10:34.650
function, equal 0.

00:10:34.650 --> 00:10:38.530
At no point will y equal
0 on this graph.

00:10:38.530 --> 00:10:41.700
So once again, the quadratic
formula seems to be working.

00:10:41.700 --> 00:10:44.710
Let's do one more example,
you can never see

00:10:44.710 --> 00:10:46.730
enough examples here.

00:10:46.730 --> 00:10:50.300
And I want to do ones that are,
you know, maybe not so

00:10:50.300 --> 00:10:52.160
obvious to factor.

00:10:52.160 --> 00:11:01.930
So let's say we get negative 3x
squared plus 12x plus 1 is

00:11:01.930 --> 00:11:03.770
equal to 0.

00:11:03.770 --> 00:11:06.450
Now let's try to do it just
having the quadratic formula

00:11:06.450 --> 00:11:07.740
in our brain.

00:11:07.740 --> 00:11:12.290
So the x's that satisfy this
equation are going to be

00:11:12.290 --> 00:11:13.690
negative b.

00:11:13.690 --> 00:11:18.580
This is b So negative b is
negative 12 plus or minus the

00:11:18.580 --> 00:11:27.300
square root of b squared, of
144, that's b squared minus 4

00:11:27.300 --> 00:11:33.870
times a, which is negative 3
times c, which is 1, all of

00:11:33.870 --> 00:11:39.540
that over 2 times a, over
2 times negative 3.

00:11:39.540 --> 00:11:43.970
So all of that over negative 6,
this is going to be equal

00:11:43.970 --> 00:11:47.630
to negative 12 plus or
minus the square root

00:11:47.630 --> 00:11:49.030
of-- What is this?

00:11:49.030 --> 00:11:51.710
It's a negative times a negative
so they cancel out.

00:11:51.710 --> 00:12:01.220
So I have 144 plus 12, so
that is 156, right?

00:12:01.220 --> 00:12:06.910
144 plus 12, all of that
over negative 6.

00:12:06.910 --> 00:12:10.350
Now, I suspect we can
simplify this 156.

00:12:10.350 --> 00:12:12.270
We could maybe bring
some things out

00:12:12.270 --> 00:12:13.850
of the radical sign.

00:12:13.850 --> 00:12:17.170
So let's attempt to do that.

00:12:17.170 --> 00:12:20.020
So let's do a prime
factorization of 156.

00:12:20.020 --> 00:12:23.600
Sometimes, this is the hardest
part, simplifying the radical.

00:12:23.600 --> 00:12:30.080
So 156 is the same thing
as 2 times 78.

00:12:30.080 --> 00:12:33.950
78 is the same thing
as 2 times what?

00:12:33.950 --> 00:12:39.800
That's 2 times 39.

00:12:39.800 --> 00:12:48.960
So the square root of 156 is
equal to the square root of 2

00:12:48.960 --> 00:12:52.800
times 2 times 39 or we could say
that's the square root of

00:12:52.800 --> 00:12:57.110
2 times 2 times the
square root of 39.

00:12:57.110 --> 00:12:58.810
And this, obviously, is just
going to be the square root of

00:12:58.810 --> 00:13:01.830
4 or this is the square root
of 2 times 2 is just 2.

00:13:01.830 --> 00:13:06.310
2 square roots of 39, if I
did that properly, let's

00:13:06.310 --> 00:13:07.710
see, 4 times 39.

00:13:07.710 --> 00:13:10.150
Yeah, it looks like
it's right.

00:13:10.150 --> 00:13:17.590
So this up here will simplify to
negative 12 plus or minus 2

00:13:17.590 --> 00:13:23.220
times the square root of 39, all
of that over negative 6.

00:13:23.220 --> 00:13:25.650
Now we can divide the numerator
and the denominator

00:13:25.650 --> 00:13:26.590
maybe by 2.

00:13:26.590 --> 00:13:32.060
So this will be equal to
negative 6 plus or minus the

00:13:32.060 --> 00:13:36.020
square root of 39
over negative 3.

00:13:36.020 --> 00:13:38.540
Or we could separate these
two terms out.

00:13:38.540 --> 00:13:42.800
We could say this is equal to
negative 6 over negative 3

00:13:42.800 --> 00:13:49.310
plus or minus the square root
of 39 over negative 3.

00:13:49.310 --> 00:13:51.400
Now, this is just a 2
right here, right?

00:13:51.400 --> 00:13:55.190
These cancel out, 6 divided
by 3 is 2, so we get 2.

00:13:55.190 --> 00:13:57.810
And now notice, if this is plus
and we use this minus

00:13:57.810 --> 00:13:59.780
sign, the plus will become
negative and the negative will

00:13:59.780 --> 00:14:01.160
become positive.

00:14:01.160 --> 00:14:02.390
But it still doesn't
matter, right?

00:14:02.390 --> 00:14:05.260
We could say minus or plus,
that's the same thing as plus

00:14:05.260 --> 00:14:10.010
or minus the square root
of 39 nine over 3.

00:14:10.010 --> 00:14:12.940
I think that's about as simple
as we can get this answered.

00:14:12.940 --> 00:14:14.620
I want to make a very clear
point of what I

00:14:14.620 --> 00:14:16.420
did that last step.

00:14:16.420 --> 00:14:19.530
I did not forget about
this negative sign.

00:14:19.530 --> 00:14:20.430
I just said it doesn't matter.

00:14:20.430 --> 00:14:22.190
It's going to turn the positive
into the negative;

00:14:22.190 --> 00:14:23.800
it's going to turn the negative
into the positive.

00:14:23.800 --> 00:14:25.140
Let me rewrite this.

00:14:25.140 --> 00:14:29.660
So this right here can be
rewritten as 2 plus the square

00:14:29.660 --> 00:14:36.210
root of 39 over negative 3 or 2
minus the square root of 39

00:14:36.210 --> 00:14:38.080
over negative 3, right?

00:14:38.080 --> 00:14:40.610
That's what the plus or minus
means, it could be this or

00:14:40.610 --> 00:14:42.600
that or both of them, really.

00:14:42.600 --> 00:14:45.430
Now in this situation, this
negative 3 will turn into 2

00:14:45.430 --> 00:14:49.290
minus the square root
of 39 over 3, right?

00:14:49.290 --> 00:14:51.150
I'm just taking this
negative out.

00:14:51.150 --> 00:14:53.130
Here the negative and the
negative will become a

00:14:53.130 --> 00:14:56.090
positive, and you get 2
plus the square root

00:14:56.090 --> 00:14:59.210
of 39 over 3, right?

00:14:59.210 --> 00:15:01.130
A negative times a negative
is a positive.

00:15:01.130 --> 00:15:04.400
So once again, you have
2 plus or minus the

00:15:04.400 --> 00:15:07.190
square of 39 over 3.

00:15:07.190 --> 00:15:11.550
2 plus or minus the square
root of 39 over 3 are

00:15:11.550 --> 00:15:16.795
solutions to this equation
right there.

00:15:16.795 --> 00:15:17.300
Let verify.

00:15:17.300 --> 00:15:19.920
I'm just curious what the
graph looks like.

00:15:19.920 --> 00:15:24.360
So let's just look at it.

00:15:24.360 --> 00:15:25.860
Let me clear this.

00:15:25.860 --> 00:15:27.670
Where is the clear button?

00:15:27.670 --> 00:15:37.280
So we have negative 3 three
squared plus 12x plus 1 and

00:15:37.280 --> 00:15:39.721
let's graph it.

00:15:39.721 --> 00:15:42.280
Let's see where it intersects
the x-axis.

00:15:42.280 --> 00:15:47.180
It goes up there and then
back down again.

00:15:47.180 --> 00:15:51.840
So 2 plus or minus the square,
you see-- The square root of

00:15:51.840 --> 00:15:56.330
39 is going to be a little
bit more than 6, right?

00:15:56.330 --> 00:15:58.090
Because 36 is 6 squared.

00:15:58.090 --> 00:16:00.910
So it's going be a little bit
more than 6, so this is going

00:16:00.910 --> 00:16:02.110
to be a little bit
more than 2.

00:16:02.110 --> 00:16:03.890
A little bit more than 6 divided
by 2 is a little bit

00:16:03.890 --> 00:16:04.615
more than 2.

00:16:04.615 --> 00:16:06.580
So you're going to get one value
that's a little bit more

00:16:06.580 --> 00:16:09.100
than 4 and then another value
that should be a little bit

00:16:09.100 --> 00:16:10.690
less than 1.

00:16:10.690 --> 00:16:13.590
And that looks like the case,
you have 1, 2, 3, 4.

00:16:13.590 --> 00:16:18.460
You have a value that's pretty
close to 4, and then you have

00:16:18.460 --> 00:16:25.020
another value that is a little
bit-- It looks close to 0 but

00:16:25.020 --> 00:16:26.540
maybe a little bit
less than that.

00:16:26.540 --> 00:16:29.090
So anyway, hopefully you found
this application of the

00:16:29.090 --> 00:16:31.260
quadratic formula helpful.