Range, variance and standard deviation as measures of dispersion

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Kind: captions
Language: en

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In the last video we talked
about different ways to

00:00:03.450 --> 00:00:07.920
represent the central tendency
or the average of a data set.

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What we're going to do in this
video is to expand that a

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little bit to understand
how spread apart

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the data is as well.

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So let's just think about
this a little bit.

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Let's say I have negative
10, 0, 10, 20 and 30.

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Let's say that's one data
set right there.

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And let's say the other data
set is 8, 9, 10, 11 and 12.

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Now let's calculate the
arithmetic mean for both of

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these data sets.

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So let's calculate the mean.

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And when you go further on in
statistics, you're going to

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understand the difference
between a

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population and a sample.

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We're assuming that
this is the entire

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population of our data.

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So we're going to be dealing
with the population mean.

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We're going to be dealing
with, as you see, the

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population measures
of dispersion.

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I know these are all
fancy words.

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In the future, you're not going
to have all of the data.

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You're just going to have some
samples of it, and you're

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going to try to estimate
things for the entire

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population.

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So I don't want you to worry too
much about that just now.

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But if you are going to go
further in statistics, I just

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want to make that
clarification.

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Now, the population mean, or
the arithmetic mean of this

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data set right here, it is
negative 10 plus 0 plus 10

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plus 20 plus 30 over-- we have
five data points-- over 5.

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And what is this equal to?

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That negative 10 cancels out
with that 10, 20 plus 30 is 50

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divided by 5, it's
equal to 10.

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Now, what's the mean
of this data set?

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8 plus 9 plus 10 plus 11 plus
12, all of that over 5.

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And the way we could think about
it, 8 plus 12 is 20, 9

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plus 11 is another 20, so
that's 40, and then

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we have a 50 there.

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Add another 10.

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So this, once again, is
going to be 50 over 5.

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So this has the exact same
population means.

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Or if you don't want to worry
about the word population or

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sample and all of that, both
of these data sets have the

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exact same arithmetic mean.

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When you average all these
numbers and divide by 5 or

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when you take the sum of these
numbers and divide by 5, you

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get 10, some of these numbers
and divide by 5,

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you get 10 as well.

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But clearly, these sets of
numbers are different.

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You know, if you just looked at
this number, you'd say, oh,

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maybe these sets are very
similar to each other.

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But when you look at these two
data sets, one thing might pop

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out at you.

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All of these numbers are
very close to 10.

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I mean, the furthest number
here is two away from 10.

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12 is only two away from 10.

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Here, these numbers are
further away from 10.

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Even the closer ones are still
10 away and these guys are 20

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away from 10.

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So this right here, this data
set right here is more

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disperse, right?

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These guys are further away from
our mean than these guys

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are from this mean.

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So let's think about different
ways we can measure

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dispersion, or how far
away we are from

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the center, on average.

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Now one way, this is
kind of the most

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simple way, is the range.

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And you won't see it used too
often, but it's kind of a very

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simple way of understanding how
far is the spread between

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the largest and the
smallest number.

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You literally take the largest
number, which is 30 in our

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example, and from that, you
subtract the smallest number.

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So 30 minus negative 10, which
is equal to 40, which tells us

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that the difference between the
largest and the smallest

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number is 40, so we have a range
of 40 for this data set.

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Here, the range is the largest
number, 12, minus the smallest

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number, which is 8, which
is equal to 4.

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So here range is actually
a pretty good measure of

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dispersion.

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We say, OK, both of these
guys have a mean of 10.

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But when I look at the range,
this guy has a much larger

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range, so that tells me this
is a more disperse set.

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But range is always not going to
tell you the whole picture.

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You might have two data sets
with the exact same range

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where still, based on how things
are bunched up, it

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could still have very different
distributions of

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where the numbers lie.

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Now, the one that you'll
see used most often

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is called the variance.

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Actually, we're going
to see the standard

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deviation in this video.

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That's probably what's used most
often, but it has a very

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close relationship
to the variance.

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So the symbol for the variance--
and we're going to

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deal with the population
variance.

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Once again, we're assuming that
this is all of the data

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for our whole population, that
we're not just sampling,

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taking a subset, of the data.

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So the variance, its symbol is
literally this sigma, this

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Greek letter, squared.

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That is the symbol
for variance.

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And we'll see that the sigma
letter actually is the symbol

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for standard deviation.

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And that is for a reason.

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But anyway, the definition of
a variance is you literally

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take each of these data points,
find the difference

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between those data points and
your mean, square them, and

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then take the average
of those squares.

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I know that sounds very
complicated, but when I

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actually calculate it, you're
going to see it's not too bad.

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So remember, the mean
here is 10.

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So I take the first
data point.

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Let me do it over here.

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Let me scroll down
a little bit.

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So I take the first
data point.

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Negative 10.

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From that, I'm going to subtract
our mean and I'm

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going to square that.

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So I just found the difference
from that first data point to

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the mean and squared it.

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And that's essentially
to make it positive.

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Plus the second data point, 0
minus 10, minus the mean--

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this is the mean; this is that
10 right there-- squared plus

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10 minus 10 squared-- that's
the middle 10 right there--

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plus 20 minus 10-- that's
the 20-- squared

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plus 30 minus 10 squared.

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So this is the squared
differences between each

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number and the mean.

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This is the mean right there.

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I'm finding the difference
between every data point and

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the mean, squaring them, summing
them up, and then

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dividing by that number
of data points.

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So I'm taking the average
of these numbers,

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of the squared distances.

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So when you say it kind of
verbally, it sounds very

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complicated.

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But you're taking each number.

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What's the difference between
that, the mean, square it,

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take the average of those.

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So I have 1, 2, 3, 4,
5, divided by 5.

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So what is this going
to be equal to?

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Negative 10 minus 10
is negative 20.

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Negative 20 squared is 400.

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0 minus 10 is negative 10
squared is 100, so plus 100.

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10 minus 10 squared, that's just
0 squared, which is 0.

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Plus 20 minus 10 is 10
squared, is 100.

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Plus 30 minus 10, which
is 20, squared is 400.

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All of that over 5.

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And what do we have here?

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400 plus 100 is 500, plus
another 500 is 1000.

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It's equal to 1000/5, which
is equal to 200.

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So in this situation, our
variance is going to be 200.

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That's our measure of
dispersion there.

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And let's compare it to this
data set over here.

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Let's compare it to the
variance of this

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less-dispersed data set.

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So let me scroll over a little
bit so we have some real

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estate, although I'm
running out.

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Maybe I could scroll up here.

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There you go.

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Let me calculate the variance
of this data set.

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So we already know its mean.

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So its variance of this data set
is going to be equal to 8

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minus 10 squared plus 9 minus
10 squared plus 10 minus 10

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squared plus 11 minus 10-- let
me scroll up a little bit--

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squared plus 12 minus
10 squared.

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Remember, that 10 is just the
mean that we calculated.

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You have to calculate the mean
first. Divided by-- we have 1,

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2, 3, 4, 5 squared
differences.

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So this is going to be equal
to-- 8 minus 10 is negative 2

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squared, is positive 4.

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9 minus 10 is negative 1
squared, is positive 1.

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10 minus 10 is 0 squared.

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You still get 0.

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11 minus 10 is 1.

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Square it, you get 1.

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12 minus 10 is 2.

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Square it, you get 4.

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And what is this equal to?

00:09:00.960 --> 00:09:02.100
All of that over 5.

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This is 10/5.

00:09:04.390 --> 00:09:09.160
So this is going to be--all
right, this is 10/5, which is

00:09:09.160 --> 00:09:11.350
equal to 2.

00:09:11.350 --> 00:09:13.420
So the variance here-- let me
make sure I got that right.

00:09:13.420 --> 00:09:15.480
Yes, we have 10/5.

00:09:15.480 --> 00:09:18.970
So the variance of this
less-dispersed data set is a

00:09:18.970 --> 00:09:19.820
lot smaller.

00:09:19.820 --> 00:09:26.150
The variance of this data set
right here is only 2.

00:09:26.150 --> 00:09:27.630
So that gave you a sense.

00:09:27.630 --> 00:09:30.540
That tells you, look, this is
definitely a less-dispersed

00:09:30.540 --> 00:09:32.300
data set then that there.

00:09:32.300 --> 00:09:34.780
Now, the problem with the
variance is you're taking

00:09:34.780 --> 00:09:36.680
these numbers, you're taking
the difference between them

00:09:36.680 --> 00:09:39.390
and the mean, then you're
squaring it.

00:09:39.390 --> 00:09:41.910
It kind of gives you a bit of
an arbitrary number, and if

00:09:41.910 --> 00:09:43.460
you're dealing with
units, let's say

00:09:43.460 --> 00:09:48.160
if these are distances.

00:09:48.160 --> 00:09:52.700
So this is negative 10 meters, 0
meters, 10 meters, this is 8

00:09:52.700 --> 00:09:55.110
meters, so on and so forth, then
when you square it, you

00:09:55.110 --> 00:09:57.250
get your variance in terms
of meters squared.

00:09:57.250 --> 00:09:59.740
It's kind of an odd
set of units.

00:09:59.740 --> 00:10:02.550
So what people like to do is
talk in terms of standard

00:10:02.550 --> 00:10:20.770
deviation, which is just the
square root of the variance,

00:10:20.770 --> 00:10:24.510
or the square root
of sigma squared.

00:10:24.510 --> 00:10:26.470
And the symbol for the standard

00:10:26.470 --> 00:10:29.200
deviation is just sigma.

00:10:29.200 --> 00:10:30.990
So now that we've figured out
the variance, it's very easy

00:10:30.990 --> 00:10:33.250
to figure out the standard
deviation of both of these

00:10:33.250 --> 00:10:33.720
characters.

00:10:33.720 --> 00:10:37.390
The standard deviation of this
first one up here, of this

00:10:37.390 --> 00:10:42.660
first data set, is going to
be the square root of 200.

00:10:42.660 --> 00:10:44.300
The square root of
200 is what?

00:10:44.300 --> 00:10:45.780
The square root of
2 times 100.

00:10:45.780 --> 00:10:48.480
This is equal to 10
square roots of 2.

00:10:48.480 --> 00:10:50.210
That's that first data set.

00:10:50.210 --> 00:10:58.410
Now the standard deviation of
the second data set is just

00:10:58.410 --> 00:11:02.820
going to be the square root of
its variance, which is just 2.

00:11:02.820 --> 00:11:08.300
So the second data set has 1/10
the standard deviation as

00:11:08.300 --> 00:11:09.170
this first data set.

00:11:09.170 --> 00:11:11.540
This is 10 roots of 2, this
is just the root of 2.

00:11:11.540 --> 00:11:18.935
So this is 10 times the
standard deviation.

00:11:25.890 --> 00:11:29.480
And this, hopefully, will make
a little bit more sense.

00:11:29.480 --> 00:11:29.960
Let's think about it.

00:11:29.960 --> 00:11:33.160
This has 10 times more the
standard deviation than this.

00:11:33.160 --> 00:11:34.480
And let's remember how
we calculated it.

00:11:34.480 --> 00:11:38.330
Variance, we just took each
data point, how far it was

00:11:38.330 --> 00:11:40.680
away from the mean,
squared that, took

00:11:40.680 --> 00:11:42.020
the average of those.

00:11:42.020 --> 00:11:44.490
Then we took the square root,
really just to make the units

00:11:44.490 --> 00:11:47.260
look nice, but the end result
is we said that that first

00:11:47.260 --> 00:11:50.850
data set has 10 times the
standard deviation as the

00:11:50.850 --> 00:11:52.290
second data set.

00:11:52.290 --> 00:11:54.980
So let's look at the
two data sets.

00:11:54.980 --> 00:12:07.170
This has 10 times the standard
deviation, which makes sense

00:12:07.170 --> 00:12:08.790
intuitively, right?

00:12:08.790 --> 00:12:12.210
I mean, they both have a 10 in
here, but each of these guys,

00:12:12.210 --> 00:12:15.520
9 is only one away from
the 10, 0 is 10 away

00:12:15.520 --> 00:12:16.640
from the 10, 10 less.

00:12:16.640 --> 00:12:17.860
8 is only two away.

00:12:17.860 --> 00:12:18.990
This guy is 20 away.

00:12:18.990 --> 00:12:21.770
So it's 10 times, on average,
further away.

00:12:21.770 --> 00:12:24.410
So the standard deviation, at
least in my sense, is giving a

00:12:24.410 --> 00:12:29.090
much better sense of how far
away, on average, we

00:12:29.090 --> 00:12:30.960
are from the mean.

00:12:30.960 --> 00:12:33.560
Anyway, hopefully, you
found that useful.


Average or Central Tendency: Arithmetic Mean, Median, and Mode

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.820
Let's say I've got
a set of numbers.

00:00:02.820 --> 00:00:08.710
2, say I've got three 3's, I've
got a couple of 4's, and

00:00:08.710 --> 00:00:10.620
I've got a 10 there.

00:00:10.620 --> 00:00:14.210
And what we want to do is find
the middle of these numbers.

00:00:14.210 --> 00:00:17.820
We want to represent these
numbers with the center of the

00:00:17.820 --> 00:00:19.900
numbers, or the middle of the
numbers, just so we have a

00:00:19.900 --> 00:00:24.350
sense of where these numbers
roughly are.

00:00:24.350 --> 00:00:27.360
And this central tendency that
we're going to try to get out

00:00:27.360 --> 00:00:31.020
of these numbers, we're going
to call the average.

00:00:31.020 --> 00:00:34.990
The average of this
set of numbers.

00:00:34.990 --> 00:00:38.840
And you've, I'm sure, heard the
word average before, but

00:00:38.840 --> 00:00:41.110
we're going to get a little
bit more detailed on the

00:00:41.110 --> 00:00:44.270
different types of averages
in this video.

00:00:44.270 --> 00:00:46.890
The one you're probably most
familiar with, although you

00:00:46.890 --> 00:00:49.950
might have not seen it referred
to in this way, is

00:00:49.950 --> 00:00:58.290
the arithmetic mean, which
literally says, look, I, the

00:00:58.290 --> 00:01:01.270
arithmetic mean of this set of
numbers, is literally the sum

00:01:01.270 --> 00:01:03.420
of all of these numbers divided
by the number of

00:01:03.420 --> 00:01:04.209
numbers there are.

00:01:04.209 --> 00:01:07.440
So the arithmetic mean for this
set right here is going

00:01:07.440 --> 00:01:17.070
to be 2 plus 3 plus 3 plus 3
plus 4 plus 4 plus 10, all of

00:01:17.070 --> 00:01:19.230
that over, how many
numbers do I have?

00:01:19.230 --> 00:01:22.870
1, 2, 3, 4, 5, 6, 7.

00:01:22.870 --> 00:01:24.720
All of that over 7.

00:01:24.720 --> 00:01:25.660
And what is this equal to?

00:01:25.660 --> 00:01:34.560
This is 2 plus 9, which is 11,
plus 8, which is 19, plus 10,

00:01:34.560 --> 00:01:35.520
which is 29.

00:01:35.520 --> 00:01:41.290
So this is going to be equal to
29/7, or you could say it's

00:01:41.290 --> 00:01:43.540
equal to 4 and 1/7.

00:01:43.540 --> 00:01:45.240
If I got my calculator out,
we could figure out

00:01:45.240 --> 00:01:46.480
the decimal of this.

00:01:46.480 --> 00:01:50.990
But this is a representation
of the central tendency, or

00:01:50.990 --> 00:01:52.450
the middle of these numbers.

00:01:52.450 --> 00:01:53.670
And it kind of makes sense.

00:01:53.670 --> 00:01:57.670
4 and 1/7, it's a little
bit higher than 4.

00:01:57.670 --> 00:02:01.070
We're kind of close to the
middle of our number range

00:02:01.070 --> 00:02:01.810
right there.

00:02:01.810 --> 00:02:03.590
And you might say, well, it's
a little skewed to the right

00:02:03.590 --> 00:02:04.700
and what caused that?

00:02:04.700 --> 00:02:06.950
And well, gee, 10 is a little
bit larger than all of the

00:02:06.950 --> 00:02:07.840
other numbers.

00:02:07.840 --> 00:02:09.320
It's kind of an outlier.

00:02:09.320 --> 00:02:13.370
Maybe that skewed this average
up, the arithmetic mean.

00:02:13.370 --> 00:02:16.680
So there are other types of
averages, although this is the

00:02:16.680 --> 00:02:19.200
one that, if people just say,
hey, let's take the average of

00:02:19.200 --> 00:02:21.660
these numbers, and they don't
really tell you more, they're

00:02:21.660 --> 00:02:23.940
probably talking about
the arithmetic mean.

00:02:23.940 --> 00:02:30.270
The other forms of average,
though, are the median, and

00:02:30.270 --> 00:02:32.290
this literally is the
middle number.

00:02:35.970 --> 00:02:38.360
If there are two middle numbers,
you actually take the

00:02:38.360 --> 00:02:40.730
arithmetic mean of those
two middle numbers.

00:02:40.730 --> 00:02:43.310
You actually find the number
halfway in between those two

00:02:43.310 --> 00:02:44.330
middle numbers.

00:02:44.330 --> 00:02:47.260
So the median of this set
right here-- let me just

00:02:47.260 --> 00:02:48.350
rewrite them.

00:02:48.350 --> 00:02:57.060
So I have a 2, a 3,
3, 3, 4, 4, 10.

00:02:57.060 --> 00:02:59.690
So, let's see, we have seven
numbers right here.

00:02:59.690 --> 00:03:02.800
The middle number, if
I go 1, 2, 3, to the

00:03:02.800 --> 00:03:04.140
right, we're there.

00:03:04.140 --> 00:03:06.470
If we go 1, 2, 3 to the
left, we're there.

00:03:06.470 --> 00:03:09.850
The middle number is
that 3 right there.

00:03:09.850 --> 00:03:13.360
I just listed them in order, and
I said, well, look, 3, you

00:03:13.360 --> 00:03:16.640
could think of it as the fourth
number from the right,

00:03:16.640 --> 00:03:19.550
and it's also the fourth
number from the left.

00:03:19.550 --> 00:03:21.360
3 is the middle number.

00:03:21.360 --> 00:03:23.940
And this case, it
is the median.

00:03:23.940 --> 00:03:29.280
So in this case, 3, if you use
the median, is our average.

00:03:29.280 --> 00:03:30.360
And that also makes sense.

00:03:30.360 --> 00:03:32.030
I mean, it's literally the
middle number, and if you look

00:03:32.030 --> 00:03:35.670
at this set of numbers, it kind
of does represent the

00:03:35.670 --> 00:03:37.790
central tendency of this set.

00:03:37.790 --> 00:03:40.880
Now just to be clear, it was
very clear what the middle

00:03:40.880 --> 00:03:45.350
number was, because I had an
odd number of numbers.

00:03:45.350 --> 00:03:48.560
I had three on each side of the
three, so it was very easy

00:03:48.560 --> 00:03:50.790
to figure out the median,
the middle number.

00:03:50.790 --> 00:03:53.800
But if I had a situation-- let's
say I have the situation

00:03:53.800 --> 00:03:57.850
where I have 2, 3, 4, and 5.

00:03:57.850 --> 00:04:01.510
Let's say that's my
set of numbers.

00:04:01.510 --> 00:04:04.350
Well, here, there is no
one middle number.

00:04:04.350 --> 00:04:07.460
The 3 is closer to the left
than it is to the right.

00:04:07.460 --> 00:04:10.040
The 4 is closer to the right
than it is to the left.

00:04:10.040 --> 00:04:12.220
There's actually two middle
numbers here.

00:04:12.220 --> 00:04:17.380
The two middle numbers here
are the 3 and the 4.

00:04:17.380 --> 00:04:20.640
And here, when you have two
middle numbers, which occurs

00:04:20.640 --> 00:04:24.430
when you have an even number
in your data set, there the

00:04:24.430 --> 00:04:27.630
median is halfway in between
these two numbers.

00:04:27.630 --> 00:04:31.070
So in this situation, the median
is going to be 3 plus 4

00:04:31.070 --> 00:04:35.670
over 2, which is equal to 3.5.

00:04:35.670 --> 00:04:37.610
And if you look at this data
set, that's not what our

00:04:37.610 --> 00:04:44.430
original problem was, but if
you look at this data set

00:04:44.430 --> 00:04:47.770
right there, you're actually
going to find that the

00:04:47.770 --> 00:04:52.070
arithmetic mean and the median
here is the exact same thing.

00:04:52.070 --> 00:04:53.230
Let's calculate it.

00:04:53.230 --> 00:04:55.470
What 's the arithmetic
mean over here?

00:04:55.470 --> 00:05:00.240
It's going to be 2 plus 3 plus
4 plus 5, which is what?

00:05:00.240 --> 00:05:09.490
5 plus 9, which is equal
to 14, over 4.

00:05:09.490 --> 00:05:10.910
And what's this equal to?

00:05:10.910 --> 00:05:16.740
14/4 is 3 and 2/4, or 3 and
1/2, the exact same thing.

00:05:16.740 --> 00:05:19.310
So for this data set, they
were the same thing.

00:05:19.310 --> 00:05:24.020
For this data set, our median
is a little bit lower.

00:05:24.020 --> 00:05:28.090
It's 3, while our arithmetic
mean is 4 and 1/7.

00:05:28.090 --> 00:05:30.080
And I really want you to think
about why that is.

00:05:30.080 --> 00:05:32.840
And it has a lot to do with this
10 that sits out there.

00:05:32.840 --> 00:05:37.190
All of these other numbers are
pretty close to whichever

00:05:37.190 --> 00:05:39.590
average you want to pick,
whether it's the arithmetic

00:05:39.590 --> 00:05:42.560
mean or it's the median.

00:05:42.560 --> 00:05:47.860
But this 10 is kind of
an outlier, or it

00:05:47.860 --> 00:05:50.650
skews the data set.

00:05:50.650 --> 00:05:53.790
Maybe it's so much larger than
the other numbers, that it

00:05:53.790 --> 00:05:57.680
makes the arithmetic mean seem
larger than maybe is

00:05:57.680 --> 00:05:59.670
representative of
this data set.

00:05:59.670 --> 00:06:01.500
And that's something important
to think about.

00:06:01.500 --> 00:06:07.570
When you're finding the average
for something, most

00:06:07.570 --> 00:06:09.740
people will immediately go
to the arithmetic mean.

00:06:09.740 --> 00:06:13.520
But in a lot of cases, median
will make a lot more sense, if

00:06:13.520 --> 00:06:16.970
you have these really large or
really small numbers that

00:06:16.970 --> 00:06:18.460
could skew the data set.

00:06:18.460 --> 00:06:21.530
I mean, you can imagine, if this
wasn't a 10-- or let's

00:06:21.530 --> 00:06:22.790
imagine adding another
number here.

00:06:22.790 --> 00:06:27.810
If I added the number 1 million,
if I added 1 million

00:06:27.810 --> 00:06:30.390
to this data set, if that was
the eighth number, the

00:06:30.390 --> 00:06:32.260
arithmetic mean is going
to be this huge number.

00:06:32.260 --> 00:06:36.400
It's going to be much larger
than what is representative of

00:06:36.400 --> 00:06:38.470
most of the numbers
in this data set.

00:06:38.470 --> 00:06:40.240
But the median is still
going to work.

00:06:40.240 --> 00:06:43.730
The median is still going to be
about 3 and a half, right?

00:06:43.730 --> 00:06:47.750
If you had 1 million here,
it would be 1, 2, 3, 4.

00:06:47.750 --> 00:06:49.200
The middle two numbers
would be that.

00:06:49.200 --> 00:06:50.580
It would be 3 and 1/2.

00:06:50.580 --> 00:06:53.900
So the median is less sensitive
to one or two

00:06:53.900 --> 00:06:58.430
numbers at the extremes that
otherwise would skew the mean.

00:06:58.430 --> 00:07:01.130
Now, the last form of average
I want to talk

00:07:01.130 --> 00:07:02.840
about is the mode.

00:07:05.400 --> 00:07:08.000
It has nothing to do
with ice cream.

00:07:08.000 --> 00:07:10.910
The mode is literally the
most frequent number.

00:07:16.800 --> 00:07:19.880
And in this data set, it's
pretty clear what the most

00:07:19.880 --> 00:07:21.130
frequent number is.

00:07:21.130 --> 00:07:25.820
I only have one 2, I have three
3's, I have two 4's, I

00:07:25.820 --> 00:07:28.445
have one 10, and even if want to
include the million, I only

00:07:28.445 --> 00:07:29.710
have one million there.

00:07:29.710 --> 00:07:31.990
So here, the number
that occurs most

00:07:31.990 --> 00:07:35.360
frequently is the 3.

00:07:35.360 --> 00:07:38.330
So, once again, the mode seems
like a pretty good measure of

00:07:38.330 --> 00:07:41.260
central tendency or a
pretty good average

00:07:41.260 --> 00:07:43.300
for this data set.

00:07:43.300 --> 00:07:45.950
Now the mode, it's a little
tricky to deal with, and you

00:07:45.950 --> 00:07:49.070
won't see it used that often,
because it becomes a little

00:07:49.070 --> 00:07:52.920
ambiguous when-- you know,
look at this data set:

00:07:52.920 --> 00:07:55.270
2, 3, 4, and 5.

00:07:55.270 --> 00:07:56.370
What is the mode there?

00:07:56.370 --> 00:07:59.200
All of these numbers are
equally frequent.

00:07:59.200 --> 00:08:01.360
So if you have a situation
like this, then you might

00:08:01.360 --> 00:08:03.930
just-- the mode really
loses its meaning.

00:08:03.930 --> 00:08:06.350
It might force you anyway to
take the median or the

00:08:06.350 --> 00:08:07.530
mean in some form.

00:08:07.530 --> 00:08:10.700
But if you really do have
numbers that one shows up a

00:08:10.700 --> 00:08:14.640
lot more than the other, then
the mode starts to make sense.

00:08:14.640 --> 00:08:18.220
So, hopefully, this has given
you a pretty good overview of

00:08:18.220 --> 00:08:25.400
how to represent the central
tendency of a data set.

00:08:25.400 --> 00:08:26.430
Very fancy word.

00:08:26.430 --> 00:08:28.180
But it's just saying, look,
we're trying to represent with

00:08:28.180 --> 00:08:30.200
one number all of this data.

00:08:30.200 --> 00:08:31.810
And you might say, hey, why do
we even worry about that?

00:08:31.810 --> 00:08:34.059
It only has seven numbers here
or eight numbers here.

00:08:34.059 --> 00:08:36.770
But you can imagine if you had 7
million numbers or 7 billion

00:08:36.770 --> 00:08:39.440
numbers, and you don't want to
show someone all of that data.

00:08:39.440 --> 00:08:42.210
You just want to give someone a
sense of what those numbers

00:08:42.210 --> 00:08:45.345
are on average.

00:08:45.345 --> 00:08:49.420
And as we said, the arithmetic
mean is what I see being used

00:08:49.420 --> 00:08:53.170
the most. But in situations
where you might have numbers

00:08:53.170 --> 00:08:55.940
that would skew the arithmetic
mean, because they're so large

00:08:55.940 --> 00:08:58.670
or they're so small,
the median might

00:08:58.670 --> 00:09:00.480
make a lot of sense.


Visual Pythagorean Theorem Proof

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:03.610
We've already seen several
videos where we, one, learned

00:00:03.610 --> 00:00:06.710
what the Pythagorean theorem is
and how it can be applied

00:00:06.710 --> 00:00:10.470
to find the sides of right
triangles, and even to find

00:00:10.470 --> 00:00:14.230
the distances between points
on the Cartesian plane, so

00:00:14.230 --> 00:00:17.290
this is the Pythagorean
theorem.

00:00:17.290 --> 00:00:21.700
And just as a bit of review, it
tells us that if I have a

00:00:21.700 --> 00:00:29.430
right triangle, and if c is the
side opposite the right

00:00:29.430 --> 00:00:30.270
angle, right?

00:00:30.270 --> 00:00:32.400
That's my right angle.
c is that side.

00:00:32.400 --> 00:00:33.950
It's the longest side.

00:00:33.950 --> 00:00:35.200
We call that the hypotenuse.

00:00:42.950 --> 00:00:46.300
Then the square of the other
sides, the sum of the squares

00:00:46.300 --> 00:00:48.700
of the other sides, will
be equal to c squared.

00:00:48.700 --> 00:00:52.580
So if this side is a and this
side is b, the Pythagorean

00:00:52.580 --> 00:00:57.100
theorem tells us that a squared
plus b squared is

00:00:57.100 --> 00:01:00.790
going to be equal
to c squared.

00:01:00.790 --> 00:01:04.400
And we did several problems
where we used this to solve

00:01:04.400 --> 00:01:05.479
different triangles.

00:01:05.479 --> 00:01:07.710
But what I want to show you
in this video is really a

00:01:07.710 --> 00:01:10.680
visual-- you can view it as
a visual proof of the

00:01:10.680 --> 00:01:14.890
Pythagorean theorem to see why
this actually makes sense.

00:01:14.890 --> 00:01:17.460
And to understand that, I'm
going to draw a little bit of

00:01:17.460 --> 00:01:18.880
a diagram here.

00:01:18.880 --> 00:01:22.910
So let me draw a little
right triangle here.

00:01:22.910 --> 00:01:26.570
So let's say that that
is my right triangle.

00:01:26.570 --> 00:01:29.340
And let's say that this
is a, this is b,

00:01:29.340 --> 00:01:31.020
that is my right angle.

00:01:31.020 --> 00:01:34.560
Then that right there, this
right here, that length right

00:01:34.560 --> 00:01:36.320
there is c.

00:01:36.320 --> 00:01:38.760
Now, what I'm going to do is
draw this exact same triangle,

00:01:38.760 --> 00:01:40.750
but I'm going to rotate
it a little bit.

00:01:40.750 --> 00:01:42.110
You're going to see what
I'm talking about.

00:01:42.110 --> 00:01:45.960
This right here is the exact
same triangle that I just

00:01:45.960 --> 00:01:49.250
drew, except I rotated it.

00:01:49.250 --> 00:01:53.970
So this is a, this is b,
and now this is c.

00:01:53.970 --> 00:01:56.030
This is the exact-- you could
do this, you know.

00:01:56.030 --> 00:01:58.650
You could take a ruler out
and actually do it.

00:01:58.650 --> 00:02:02.040
Pick an a, b, and c, and you
can actually rotate this

00:02:02.040 --> 00:02:03.950
triangle around to the
right like this.

00:02:03.950 --> 00:02:06.370
You could just rotate it like
that, and you would get this

00:02:06.370 --> 00:02:08.740
triangle, just like that.

00:02:08.740 --> 00:02:10.089
Now, I'm going to
rotate it again.

00:02:12.860 --> 00:02:15.780
And I'm just going to imagine
copy and pasting it.

00:02:15.780 --> 00:02:20.730
And then I'll get this triangle,
just like that.

00:02:20.730 --> 00:02:26.460
And once again, this is a,
this is b, and this is c.

00:02:26.460 --> 00:02:30.260
These triangles are all the
exact same triangle, the exact

00:02:30.260 --> 00:02:31.480
same measure.

00:02:31.480 --> 00:02:34.510
Now let me rotate this again,
and I'll get this triangle.

00:02:37.500 --> 00:02:41.970
I will get this triangle
right here.

00:02:41.970 --> 00:02:43.970
And once again, this is a.

00:02:43.970 --> 00:02:45.500
This length right here is a.

00:02:45.500 --> 00:02:47.430
This length right here is b.

00:02:47.430 --> 00:02:51.670
And this is the hypotenuse of
this right angle, this is c.

00:02:51.670 --> 00:02:54.780
Now what do we have here?

00:02:54.780 --> 00:02:58.100
We have all of these right
angles, and inside, I have

00:02:58.100 --> 00:03:01.510
this other square that
is inscribed

00:03:01.510 --> 00:03:02.900
into this right angle.

00:03:02.900 --> 00:03:08.960
And I can prove to you, if you
like, that these are also

00:03:08.960 --> 00:03:10.090
right angles.

00:03:10.090 --> 00:03:12.150
And you know, why don't
I just show it to you.

00:03:12.150 --> 00:03:15.900
You may or may not already be
familiar that the angles in a

00:03:15.900 --> 00:03:17.990
triangle add up to 180.

00:03:17.990 --> 00:03:24.550
So if you call this angle x, you
call this angle y, we know

00:03:24.550 --> 00:03:29.990
that x plus y plus 90 degrees
have to be equal to 180.

00:03:29.990 --> 00:03:36.300
So x plus y plus 90 have
to be equal to 180.

00:03:36.300 --> 00:03:39.830
Or you could say that if you
subtract 90 from both sides of

00:03:39.830 --> 00:03:44.680
this equation, x plus y are
going to be equal to 90.

00:03:44.680 --> 00:03:46.520
I just subtracted 90
from both sides.

00:03:46.520 --> 00:03:47.260
This disappears.

00:03:47.260 --> 00:03:50.060
180 minus 90 is 90.

00:03:50.060 --> 00:03:52.420
And this x and y, they're
all over.

00:03:52.420 --> 00:03:55.726
If this angle is x right here,
so is this angle because this

00:03:55.726 --> 00:03:56.750
is the exact same triangle.

00:03:56.750 --> 00:03:58.100
So is this angle.

00:03:58.100 --> 00:03:59.850
So is this angle.

00:03:59.850 --> 00:04:03.720
If this angle is y, so is that
angle, so is that angle, and

00:04:03.720 --> 00:04:06.120
so is that angle.

00:04:06.120 --> 00:04:12.250
Now, you may or may not already
know that if you have

00:04:12.250 --> 00:04:16.269
three angles that form a line--
so let me draw it.

00:04:16.269 --> 00:04:19.450
If you have three angles that
form a line like this, so

00:04:19.450 --> 00:04:24.650
let's say we have angle a,
angle b-- let me draw b a

00:04:24.650 --> 00:04:29.850
little bit different-- and angle
c, just like that, and

00:04:29.850 --> 00:04:33.600
together they kind of form you
could imagine a semi-circle or

00:04:33.600 --> 00:04:36.900
they go from one line all the
way back, that a, b plus c are

00:04:36.900 --> 00:04:38.180
going to be equal to 180.

00:04:38.180 --> 00:04:43.890
a plus b plus c is going to
be equal to 180 degrees.

00:04:43.890 --> 00:04:47.080
They complete a 180-degree
arc.

00:04:47.080 --> 00:04:51.750
Now, if you look at, let's say,
this point right here on

00:04:51.750 --> 00:04:56.560
our diagram, you can see that
angle x plus this angle right

00:04:56.560 --> 00:05:07.490
here, which I'll call angle z, x
plus z plus y is going to be

00:05:07.490 --> 00:05:09.050
equal to 180 degrees.

00:05:09.050 --> 00:05:09.880
They form a line.

00:05:09.880 --> 00:05:15.900
So x plus z plus y, or maybe I
should write it as x plus y

00:05:15.900 --> 00:05:18.540
plus z is equal to 180.

00:05:18.540 --> 00:05:22.750
Now, we already know x plus y
is equal to 90, so this is

00:05:22.750 --> 00:05:23.440
equal to 90.

00:05:23.440 --> 00:05:26.770
So 90 plus z is equal to 180.

00:05:26.770 --> 00:05:30.200
z minus-- well, both sides
of this equation minus 90

00:05:30.200 --> 00:05:34.740
degrees, you get 180 minus 90.

00:05:34.740 --> 00:05:37.960
I just subtract 90 from
both sides, so z

00:05:37.960 --> 00:05:40.370
is equal to 90 degrees.

00:05:40.370 --> 00:05:44.620
So these are all 90-degree
angles.

00:05:44.620 --> 00:05:48.020
Their sides are all the same
length of c, so this is a

00:05:48.020 --> 00:05:52.000
square in the middle
of this diagram.

00:05:52.000 --> 00:05:56.850
Now, let's think a little
bit about what the area

00:05:56.850 --> 00:05:57.610
of the square is.

00:05:57.610 --> 00:06:01.110
And there's two ways we
can come up with the

00:06:01.110 --> 00:06:01.995
area of that square.

00:06:01.995 --> 00:06:03.330
So let me draw something here.

00:06:03.330 --> 00:06:05.200
So the first easy way to think
of the area of the square,

00:06:05.200 --> 00:06:06.280
well, it's a square.

00:06:06.280 --> 00:06:08.900
This length is c, that
length is c.

00:06:08.900 --> 00:06:10.710
To figure out the area, I
just multiply one side

00:06:10.710 --> 00:06:11.470
by the other side.

00:06:11.470 --> 00:06:14.490
It's c times c, or c squared.

00:06:14.490 --> 00:06:19.750
So if I want the area of that
right there, it is c

00:06:19.750 --> 00:06:22.260
squared, c times c.

00:06:22.260 --> 00:06:26.120
Area is equal to c squared.

00:06:26.120 --> 00:06:30.230
But how else could we figure
out that area?

00:06:30.230 --> 00:06:35.400
Well, we could say that would
be the area of this larger

00:06:35.400 --> 00:06:43.680
square, of this whole square
over here minus the area of

00:06:43.680 --> 00:06:46.400
these triangles.

00:06:46.400 --> 00:06:51.310
That also would give us the area
of that inside square.

00:06:51.310 --> 00:06:55.830
So what is the area of this
outside, bigger square?

00:06:55.830 --> 00:07:00.570
Well, each of its sides
are a plus b.

00:07:00.570 --> 00:07:02.140
And so you take a plus b.

00:07:02.140 --> 00:07:04.670
If you want the area, you could
say, a plus b times a

00:07:04.670 --> 00:07:08.800
plus b, or a plus b
squared, right?

00:07:08.800 --> 00:07:12.920
That's the area of the outer
square, but we still have to

00:07:12.920 --> 00:07:15.130
get the area of our
inner square.

00:07:15.130 --> 00:07:16.565
Area of small square.

00:07:19.970 --> 00:07:22.700
We still have to subtract out
the area of these triangles.

00:07:22.700 --> 00:07:24.790
And what are the areas
of these triangles?

00:07:24.790 --> 00:07:28.020
Well, we have to subtract out.

00:07:28.020 --> 00:07:31.420
The area of each of them is 1/2
times base times height,

00:07:31.420 --> 00:07:35.120
or base times height,
ab, 1/2 ab.

00:07:35.120 --> 00:07:38.160
That's the area of each
of these triangles.

00:07:38.160 --> 00:07:41.750
Now we have 1, 2, 3, 4
of these triangles.

00:07:41.750 --> 00:07:45.190
So it's 4 times 1/2 ab.

00:07:45.190 --> 00:07:50.280
This is another way of getting
the area of this inside

00:07:50.280 --> 00:07:51.900
smaller square.

00:07:51.900 --> 00:07:53.740
Now let's just do a little bit
of algebra-- we're pretty

00:07:53.740 --> 00:07:57.150
familiar with this-- to
essentially simplify the

00:07:57.150 --> 00:07:59.900
right-hand side of this
equation right here.

00:07:59.900 --> 00:08:03.690
So a plus b times times a plus
b, a times a is a squared,

00:08:03.690 --> 00:08:08.120
plus 2 ab, plus b squared.

00:08:08.120 --> 00:08:10.900
And then we have minus-- what
do we have right here?

00:08:10.900 --> 00:08:12.320
Minus 2ab, right?

00:08:12.320 --> 00:08:16.560
4 times 1/2 is 2,
so minus 2 ab.

00:08:16.560 --> 00:08:17.540
Well, these cancel out.

00:08:17.540 --> 00:08:22.530
2ab minus 2ab, we're just left
with a squared and b squared.

00:08:22.530 --> 00:08:26.620
So the area of this inside
circle, which is c squared,

00:08:26.620 --> 00:08:31.950
it's also equal to a squared
plus b squared.

00:08:31.950 --> 00:08:35.299
And I've just to shown you
a visual proof for the

00:08:35.299 --> 00:08:38.400
Pythagorean theorem, because
remember, a and b were the

00:08:38.400 --> 00:08:42.330
smaller sides of a right
triangle having c as its

00:08:42.330 --> 00:08:42.940
hypotenuse.

00:08:42.940 --> 00:08:46.700
And from there, we constructed
this bigger diagram.

00:08:46.700 --> 00:08:49.080
Hopefully, you found that fun.


Midpoint formula

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:05.630
Let's say I have the point
3 comma negative 4.

00:00:05.630 --> 00:00:08.740
So that would be 1, 2,
3, and then down 4.

00:00:08.740 --> 00:00:10.530
1, 2, 3, 4.

00:00:10.530 --> 00:00:14.910
So that's 3 comma negative 4.

00:00:14.910 --> 00:00:17.310
And I also had the
point 6 comma 1.

00:00:17.310 --> 00:00:22.050
So 1, 2, 3, 4, 5, 6 comma 1.

00:00:22.050 --> 00:00:23.510
So just like that.

00:00:23.510 --> 00:00:25.560
6 comma 1.

00:00:25.560 --> 00:00:28.750
In the last video, we figured
out that we could just use the

00:00:28.750 --> 00:00:30.530
Pythagorean theorem if we
wanted to figure out the

00:00:30.530 --> 00:00:32.270
distance between these
two points.

00:00:32.270 --> 00:00:35.000
We just drew a triangle there
and realized that this was the

00:00:35.000 --> 00:00:35.870
hypotenuse.

00:00:35.870 --> 00:00:38.570
In this video, we're going to
try to figure out what is the

00:00:38.570 --> 00:00:42.980
coordinate of the point that
is exactly halfway between

00:00:42.980 --> 00:00:44.170
this point and that point?

00:00:44.170 --> 00:00:46.510
So this right here is kind of
the distance, the line that

00:00:46.510 --> 00:00:48.000
connects them.

00:00:48.000 --> 00:00:54.500
Now what is the coordinate of
the point that is exactly

00:00:54.500 --> 00:00:56.520
halfway in between the two?

00:00:56.520 --> 00:00:58.910
What is this coordinate
right here?

00:00:58.910 --> 00:01:01.830
It's something comma
something.

00:01:01.830 --> 00:01:04.280
And to do that-- let me draw
it really big here.

00:01:04.280 --> 00:01:05.590
Because I think you're going to
find out that it's actually

00:01:05.590 --> 00:01:06.255
pretty straightforward.

00:01:06.255 --> 00:01:08.050
At first it seems like a
really tough problem.

00:01:08.050 --> 00:01:10.620
Gee, let me use the distance
formula with some variables.

00:01:10.620 --> 00:01:12.490
But you're going to see, it's
actually one of the simplest

00:01:12.490 --> 00:01:15.210
things you'll learn in
algebra and geometry.

00:01:15.210 --> 00:01:17.430
So let's say that this is
my triangle right there.

00:01:22.020 --> 00:01:24.830
This right here is the
point 6 comma 1.

00:01:24.830 --> 00:01:28.240
This down here is the point
3 comma negative 4.

00:01:28.240 --> 00:01:32.310
And we're looking for the point
that is smack dab in

00:01:32.310 --> 00:01:33.840
between those two points.

00:01:33.840 --> 00:01:35.090
What are its coordinates?

00:01:35.090 --> 00:01:38.110
It seems very hard at first. But
it's easy when you think

00:01:38.110 --> 00:01:41.360
about it in terms of just the
x and the y coordinates.

00:01:41.360 --> 00:01:43.700
What's this guy's x-coordinate
going to be?

00:01:43.700 --> 00:01:48.000
This line out here represents
x is equal to 6.

00:01:48.000 --> 00:01:50.530
This over here-- let me do it
in a little darker color--

00:01:50.530 --> 00:01:52.460
this over here represents
x is equal to 6.

00:01:52.460 --> 00:01:55.120
This over here represents
x is equal to 3.

00:01:55.120 --> 00:01:57.690
What will this guy's
x-coordinate be?

00:01:57.690 --> 00:02:01.530
Well, his x-coordinate is
going to be smack dab in

00:02:01.530 --> 00:02:03.660
between the two x-coordinates.

00:02:03.660 --> 00:02:05.980
This is x is equal to 3, this
is x is equal to 6.

00:02:05.980 --> 00:02:07.760
He's going to be right
in between.

00:02:07.760 --> 00:02:09.830
This distance is going to be
equal to that distance.

00:02:09.830 --> 00:02:12.020
His x-coordinate is going
to be right in between

00:02:12.020 --> 00:02:13.770
the 3 and the 6.

00:02:13.770 --> 00:02:16.800
So what do we call the number
that's right in between

00:02:16.800 --> 00:02:18.710
the 3 and the 6?

00:02:18.710 --> 00:02:22.650
Well we could call that the
midpoint, or we could call it

00:02:22.650 --> 00:02:24.540
the mean, or the average,
or however you want

00:02:24.540 --> 00:02:25.180
to talk about it.

00:02:25.180 --> 00:02:27.980
We just want to know, what's
the average of 3 and 6?

00:02:27.980 --> 00:02:30.730
So to figure out this point,
the point halfway between 3

00:02:30.730 --> 00:02:35.700
and 6, you literally just figure
out, 3 plus 6 over 2.

00:02:35.700 --> 00:02:38.280
Which is equal to 4.5.

00:02:38.280 --> 00:02:42.000
So this x-coordinate
is going to be 4.5.

00:02:42.000 --> 00:02:45.050
Let me draw that
on this graph.

00:02:45.050 --> 00:02:48.330
1, 2, 3, 4.5.

00:02:48.330 --> 00:02:50.050
And you see, it's smack
dab in between.

00:02:50.050 --> 00:02:51.610
That's its x-coordinate.

00:02:51.610 --> 00:02:57.440
Now, by the exact same logic,
this guy's y-coordinate is

00:02:57.440 --> 00:03:02.740
going to be smack dab between y
is equal to negative 4 and y

00:03:02.740 --> 00:03:05.650
is equal to 1.

00:03:05.650 --> 00:03:07.520
So it's going to be right
in between those.

00:03:07.520 --> 00:03:09.450
So this is the x right there.

00:03:09.450 --> 00:03:12.810
The y-coordinate is going to be
right in between y is equal

00:03:12.810 --> 00:03:14.470
to negative 4 and
y is equal to 1.

00:03:14.470 --> 00:03:16.040
So you just take the average.

00:03:16.040 --> 00:03:19.390
1 plus negative 4 over 2.

00:03:19.390 --> 00:03:23.040
That's equal to negative 3
over 2 or you could say

00:03:23.040 --> 00:03:25.940
negative 1.5.

00:03:25.940 --> 00:03:27.630
So you go down 1.5.

00:03:27.630 --> 00:03:30.450
It is literally right there.

00:03:30.450 --> 00:03:31.200
So just like that.

00:03:31.200 --> 00:03:34.260
You literally take the average
of the x's, take the average

00:03:34.260 --> 00:03:36.370
of the y's, or maybe I should
say the mean to be a little

00:03:36.370 --> 00:03:37.150
bit more specific.

00:03:37.150 --> 00:03:38.840
A mean of only two points.

00:03:38.840 --> 00:03:43.320
And you will get the midpoint
of those two points.

00:03:43.320 --> 00:03:45.840
The point that's equidistant
from both of them.

00:03:45.840 --> 00:03:48.380
It's the midpoint of the line
that connects them.

00:03:48.380 --> 00:03:53.150
So the coordinates are 4.5
comma negative 1.5.

00:03:53.150 --> 00:03:54.200
Let's do a couple
more of these.

00:03:54.200 --> 00:03:56.560
These, actually, you're going
to find are very, very

00:03:56.560 --> 00:03:57.970
straightforward.

00:03:57.970 --> 00:04:00.690
But just to visualize
it, let me graph it.

00:04:00.690 --> 00:04:02.660
Let's say I have the point
4, negative 5.

00:04:02.660 --> 00:04:04.270
So 1, 2, 3, 4.

00:04:04.270 --> 00:04:05.100
And then go down 5.

00:04:05.100 --> 00:04:08.160
1, 2, 3, 4, 5.

00:04:08.160 --> 00:04:10.420
So that's 4, negative 5.

00:04:10.420 --> 00:04:11.960
And I have the point
8 comma 2.

00:04:11.960 --> 00:04:17.100
So 1, 2, 3, 4, 5,
6, 7, 8 comma 2.

00:04:17.100 --> 00:04:18.410
8 comma 2.

00:04:18.410 --> 00:04:20.149
So what is the coordinate
of the midpoint

00:04:20.149 --> 00:04:20.899
of these two points?

00:04:20.899 --> 00:04:23.600
The point that is smack
dab in between them?

00:04:23.600 --> 00:04:26.970
Well, we just average the
x's, average the y's.

00:04:26.970 --> 00:04:30.020
So the midpoint is going to be--
the x values are 8 and 4.

00:04:30.020 --> 00:04:33.260
It's going to be 8
plus 4 over 2.

00:04:33.260 --> 00:04:35.660
And the y value is going to be--
well, we have a 2 and a

00:04:35.660 --> 00:04:36.510
negative 5.

00:04:36.510 --> 00:04:40.220
So you get 2 plus negative
5 over 2.

00:04:40.220 --> 00:04:41.460
And what is this equal to?

00:04:41.460 --> 00:04:48.750
This is 12 over 2, which is 6
comma 2 minus 5 is negative 3.

00:04:48.750 --> 00:04:53.280
Negative 3 over 2
is negative 1.5.

00:04:53.280 --> 00:04:58.610
So that right there
is the midpoint.

00:04:58.610 --> 00:05:01.040
You literally just average the
x's and average the y's, or

00:05:01.040 --> 00:05:01.660
find their means.

00:05:01.660 --> 00:05:02.820
So let's graph it,
just to make sure

00:05:02.820 --> 00:05:04.220
it looks like midpoint.

00:05:04.220 --> 00:05:05.170
6, negative 5.

00:05:05.170 --> 00:05:08.200
1, 2, 3, 4, 5, 6.

00:05:08.200 --> 00:05:09.520
Negative 1.5.

00:05:09.520 --> 00:05:11.520
Negative 1, negative 1.5.

00:05:11.520 --> 00:05:13.100
Yep, looks pretty good.

00:05:13.100 --> 00:05:16.020
It looks like it's equidistant
from this point and

00:05:16.020 --> 00:05:17.550
that point up there.

00:05:17.550 --> 00:05:19.090
Now that's all you
have to remember.

00:05:19.090 --> 00:05:21.485
Average the x, or take the mean
of the x, or find the x

00:05:21.485 --> 00:05:22.840
that's right in between
the two.

00:05:22.840 --> 00:05:23.800
Average the y's.

00:05:23.800 --> 00:05:25.050
You've got the midpoint.

00:05:25.050 --> 00:05:28.160
What I'm going to show you now
is what's in many textbooks.

00:05:28.160 --> 00:05:33.530
They'll write, oh, if I have
the point x1 y1, and then I

00:05:33.530 --> 00:05:36.250
have the point-- actually, I'll
just stick it in yellow.

00:05:36.250 --> 00:05:38.950
It's kind of painful to switch
colors all the time-- and then

00:05:38.950 --> 00:05:43.390
I have the point x2 y2, many
books will give you something

00:05:43.390 --> 00:05:44.900
called the midpoint formula.

00:05:44.900 --> 00:05:47.820
Which once again, I think is
kind of silly to memorize.

00:05:47.820 --> 00:05:49.460
Just remember, you
just average.

00:05:49.460 --> 00:05:51.780
Find the x in between, find
the y in between.

00:05:51.780 --> 00:05:53.675
So midpoint formula.

00:05:57.080 --> 00:05:59.880
What they'll really say is the
midpoint-- so maybe we'll say

00:05:59.880 --> 00:06:04.410
the midpoint x-- or maybe
I'll call it this way.

00:06:04.410 --> 00:06:05.560
I'm just making up notation.

00:06:05.560 --> 00:06:10.080
The x midpoint and the y
midpoint is going to be equal

00:06:10.080 --> 00:06:16.170
to-- and they'll give you this
formula. x1 plus x2 over 2,

00:06:16.170 --> 00:06:19.580
and then y1 plus y2 over 2.

00:06:19.580 --> 00:06:21.070
And it looks like something
you have to memorize.

00:06:21.070 --> 00:06:22.140
But all you have to
say is, look.

00:06:22.140 --> 00:06:27.090
That's just the average,
or the mean,

00:06:27.090 --> 00:06:28.340
of these two numbers.

00:06:32.040 --> 00:06:35.050
I'm adding the two together,
dividing by two, adding these

00:06:35.050 --> 00:06:36.730
two together, dividing by two.

00:06:36.730 --> 00:06:37.900
And then I get the midpoint.

00:06:37.900 --> 00:06:40.770
That's all the midpoint
formula is.


Distance formula

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:02.890
In this video, we're going to
learn how to take the distance

00:00:02.890 --> 00:00:08.640
between any two points in our
x, y coordinate plane, and

00:00:08.640 --> 00:00:11.060
we're going to see, it's really
just an application of

00:00:11.060 --> 00:00:12.780
the Pythagorean theorem.

00:00:12.780 --> 00:00:14.040
So let's start with
an example.

00:00:14.040 --> 00:00:16.070
Let's say I have the point, I'll
do it in a darker color

00:00:16.070 --> 00:00:17.950
so we can see it on
the graph paper.

00:00:17.950 --> 00:00:22.070
Let's say I have the point
3 comma negative 4.

00:00:22.070 --> 00:00:25.180
So if I were to graph it,
I'd go 1, 2, 3, and

00:00:25.180 --> 00:00:26.480
then I'd go down 4.

00:00:26.480 --> 00:00:33.440
1, 2, 3, 4, right there,
is 3 comma negative 4.

00:00:33.440 --> 00:00:40.110
And let's say I also have
the point 6 comma 0.

00:00:40.110 --> 00:00:45.660
So 1, 2, 3, 4, 5, 6, and then
there's no movement in the

00:00:45.660 --> 00:00:46.310
y-direction.

00:00:46.310 --> 00:00:47.610
We're just sitting
on the x-axis.

00:00:47.610 --> 00:00:52.086
The y-coordinate is 0,
so that's 6 comma 0.

00:00:52.086 --> 00:00:55.810
And what I want to figure out is
the distance between these

00:00:55.810 --> 00:00:56.410
two points.

00:00:56.410 --> 00:01:00.350
How far is this blue point away
from this orange point?

00:01:00.350 --> 00:01:01.910
And at first, you're like, gee,
Sal, I don't think I've

00:01:01.910 --> 00:01:03.570
ever seen anything about
how to solve for a

00:01:03.570 --> 00:01:04.440
distance like this.

00:01:04.440 --> 00:01:06.290
And what are you even talking
about the Pythagorean theorem?

00:01:06.290 --> 00:01:08.180
I don't see a triangle there!

00:01:08.180 --> 00:01:09.690
And if you don't see
a triangle, let

00:01:09.690 --> 00:01:10.940
me draw it for you.

00:01:13.770 --> 00:01:18.060
Let me draw this triangle right
there, just like that.

00:01:18.060 --> 00:01:20.690
Let me actually do several
colors here, just to really

00:01:20.690 --> 00:01:23.090
hit the point home.

00:01:23.090 --> 00:01:24.650
So there is our triangle.

00:01:24.650 --> 00:01:25.990
And you might immediately
recognize

00:01:25.990 --> 00:01:27.950
this is a right triangle.

00:01:27.950 --> 00:01:29.810
This is a right angle
right there.

00:01:29.810 --> 00:01:33.520
The base goes straight left to
right, the right side goes

00:01:33.520 --> 00:01:36.710
straight up and down, so we're
dealing with a right triangle.

00:01:36.710 --> 00:01:39.210
So if we could just figure out
what the base length is and

00:01:39.210 --> 00:01:42.350
what this height is, we could
use the Pythagorean theorem to

00:01:42.350 --> 00:01:47.690
figure out this long side, the
side that is opposite the

00:01:47.690 --> 00:01:49.700
right angle, the hypotenuse.

00:01:49.700 --> 00:01:55.890
This right here, the distance
is the hypotenuse of this

00:01:55.890 --> 00:01:57.180
right triangle.

00:01:57.180 --> 00:01:58.180
Let me write that down.

00:01:58.180 --> 00:02:04.050
The distance is equal
to the hypotenuse

00:02:04.050 --> 00:02:05.660
of this right triangle.

00:02:05.660 --> 00:02:07.480
So let me draw it a
little bit bigger.

00:02:07.480 --> 00:02:10.710
So this is the hypotenuse
right there.

00:02:10.710 --> 00:02:14.880
And then we have the side on the
right, the side that goes

00:02:14.880 --> 00:02:16.070
straight up and down.

00:02:16.070 --> 00:02:18.500
And then we have our base.

00:02:18.500 --> 00:02:19.860
Now, how do we figure
out-- let's

00:02:19.860 --> 00:02:22.550
call this d for distance.

00:02:22.550 --> 00:02:24.540
That's the length of
our hypotenuse.

00:02:24.540 --> 00:02:27.640
How do we figure out the lengths
of this up and down

00:02:27.640 --> 00:02:30.270
side and the base
side right here?

00:02:30.270 --> 00:02:32.940
So let's look at the base first.
What is this distance?

00:02:32.940 --> 00:02:35.010
You could even count it on this
graph paper, but here,

00:02:35.010 --> 00:02:38.150
where x is equal to-- let
me do it in the green.

00:02:38.150 --> 00:02:41.530
Here, we're at x is equal to 3
and here we're at x is equal

00:02:41.530 --> 00:02:42.800
to 6, right?

00:02:42.800 --> 00:02:44.550
We're just moving
straight right.

00:02:44.550 --> 00:02:47.570
This is the same distance as
that distance right there.

00:02:47.570 --> 00:02:49.910
So to figure out that distance,
it's literally the

00:02:49.910 --> 00:02:50.710
end x point.

00:02:50.710 --> 00:02:52.240
And you could actually go either
way, because you're

00:02:52.240 --> 00:02:53.770
going to square everything, so
it doesn't matter if you get

00:02:53.770 --> 00:02:59.230
negative numbers, so the
distance here is going to be 6

00:02:59.230 --> 00:03:03.360
minus 3, right?

00:03:03.360 --> 00:03:04.680
6 minus 3.

00:03:04.680 --> 00:03:08.520
That's this distance right here,
which is equal to 3.

00:03:08.520 --> 00:03:10.140
So we figured out the base.

00:03:10.140 --> 00:03:12.610
And to just remind ourselves,
that is equal to

00:03:12.610 --> 00:03:14.200
the change in x.

00:03:14.200 --> 00:03:17.100
That was equal to your finishing
x minus your

00:03:17.100 --> 00:03:17.890
starting x.

00:03:17.890 --> 00:03:20.020
6 minus 3.

00:03:20.020 --> 00:03:21.860
This is our delta x.

00:03:21.860 --> 00:03:25.990
Now, by the same exact line of
reasoning, this height right

00:03:25.990 --> 00:03:29.640
here is going to be
your change in y.

00:03:29.640 --> 00:03:31.290
Up here, you're at
y is equal to 0.

00:03:31.290 --> 00:03:32.670
That's kind of where
you finish.

00:03:32.670 --> 00:03:34.360
That's your higher y point.

00:03:34.360 --> 00:03:37.900
And over here, you're at y
is equal to negative 4.

00:03:37.900 --> 00:03:42.460
So change in y is equal
to 0 minus negative 4.

00:03:42.460 --> 00:03:44.930
I'm just taking the larger
y-value minus the smaller

00:03:44.930 --> 00:03:48.810
y-value, the larger x-value
minus the smaller x-value.

00:03:48.810 --> 00:03:49.870
But you're going to see we're
going to square it in a

00:03:49.870 --> 00:03:51.830
second, so even if you did it
the other way around, you'd

00:03:51.830 --> 00:03:53.410
get a negative number, but
you'd still get the same

00:03:53.410 --> 00:03:56.030
answer, so this is equal to 4.

00:03:56.030 --> 00:03:57.890
So this side is equal to 4.

00:03:57.890 --> 00:04:00.370
You can even count it on the
graph paper if you like.

00:04:00.370 --> 00:04:02.310
And this side is equal to 3.

00:04:02.310 --> 00:04:07.040
And now we can do the
Pythagorean theorem.

00:04:07.040 --> 00:04:11.730
This distance is the
distance squared.

00:04:11.730 --> 00:04:12.400
Be careful.

00:04:12.400 --> 00:04:15.270
The distance squared is going
to be equal to this delta x

00:04:15.270 --> 00:04:19.649
squared, the change
in x squared plus

00:04:19.649 --> 00:04:23.440
the change in y squared.

00:04:23.440 --> 00:04:24.620
This is nothing fancy.

00:04:24.620 --> 00:04:26.770
Sometimes people will call this
the distance formula.

00:04:26.770 --> 00:04:29.380
It's just the Pythagorean
theorem.

00:04:29.380 --> 00:04:31.660
This side squared plus that
side squared is equal to

00:04:31.660 --> 00:04:34.940
hypotenuse squared, because
this is a right triangle.

00:04:34.940 --> 00:04:38.260
So let's apply it with these
numbers, the numbers that we

00:04:38.260 --> 00:04:39.410
have at hand.

00:04:39.410 --> 00:04:43.020
So the distance squared is going
to be equal to delta x

00:04:43.020 --> 00:04:47.580
squared is 3 squared plus
delta y squared plus 4

00:04:47.580 --> 00:04:53.080
squared, which is equal to 9
plus 16, which is equal to 25.

00:04:53.080 --> 00:04:56.050
So the distance is equal to--
let me write that-- d squared

00:04:56.050 --> 00:04:57.550
is equal to 25.

00:04:57.550 --> 00:05:01.600
d, our distance, is equal to--
you don't want to take the

00:05:01.600 --> 00:05:03.160
negative square root, because
you can't have a negative

00:05:03.160 --> 00:05:05.370
distance, So it's only the
principal root, the positive

00:05:05.370 --> 00:05:09.020
square root of 25, which
is equal to 5.

00:05:09.020 --> 00:05:10.780
So this distance right
here is 5.

00:05:10.780 --> 00:05:13.130
Or if we look at this distance
right here, that was the

00:05:13.130 --> 00:05:13.920
original problem.

00:05:13.920 --> 00:05:16.170
How far is this point
from that point?

00:05:16.170 --> 00:05:20.180
It is 5 units away.

00:05:20.180 --> 00:05:22.930
So what you'll see here, they
call it the distance formula,

00:05:22.930 --> 00:05:24.730
but it's just the Pythagorean
theorem.

00:05:24.730 --> 00:05:27.000
And just so you're exposed to
all of the ways that you'll

00:05:27.000 --> 00:05:29.770
see the distance formula,
sometimes people will say, oh,

00:05:29.770 --> 00:05:33.340
if I have two points, if I have
one point, let's call it

00:05:33.340 --> 00:05:37.580
x1 and y1, so that's just
a particular point.

00:05:37.580 --> 00:05:42.880
And let's say I have another
point that is x2 comma y2.

00:05:42.880 --> 00:05:45.760
Sometimes, you'll see this
formula, that the distance--

00:05:45.760 --> 00:05:46.940
you'll see it in
different ways.

00:05:46.940 --> 00:05:49.850
But you'll see that the distance
is equal to-- and it

00:05:49.850 --> 00:05:52.170
looks as though there's this
really complicated formula,

00:05:52.170 --> 00:05:53.930
but I want you to see that
this is really just the

00:05:53.930 --> 00:05:55.370
Pythagorean theorem.

00:05:55.370 --> 00:06:05.340
You see that the distance is
equal to x2 minus x1 minus x1

00:06:05.340 --> 00:06:18.650
squared plus y2 minus
y1 squared.

00:06:18.650 --> 00:06:20.950
You'll see this written in
a lot of textbooks as the

00:06:20.950 --> 00:06:22.200
distance formula.

00:06:24.855 --> 00:06:27.250
And it's a complete waste of
your time to memorize it

00:06:27.250 --> 00:06:30.060
because it's really just the
Pythagorean theorem.

00:06:30.060 --> 00:06:32.370
This is your change in x.

00:06:32.370 --> 00:06:34.710
And it really doesn't matter
which x you pick to be first

00:06:34.710 --> 00:06:36.730
or second, because even if you
get the negative of this

00:06:36.730 --> 00:06:39.470
value, when you square it,
the negative disappears.

00:06:39.470 --> 00:06:42.900
This right here is
your change in y.

00:06:42.900 --> 00:06:44.693
So it's just saying that the
distance squared-- remember,

00:06:44.693 --> 00:06:48.710
if you square both sides of
this equation, the radical

00:06:48.710 --> 00:06:51.010
will disappear and this will
be the distance squared is

00:06:51.010 --> 00:06:54.900
equal to this expression
squared, delta x squared,

00:06:54.900 --> 00:06:59.090
change in x-- delta means change
in-- delta x squared

00:06:59.090 --> 00:07:01.540
plus delta y squared.

00:07:01.540 --> 00:07:02.190
I don't want to confuse you.

00:07:02.190 --> 00:07:04.170
Delta y just means
change in y.

00:07:04.170 --> 00:07:06.380
I should have probably said
that earlier in the video.

00:07:06.380 --> 00:07:09.290
But let's apply it to a couple
more, and I'll just pick some

00:07:09.290 --> 00:07:10.030
points at random.

00:07:10.030 --> 00:07:14.850
Let's say I have the point,
let's see, 1, 2, 3, 4, 5, 6.

00:07:14.850 --> 00:07:17.660
Negative 6 comma negative 4.

00:07:22.370 --> 00:07:24.430
And let's say I want to find the
distance between that and

00:07:24.430 --> 00:07:32.290
1 comma 1, 2, 3, 4, 5, 6, 7, and
the point 1 comma 7, so I

00:07:32.290 --> 00:07:34.370
want to find this distance
right here.

00:07:34.370 --> 00:07:36.420
So it's the exact same idea.

00:07:36.420 --> 00:07:38.610
We just use the Pythagorean
theorem.

00:07:38.610 --> 00:07:41.830
You figure out this distance,
which is our change in x, this

00:07:41.830 --> 00:07:43.790
distance, which is
our change in y.

00:07:43.790 --> 00:07:45.955
This distance squared plus this
distance squared is going

00:07:45.955 --> 00:07:48.750
to equal that distance
squared.

00:07:48.750 --> 00:07:50.600
So let's do it.

00:07:50.600 --> 00:07:53.160
So our change in x, you
just take-- you

00:07:53.160 --> 00:07:54.130
know, it doesn't matter.

00:07:54.130 --> 00:07:56.170
In general, you want to take the
larger x-value minus the

00:07:56.170 --> 00:07:58.400
smaller x-value, but you
could do it either way.

00:07:58.400 --> 00:08:02.180
So we could write the distance
squared is equal to-- what's

00:08:02.180 --> 00:08:04.430
our change in x?

00:08:04.430 --> 00:08:08.650
So let's take the larger x minus
the smaller x, 1 minus

00:08:08.650 --> 00:08:09.430
negative 6.

00:08:09.430 --> 00:08:14.960
1 minus negative 6 squared
plus the change in y.

00:08:14.960 --> 00:08:16.120
The larger y is here.

00:08:16.120 --> 00:08:17.530
It's 7.

00:08:17.530 --> 00:08:19.210
7 minus negative 4.

00:08:19.210 --> 00:08:22.550
7 minus negative 4 squared.

00:08:22.550 --> 00:08:24.430
And I just picked these numbers
at random, so they're

00:08:24.430 --> 00:08:26.260
probably not going to come
out too cleanly.

00:08:26.260 --> 00:08:30.490
So we get that the distance
squared is equal to 1 minus

00:08:30.490 --> 00:08:31.280
negative 6.

00:08:31.280 --> 00:08:35.000
That is 7, 7 squared, and you'll
even see it over here,

00:08:35.000 --> 00:08:36.169
if you count it.

00:08:36.169 --> 00:08:39.690
You go, 1, 2, 3, 4, 5, 6, 7.

00:08:39.690 --> 00:08:41.470
That's that number right here.

00:08:41.470 --> 00:08:43.400
That's what your
change in x is.

00:08:43.400 --> 00:08:46.530
Plus 7 minus negative 4.

00:08:46.530 --> 00:08:47.720
That's 11.

00:08:47.720 --> 00:08:51.010
That's this distance right here,
and you can count it on

00:08:51.010 --> 00:08:51.490
the blocks.

00:08:51.490 --> 00:08:52.650
We're going up 11.

00:08:52.650 --> 00:08:55.860
We're just taking 7 minus
negative 4 to get

00:08:55.860 --> 00:08:57.740
a distance of 11.

00:08:57.740 --> 00:09:01.750
So plus 11 squared is
equal to d squared.

00:09:01.750 --> 00:09:05.000
So let me just take the
calculator out.

00:09:05.000 --> 00:09:13.266
So the distance if we take 7
squared plus 11 squared is

00:09:13.266 --> 00:09:16.170
equal to 170, that distance is
going to be the square root of

00:09:16.170 --> 00:09:18.920
that, right? d squared
is equal to 170.

00:09:18.920 --> 00:09:25.250
So let's take the square root
of 170 and we get 13.0,

00:09:25.250 --> 00:09:28.250
roughly 13.04.

00:09:28.250 --> 00:09:30.550
So this distance right
here that we tried to

00:09:30.550 --> 00:09:35.920
figure out is 13.04.

00:09:35.920 --> 00:09:38.540
Hopefully, you found
that helpful.


Pythagorean theorem

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:04.650
Let's now talk about what is
easily one of the most famous

00:00:04.650 --> 00:00:07.230
theorems in all of
mathematics.

00:00:07.230 --> 00:00:09.230
And that's the Pythagorean
theorem.

00:00:18.440 --> 00:00:20.980
And it deals with
right triangles.

00:00:20.980 --> 00:00:28.040
So a right triangle is a
triangle that has a 90 degree

00:00:28.040 --> 00:00:29.000
angle in it.

00:00:29.000 --> 00:00:31.340
So the way I drew it
right here, this is

00:00:31.340 --> 00:00:33.330
our 90 degree angle.

00:00:33.330 --> 00:00:36.320
If you've never seen a 90 degree
angle before, the way

00:00:36.320 --> 00:00:39.700
to think about it is, if this
side goes straight left to

00:00:39.700 --> 00:00:42.600
right, this side goes straight
up and down.

00:00:42.600 --> 00:00:46.970
These sides are perpendicular,
or the angle between them is

00:00:46.970 --> 00:00:50.110
90 degrees, or it is
a right angle.

00:00:50.110 --> 00:00:52.990
And the Pythagorean theorem
tells us that if we're dealing

00:00:52.990 --> 00:00:56.180
with a right triangle-- let me
write that down-- if we're

00:00:56.180 --> 00:01:02.340
dealing with a right triangle--
not a wrong

00:01:02.340 --> 00:01:05.640
triangle-- if we're dealing with
a right triangle, which

00:01:05.640 --> 00:01:11.140
is a triangle that has a right
angle, or a 90 degree angle in

00:01:11.140 --> 00:01:17.060
it, then the relationship
between their sides is this.

00:01:17.060 --> 00:01:23.670
So this side is a, this side
is b, and this side is c.

00:01:23.670 --> 00:01:26.190
And remember, the c that we're
dealing with right here is the

00:01:26.190 --> 00:01:30.510
side opposite the
90 degree angle.

00:01:30.510 --> 00:01:33.430
It's important to keep track
of which side is which.

00:01:33.430 --> 00:01:38.030
The Pythagorean theorem tells us
that if and only if this is

00:01:38.030 --> 00:01:44.330
a right triangle, then a squared
plus b squared is

00:01:44.330 --> 00:01:48.270
going to be equal
to c squared.

00:01:48.270 --> 00:01:49.630
And we can use this
information.

00:01:49.630 --> 00:01:51.960
If we know two of these, we
can then use this theorem,

00:01:51.960 --> 00:01:54.440
this formula to solve
for the third.

00:01:54.440 --> 00:01:58.640
And I'll give you one more piece
of terminology here.

00:01:58.640 --> 00:02:02.600
This long side, the side that
is the longest side of our

00:02:02.600 --> 00:02:05.650
right triangle, the side that
is opposite of our right

00:02:05.650 --> 00:02:09.590
angle, this right here-- it's
c in this example-- this is

00:02:09.590 --> 00:02:10.840
called a hypotenuse.

00:02:13.400 --> 00:02:16.090
A very fancy word for
a very simple idea.

00:02:16.090 --> 00:02:19.050
The longest side of a right
triangle, the side that is

00:02:19.050 --> 00:02:22.770
opposite the 90 degree angle,
is called the hypotentuse.

00:02:22.770 --> 00:02:26.440
Now that we know the Pythagorean
theorem, let's

00:02:26.440 --> 00:02:27.620
actually use it.

00:02:27.620 --> 00:02:30.070
Because it's one thing to know
something, but it's a lot more

00:02:30.070 --> 00:02:31.250
fun to use it.

00:02:31.250 --> 00:02:35.860
So let's say I have the
following right triangle.

00:02:35.860 --> 00:02:37.800
Let me draw it a little
bit neater than that.

00:02:42.240 --> 00:02:43.770
It's a right triangle.

00:02:43.770 --> 00:02:47.080
This side over here
has length 9.

00:02:47.080 --> 00:02:49.950
This side over here
has length 7.

00:02:49.950 --> 00:02:52.900
And my question is, what
is this side over here?

00:02:52.900 --> 00:02:56.130
Maybe we can call that--
we'll call that c.

00:02:56.130 --> 00:02:58.710
Well, c, in this case, once
again, it is the hypotenuse.

00:02:58.710 --> 00:03:00.640
It is the longest side.

00:03:00.640 --> 00:03:03.840
So we know that the sum of the
squares of the other side is

00:03:03.840 --> 00:03:06.140
going to be equal
to c squared.

00:03:06.140 --> 00:03:13.900
So by the Pythagorean theorem,
9 squared plus 7 squared is

00:03:13.900 --> 00:03:17.510
going to be equal
to c squared.

00:03:17.510 --> 00:03:24.260
9 squared is 81, plus
7 squared is 49.

00:03:24.260 --> 00:03:28.130
80 plus 40 is 120.

00:03:28.130 --> 00:03:30.630
Then we're going to have the 1
plus the 9, that's another 10,

00:03:30.630 --> 00:03:34.260
so this is going to
be equal to 130.

00:03:34.260 --> 00:03:36.660
So let me write it this way.

00:03:36.660 --> 00:03:40.290
The left-hand side is going to
be equal to 130, and that is

00:03:40.290 --> 00:03:43.600
equal to c squared.

00:03:43.600 --> 00:03:45.280
So what's c going
to be equal to?

00:03:45.280 --> 00:03:47.170
Let me rewrite it over here.

00:03:47.170 --> 00:03:52.550
c squared is equal to 130, or
we could say that c is equal

00:03:52.550 --> 00:03:56.330
to the square root of 130.

00:03:56.330 --> 00:03:58.610
And notice, I'm only taking
the principal root here,

00:03:58.610 --> 00:04:00.210
because c has to be positive.

00:04:00.210 --> 00:04:03.580
We're dealing with a distance,
so we can't take the negative

00:04:03.580 --> 00:04:04.120
square root.

00:04:04.120 --> 00:04:05.400
So we'll only take
the principal

00:04:05.400 --> 00:04:06.600
square root right here.

00:04:06.600 --> 00:04:08.716
And if we want to simplify this
a little bit, we know how

00:04:08.716 --> 00:04:09.960
to simplify our radicals.

00:04:09.960 --> 00:04:18.899
130 is 2 times 65, which
is 5 times 13.

00:04:18.899 --> 00:04:20.910
Well, these are all prime
numbers, so that's about as

00:04:20.910 --> 00:04:23.740
simple as I can get.
c is equal to the

00:04:23.740 --> 00:04:27.820
square root of 130.

00:04:27.820 --> 00:04:31.130
Let's do another one of these.

00:04:31.130 --> 00:04:33.930
Maybe I want to keep this
Pythagorean theorem right

00:04:33.930 --> 00:04:35.830
there, just so we always
remember what

00:04:35.830 --> 00:04:37.400
we're referring to.

00:04:37.400 --> 00:04:40.490
So let's say I have a triangle
that looks like this.

00:04:40.490 --> 00:04:41.260
Let's see.

00:04:41.260 --> 00:04:42.720
Let's say it looks like that.

00:04:45.460 --> 00:04:49.720
And this is the right
angle, up here.

00:04:49.720 --> 00:04:52.160
Let's say that this side,
I'm going to call it a.

00:04:52.160 --> 00:04:56.140
The side, it's going
to have length 21.

00:04:56.140 --> 00:04:59.450
And this side right here is
going to be of length 35.

00:04:59.450 --> 00:05:02.800
So your instinct to solve for a,
might say, hey, 21 squared

00:05:02.800 --> 00:05:04.710
plus 35 squared is going to
be equal to a squared.

00:05:04.710 --> 00:05:09.930
But notice, in this situation,
35 is a hypotenuse.

00:05:09.930 --> 00:05:11.390
35 is our c.

00:05:11.390 --> 00:05:15.590
It's the longest side of
our right triangle.

00:05:15.590 --> 00:05:19.010
So what the Pythagorean theorem
tells us is that a

00:05:19.010 --> 00:05:23.800
squared plus the other
non-longest side-- the other

00:05:23.800 --> 00:05:28.970
non-hypotenuse squared-- so a
squared plus 21 squared is

00:05:28.970 --> 00:05:33.190
going to be equal
to 35 squared.

00:05:33.190 --> 00:05:36.540
You always have to remember, the
c squared right here, the

00:05:36.540 --> 00:05:39.500
c that we're talking about, is
always going to be the longest

00:05:39.500 --> 00:05:41.450
side of your right triangle.

00:05:41.450 --> 00:05:46.020
The side that is opposite
of our right angle.

00:05:46.020 --> 00:05:48.060
This is the side that's opposite
of the right angle.

00:05:48.060 --> 00:05:52.610
So a squared plus 21 squared
is equal to 35 squared.

00:05:52.610 --> 00:05:54.100
And what do we have here?

00:05:54.100 --> 00:05:58.740
So 21 squared-- I'm tempted to
use a calculator, but I won't.

00:05:58.740 --> 00:06:06.530
So 21 times 21: 1 times 21
is 21, 2 times 21 is 42.

00:06:06.530 --> 00:06:09.510
It is 441.

00:06:09.510 --> 00:06:10.600
35 squared.

00:06:10.600 --> 00:06:13.860
Once again, I'm tempted to use
a calculator, but I won't.

00:06:13.860 --> 00:06:20.700
35 times 35: 5 times 5 is 25.

00:06:20.700 --> 00:06:22.280
Carry the 2.

00:06:22.280 --> 00:06:28.400
5 times 3 is 15, plus 2 is 17.

00:06:28.400 --> 00:06:30.840
Put a 0 here, get rid
of that thing.

00:06:30.840 --> 00:06:34.130
3 times 5 is 15.

00:06:34.130 --> 00:06:38.030
3 times 3 is 9, plus 1 is 10.

00:06:38.030 --> 00:06:44.210
So it is 11-- let me do it in
order-- 5 plus 0 is 5, 7 plus

00:06:44.210 --> 00:06:49.070
5 is 12, 1 plus 1 is 2,
bring down the 1.

00:06:49.070 --> 00:06:50.670
1225.

00:06:50.670 --> 00:06:58.540
So this tells us that a squared
plus 441 is going to

00:06:58.540 --> 00:07:03.450
be equal to 35 squared,
which is 1225.

00:07:03.450 --> 00:07:06.300
Now, we could subtract
441 from both

00:07:06.300 --> 00:07:07.550
sides of this equation.

00:07:11.350 --> 00:07:14.760
The left-hand side just
becomes a squared.

00:07:14.760 --> 00:07:17.510
The right-hand side,
what do we get?

00:07:17.510 --> 00:07:22.210
We get 5 minus 1 is 4.

00:07:22.210 --> 00:07:25.793
We want to-- let me write this
a little bit neater here.

00:07:29.980 --> 00:07:32.810
Minus 441.

00:07:32.810 --> 00:07:35.260
So the left-hand side, once
again, they cancel out. a

00:07:35.260 --> 00:07:39.110
squared is equal to-- and then
on the right-hand side, what

00:07:39.110 --> 00:07:40.440
do we have to do?

00:07:40.440 --> 00:07:43.380
That's larger than that, but 2
is not larger than 4, so we're

00:07:43.380 --> 00:07:45.020
going to have to borrow.

00:07:45.020 --> 00:07:47.660
So that becomes a 12, or
regrouped, depending on how

00:07:47.660 --> 00:07:48.320
you want to view it.

00:07:48.320 --> 00:07:49.710
That becomes a 1.

00:07:49.710 --> 00:07:52.780
1 is not greater than
4, so we're going to

00:07:52.780 --> 00:07:54.110
have to borrow again.

00:07:54.110 --> 00:07:54.800
Get rid of that.

00:07:54.800 --> 00:07:57.170
And then this becomes an 11.

00:07:57.170 --> 00:08:00.100
5 minus 1 is 4.

00:08:00.100 --> 00:08:02.780
12 minus 4 is 8.

00:08:02.780 --> 00:08:05.850
11 minus 4 is 7.

00:08:05.850 --> 00:08:08.640
So a squared is equal to 784.

00:08:08.640 --> 00:08:14.380
And we could write, then,
that a is equal to the

00:08:14.380 --> 00:08:17.910
square root of 784.

00:08:17.910 --> 00:08:20.800
And once again, I'm very tempted
to use a calculator,

00:08:20.800 --> 00:08:24.710
but let's, well, let's not.

00:08:24.710 --> 00:08:25.630
Let's not use it.

00:08:25.630 --> 00:08:28.880
So this is 2 times, what?

00:08:28.880 --> 00:08:35.400
392.

00:08:35.400 --> 00:08:41.890
And then this-- 390 times
2 is 78, yeah.

00:08:41.890 --> 00:08:44.390
And then this is
2 times, what?

00:08:44.390 --> 00:08:49.790
This is 2 times 196.

00:08:49.790 --> 00:08:50.900
That's right.

00:08:50.900 --> 00:08:55.680
190 times 2 is-- yeah,
that's 2 times 196.

00:08:55.680 --> 00:09:00.060
196 is 2 times-- I want to
make sure I don't make a

00:09:00.060 --> 00:09:00.760
careless mistake.

00:09:00.760 --> 00:09:06.200
196 is 2 times 98.

00:09:06.200 --> 00:09:08.230
Let's keep going down here.

00:09:08.230 --> 00:09:16.850
98 is 2 times 49.

00:09:16.850 --> 00:09:18.540
And, of course, we know
what that is.

00:09:18.540 --> 00:09:21.120
So notice, we have 2 times
2, times 2, times 2.

00:09:21.120 --> 00:09:23.000
So this is 2 to the
fourth power.

00:09:23.000 --> 00:09:24.680
So it's 16 times 49.

00:09:24.680 --> 00:09:30.220
So a is equal to the square
root of 16 times 49.

00:09:30.220 --> 00:09:32.990
I picked those numbers because
they're both perfect squares.

00:09:32.990 --> 00:09:37.480
So this is equal to the square
root of 16 is 4, times the

00:09:37.480 --> 00:09:39.930
square root of 49 is 7.

00:09:39.930 --> 00:09:43.260
It's equal to 28.

00:09:43.260 --> 00:09:47.800
So this side right here is going
to be equal to 28, by

00:09:47.800 --> 00:09:49.980
the Pythagorean theorem.

00:09:49.980 --> 00:09:52.650
Let's do one more of these.

00:09:52.650 --> 00:09:55.770
Can never get enough practice.

00:09:55.770 --> 00:09:57.760
So let's say I have
another triangle.

00:09:57.760 --> 00:09:59.680
I'll draw this one big.

00:09:59.680 --> 00:10:01.330
There you go.

00:10:01.330 --> 00:10:02.380
That's my triangle.

00:10:02.380 --> 00:10:04.160
That is the right angle.

00:10:04.160 --> 00:10:05.870
This side is 24.

00:10:05.870 --> 00:10:07.390
This side is 12.

00:10:07.390 --> 00:10:10.000
We'll call this side
right here b.

00:10:10.000 --> 00:10:12.900
Now, once again, always identify
the hypotenuse.

00:10:12.900 --> 00:10:15.270
That's the longest side,
the side opposite

00:10:15.270 --> 00:10:16.070
the 90 degree angle.

00:10:16.070 --> 00:10:17.620
You might say, hey, I don't know
that's the longest side.

00:10:17.620 --> 00:10:19.020
I don't know what b is yet.

00:10:19.020 --> 00:10:19.980
How do I know this is longest?

00:10:19.980 --> 00:10:22.670
And there, in that situation,
you say, well, it's the side

00:10:22.670 --> 00:10:25.380
opposite the 90 degree angle.

00:10:25.380 --> 00:10:31.070
So if that's the hypotenuse,
then this squared plus that

00:10:31.070 --> 00:10:33.680
squared is going to be
equal to 24 squared.

00:10:33.680 --> 00:10:38.370
So the Pythagorean theorem-- b
squared plus 12 squared is

00:10:38.370 --> 00:10:42.100
equal to 24 squared.

00:10:42.100 --> 00:10:44.930
Or we could subtract 12 squared
from both sides.

00:10:44.930 --> 00:10:50.350
We say, b squared is equal to
24 squared minus 12 squared,

00:10:50.350 --> 00:10:55.200
which we know is 144, and that
b is equal to the square root

00:10:55.200 --> 00:10:59.660
of 24 squared minus
12 squared.

00:10:59.660 --> 00:11:03.490
Now I'm tempted to use a
calculator, and I'll give into

00:11:03.490 --> 00:11:04.990
the temptation.

00:11:04.990 --> 00:11:07.770
So let's do it.

00:11:07.770 --> 00:11:10.550
The last one was so painful,
I'm still recovering.

00:11:10.550 --> 00:11:20.350
So 24 squared minus 12 squared
is equal to 24.78.

00:11:20.350 --> 00:11:23.110
So this actually turns into--
let me do it without a-- well,

00:11:23.110 --> 00:11:23.730
I'll do it halfway.

00:11:23.730 --> 00:11:30.390
24 squared minus 12 squared
is equal to 432.

00:11:30.390 --> 00:11:37.000
So b is equal to the
square root of 432.

00:11:37.000 --> 00:11:38.080
And let's factor this again.

00:11:38.080 --> 00:11:40.680
We saw what the answer is, but
maybe we can write it in kind

00:11:40.680 --> 00:11:42.830
of a simplified radical form.

00:11:42.830 --> 00:11:47.090
So this is 2 times 216.

00:11:47.090 --> 00:11:49.740
216, I believe, is
a-- let me see.

00:11:49.740 --> 00:11:51.570
I believe that's a
perfect square.

00:11:51.570 --> 00:11:56.460
So let me take the square
root of 216.

00:11:56.460 --> 00:11:57.550
Nope, not a perfect square.

00:11:57.550 --> 00:11:59.940
So 216, let's just keep going.

00:11:59.940 --> 00:12:03.700
216 is 2 times 108.

00:12:03.700 --> 00:12:10.220
108 is, we could say,
4 times what?

00:12:10.220 --> 00:12:18.050
25 plus another 2-- 4 times
27, which is 9 times 3.

00:12:18.050 --> 00:12:19.470
So what do we have here?

00:12:19.470 --> 00:12:24.550
We have 2 times 2, times 4, so
this right here is a 16.

00:12:24.550 --> 00:12:26.430
16 times 9 times 3.

00:12:26.430 --> 00:12:27.720
Is that right?

00:12:27.720 --> 00:12:29.260
I'm using a different
calculator.

00:12:29.260 --> 00:12:35.580
16 times 9 times 3
is equal to 432.

00:12:35.580 --> 00:12:39.700
So this is going to be equal
to-- b is equal to the square

00:12:39.700 --> 00:12:45.490
root of 16 times 9, times 3,
which is equal to the square

00:12:45.490 --> 00:12:49.450
root of 16, which is 4 times the
square root of 9, which is

00:12:49.450 --> 00:12:52.660
3, times the square root
of 3, which is equal

00:12:52.660 --> 00:12:56.420
to 12 roots of 3.

00:12:56.420 --> 00:13:00.540
So b is 12 times the
square root of 3.

00:13:00.540 --> 00:13:02.560
Hopefully you found
that useful.


More Involved Radical Equation Example

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.090
Let's do one more radical
equation example.

00:00:03.090 --> 00:00:06.400
And we'll make this one
a slightly hairy one.

00:00:06.400 --> 00:00:10.600
So let's say we have the
principal square root of 10

00:00:10.600 --> 00:00:20.640
minus 5x plus the principal
square root of 1 minus x is

00:00:20.640 --> 00:00:23.410
equal to 7.

00:00:23.410 --> 00:00:26.470
Now, just like I've said over
the last two videos, your goal

00:00:26.470 --> 00:00:29.450
is always to isolate one of the
radicals and then square

00:00:29.450 --> 00:00:31.060
both sides of the equation.

00:00:31.060 --> 00:00:34.690
So we can isolate this one right
here by subtracting this

00:00:34.690 --> 00:00:36.710
radical from both sides
of this equation.

00:00:36.710 --> 00:00:40.790
So you get minus the square root
of 1 minus x, minus the

00:00:40.790 --> 00:00:43.720
square root of 1 minus x.

00:00:43.720 --> 00:00:46.180
The left-hand side,
these cancel out.

00:00:46.180 --> 00:00:49.690
The left-hand side, you're just
left with the square root

00:00:49.690 --> 00:00:54.870
of 10 minus 5x is equal to--
on the right-hand side, you

00:00:54.870 --> 00:01:01.880
have 7 minus the square
root of 1 minus x.

00:01:01.880 --> 00:01:04.090
All I did is subtracted this
from both sides, essentially

00:01:04.090 --> 00:01:06.060
moving it over to the
right-hand side.

00:01:06.060 --> 00:01:08.820
Now we've isolated one
of the radicals.

00:01:08.820 --> 00:01:11.680
Let's square both sides
of this equation.

00:01:11.680 --> 00:01:15.170
So this squared, 10 minus 5x
squared, is just going to be--

00:01:15.170 --> 00:01:18.980
or the square root of 10 minus
5x squared, so if I square

00:01:18.980 --> 00:01:22.400
that, that's just going to
be equal to 10 minus 5x.

00:01:22.400 --> 00:01:26.210
And this is going to be equal
to this expression squared.

00:01:26.210 --> 00:01:29.460
Now, this might look a little
daunting, but you just have to

00:01:29.460 --> 00:01:37.710
remember that a plus b squared
is equal to a squared plus 2ab

00:01:37.710 --> 00:01:38.890
plus b squared.

00:01:38.890 --> 00:01:39.740
In fact, you don't have
to remember it.

00:01:39.740 --> 00:01:42.600
You could actually multiply it
out if you ever forget this.

00:01:42.600 --> 00:01:47.110
Now, this is just the same
situation where a is 7, and

00:01:47.110 --> 00:01:51.510
our b is negative the square
root of 1 minus x.

00:01:51.510 --> 00:01:54.840
So if we expand this out or
multiply this out, this is

00:01:54.840 --> 00:02:00.250
going to be 7 squared,
which is 49.

00:02:00.250 --> 00:02:03.920
And then plus 2 times
7 times this.

00:02:03.920 --> 00:02:05.660
So this is a negative right
here, so it's going to be

00:02:05.660 --> 00:02:15.220
minus 2 times 7 times the square
root of 1 minus x.

00:02:15.220 --> 00:02:18.225
So we've got the negative side
from that negative sign, the 7

00:02:18.225 --> 00:02:20.740
is over there, and 1 minus the
square root of 1 minus x is

00:02:20.740 --> 00:02:22.740
there, 2 times it.

00:02:22.740 --> 00:02:24.520
When you take a perfect square,
that just ends up,

00:02:24.520 --> 00:02:26.000
when you actually take
the product.

00:02:26.000 --> 00:02:30.100
And then finally, plus
this guy squared.

00:02:30.100 --> 00:02:31.760
This guy over here squared.

00:02:31.760 --> 00:02:34.370
Now the negative part, negative
1 squared, that just

00:02:34.370 --> 00:02:35.550
becomes positive.

00:02:35.550 --> 00:02:38.780
Square root of 1 minus x
squared is just going

00:02:38.780 --> 00:02:41.570
to be 1 minus x.

00:02:41.570 --> 00:02:43.390
And let's simplify this.

00:02:43.390 --> 00:02:49.430
So this is going to be equal to
10 minus 5x is equal to--

00:02:49.430 --> 00:02:52.720
well, one thing we can do, let's
add the 49 and the 1.

00:02:52.720 --> 00:02:58.830
So you add the 49 and the 1, you
get 50, and then minus--

00:02:58.830 --> 00:03:05.020
this is 2 times 7 is 14 times
the square root of 1 minus x.

00:03:05.020 --> 00:03:09.440
And then we have this final
minus x out here, minus x.

00:03:09.440 --> 00:03:12.090
And let's see what else
we can do here.

00:03:12.090 --> 00:03:14.960
We could subtract 10 from both
sides of this equation.

00:03:14.960 --> 00:03:18.010
I don't want to do too many
steps at once, because this is

00:03:18.010 --> 00:03:20.190
a slightly involved problem.

00:03:20.190 --> 00:03:23.710
So if you subtract 10 from both
sides, you get negative

00:03:23.710 --> 00:03:30.080
5x is equal to 40 minus 14 times
the square root of 1

00:03:30.080 --> 00:03:32.790
minus x minus x.

00:03:32.790 --> 00:03:37.050
And then we can add x to both
sides, essentially getting rid

00:03:37.050 --> 00:03:38.680
of that character over there.

00:03:38.680 --> 00:03:43.120
So if you add x to both sides
of this equation, the

00:03:43.120 --> 00:03:47.650
left-hand side becomes
negative 4x, and the

00:03:47.650 --> 00:03:50.750
right-hand side, this guy
disappears, and you're just

00:03:50.750 --> 00:03:58.870
left with 40 minus 14 times the
square root of 1 minus x.

00:03:58.870 --> 00:04:01.220
And now let's subtract
40 from both sides.

00:04:01.220 --> 00:04:03.780
I'm just trying to isolate this
expression right here.

00:04:03.780 --> 00:04:06.590
So let's subtract 40
from both sides.

00:04:06.590 --> 00:04:14.360
We get negative 4x minus 40 is
equal to negative 14 times the

00:04:14.360 --> 00:04:16.279
square root of 1 minus x.

00:04:16.279 --> 00:04:18.589
Just subtracted this from both
sides, so it disappeared on

00:04:18.589 --> 00:04:21.519
the right, shows up in negative
form on the left.

00:04:21.519 --> 00:04:25.180
Now, we could square both sides,
but let's simplify it a

00:04:25.180 --> 00:04:25.920
little bit.

00:04:25.920 --> 00:04:27.780
Let's divide both sides
by negative 2.

00:04:27.780 --> 00:04:30.050
That'll get rid of all of these
negative numbers and

00:04:30.050 --> 00:04:31.520
make the numbers a little
bit smaller.

00:04:31.520 --> 00:04:33.580
So you divide everything
by negative 2.

00:04:33.580 --> 00:04:38.200
We get 2x-- let me
switch colors.

00:04:38.200 --> 00:04:46.170
We get 2x plus the 20 is equal
to 7 times the square

00:04:46.170 --> 00:04:48.450
root of 1 minus x.

00:04:48.450 --> 00:04:50.540
Just divide everything
by negative 2.

00:04:50.540 --> 00:04:54.030
40 divided by negative 2 is 20,
14 divided by negative 2

00:04:54.030 --> 00:04:55.560
is positive 7.

00:04:55.560 --> 00:04:59.350
Now we could divide by 7, but
that makes things awkward.

00:04:59.350 --> 00:05:01.280
We've essentially isolated
the radical.

00:05:01.280 --> 00:05:02.670
We're scaling it by
a little bit.

00:05:02.670 --> 00:05:06.350
But if we square both sides of
this equation, that radical

00:05:06.350 --> 00:05:07.930
will disappear.

00:05:07.930 --> 00:05:10.030
So let's square both sides
of that equation.

00:05:14.060 --> 00:05:22.520
The left-hand side becomes 4x
squared plus 2x time 20 is 40x

00:05:22.520 --> 00:05:29.330
times 2 is 80x plus 400 is
going to be equal to 7

00:05:29.330 --> 00:05:33.730
squared, which is 49, times
this expression squared.

00:05:33.730 --> 00:05:36.760
Well, the square root of 1 minus
x squared is just going

00:05:36.760 --> 00:05:41.435
to be 1 minus x.

00:05:44.800 --> 00:05:46.590
Let's just keep going
down here.

00:05:46.590 --> 00:05:47.730
I have space down here.

00:05:47.730 --> 00:05:50.490
I can just keep going.

00:05:50.490 --> 00:05:52.370
So what can we do here?

00:05:52.370 --> 00:05:54.220
I don't want to skip
too many steps.

00:05:54.220 --> 00:05:58.790
So 4x squared plus 80x plus
400 is equal to--

00:05:58.790 --> 00:05:59.700
distribute the 49.

00:05:59.700 --> 00:06:04.040
49 minus 49x.

00:06:04.040 --> 00:06:09.120
And now, let's see, we
could add 49x to both

00:06:09.120 --> 00:06:10.010
sides of this equation.

00:06:10.010 --> 00:06:11.130
Actually, let's do this.

00:06:11.130 --> 00:06:21.930
Let's add 49x to both sides,
and let's subtract 49 from

00:06:21.930 --> 00:06:23.570
both sides of the equation.

00:06:23.570 --> 00:06:27.100
And we are left with--
the left-hand

00:06:27.100 --> 00:06:30.840
side becomes 4x squared.

00:06:30.840 --> 00:06:40.160
Now, 80 plus 49 is-- 80 plus 40
is 120, so it's going to be

00:06:40.160 --> 00:06:43.830
129, plus 129x.

00:06:43.830 --> 00:06:51.500
And then 400 minus 49 is 351,
plus 351 is equal to-- these

00:06:51.500 --> 00:06:55.520
guys cancel out, those guys
cancel out-- is equal to 0.

00:06:55.520 --> 00:06:57.920
Now, let me just make sure that
I haven't made some crazy

00:06:57.920 --> 00:07:00.460
mistake because these are
strange-looking numbers.

00:07:00.460 --> 00:07:03.230
80 plus 49-- yep, that's right.

00:07:03.230 --> 00:07:03.980
129.

00:07:03.980 --> 00:07:07.020
400 minus 49, that's 351.

00:07:07.020 --> 00:07:09.900
Now we have just a straight-up
quadratic equation.

00:07:09.900 --> 00:07:11.710
Let's use the quadratic formula
since we have all

00:07:11.710 --> 00:07:13.100
these crazy numbers.

00:07:13.100 --> 00:07:16.220
So our solutions, we're
going to get x is

00:07:16.220 --> 00:07:18.330
equal to negative b.

00:07:18.330 --> 00:07:29.230
So it's negative 129 plus or
minus the square root of 129

00:07:29.230 --> 00:07:41.490
squared minus 4 times a, which
is 4, times c, which is 351--

00:07:41.490 --> 00:07:43.920
definitely going to need a
calculator for this one-- all

00:07:43.920 --> 00:07:49.990
of that over 2 times a, which
is 2 times 4, which is 8.

00:07:49.990 --> 00:07:52.430
Well, let's get our
calculator out to

00:07:52.430 --> 00:07:54.660
actually calculate this.

00:07:54.660 --> 00:07:58.430
So let's figure out the
discriminant first. So we have

00:07:58.430 --> 00:08:07.570
129 squared minus-- we could
say 16 times 351.

00:08:07.570 --> 00:08:11.450
Nope, that's not what
I wanted to do.

00:08:11.450 --> 00:08:14.160
16 times 351.

00:08:14.160 --> 00:08:19.070
4 times 4 is 16, times 351,
is that crazy number.

00:08:19.070 --> 00:08:22.790
Let's take the square
root of it.

00:08:22.790 --> 00:08:26.870
And on this calculator,
you press second and

00:08:26.870 --> 00:08:28.380
answer right here.

00:08:28.380 --> 00:08:30.480
That'll use the last answer,
so it's going to take the

00:08:30.480 --> 00:08:33.080
square root of that
right there.

00:08:33.080 --> 00:08:34.049
It's 105.

00:08:34.049 --> 00:08:36.400
So lucky for us, it's not
that crazy of a number.

00:08:36.400 --> 00:08:41.650
So we get x is equal to
negative 129 plus or

00:08:41.650 --> 00:08:46.050
minus 105 over 8.

00:08:46.050 --> 00:08:47.980
Let's just use a calculator
for this.

00:08:47.980 --> 00:08:52.780
So the first one, if we have
a negative 129 plus

00:08:52.780 --> 00:08:56.600
105, what do we get?

00:08:56.600 --> 00:08:59.580
Negative 24, then you divide
that by 8, that's going to be

00:08:59.580 --> 00:09:00.800
negative 3.

00:09:00.800 --> 00:09:02.740
So one of our answers
is negative 3.

00:09:02.740 --> 00:09:12.130
Now, what happens if we take
negative 129 minus 105?

00:09:12.130 --> 00:09:14.980
We get negative 234.

00:09:14.980 --> 00:09:16.385
Divide that by 8.

00:09:19.100 --> 00:09:22.250
We get negative 29.25.

00:09:22.250 --> 00:09:26.420
So our two answers are negative
3 and negative 29.25.

00:09:26.420 --> 00:09:29.520
So x is equal to negative
3 and x is

00:09:29.520 --> 00:09:33.580
equal to negative 29.25.

00:09:33.580 --> 00:09:36.440
Now, remember, this is
a radical equation.

00:09:36.440 --> 00:09:38.970
We have to validate that both
of these work, or if they

00:09:38.970 --> 00:09:41.550
don't work, they're going to
be extraneous solutions, or

00:09:41.550 --> 00:09:43.650
one of them might be an
extraneous solution.

00:09:43.650 --> 00:09:47.520
So let's copy these and go
back up to our original

00:09:47.520 --> 00:09:50.210
problem up here, right there.

00:09:50.210 --> 00:09:54.930
Let me paste them, just so we
remember what our solutions

00:09:54.930 --> 00:09:58.970
were, and let's just evaluate
them in the calculator, to see

00:09:58.970 --> 00:10:00.220
if they work.

00:10:03.100 --> 00:10:04.502
I don't want to use
that calculator.

00:10:04.502 --> 00:10:05.720
I'll Use this calculator.

00:10:05.720 --> 00:10:06.990
Much better.

00:10:06.990 --> 00:10:08.170
All right.

00:10:08.170 --> 00:10:10.140
So let's try negative 3.

00:10:10.140 --> 00:10:19.580
So we're going to take the
square root of 10 minus 5

00:10:19.580 --> 00:10:24.740
times negative 3-- that's that
right there; we got out of the

00:10:24.740 --> 00:10:34.790
radical-- plus the square root
of 1 minus negative 3.

00:10:34.790 --> 00:10:35.890
I'm doing nothing in my head.

00:10:35.890 --> 00:10:38.060
Obviously, that would be
positive 3, and all of that.

00:10:38.060 --> 00:10:40.200
What is that going
to be equal to?

00:10:40.200 --> 00:10:41.480
It's equal to 7.

00:10:41.480 --> 00:10:43.640
So negative 3 definitely
works.

00:10:43.640 --> 00:10:47.920
Now let's try negative 29.25
and see what happens.

00:10:47.920 --> 00:10:56.160
So we take-- once again-- the
square root of 10 minus 5

00:10:56.160 --> 00:11:02.360
times negative 29.25 right?

00:11:02.360 --> 00:11:10.510
Close parentheses plus the
square root of 1 minus

00:11:10.510 --> 00:11:19.040
negative 29.25 is equal to 18.

00:11:19.040 --> 00:11:20.220
This didn't work.

00:11:20.220 --> 00:11:21.540
It's not even equal
to negative 7.

00:11:21.540 --> 00:11:25.040
That's because we took the
principal square roots twice

00:11:25.040 --> 00:11:27.440
or we squared it twice, or we
lost information twice.

00:11:27.440 --> 00:11:30.370
But the simple answer is
this does not work.

00:11:30.370 --> 00:11:32.350
It is an extraneous solution.

00:11:32.350 --> 00:11:35.090
We got that solution assuming
not the principal roots,

00:11:35.090 --> 00:11:37.580
assuming the negative square
roots somewhere along here.

00:11:37.580 --> 00:11:39.940
So for this equation that deals
with just the principal

00:11:39.940 --> 00:11:44.200
roots, we can say that
this right here is

00:11:44.200 --> 00:11:47.580
an extraneous solution.

00:11:47.580 --> 00:11:51.650
And the only valid solution to
this equation is x is equal to

00:11:51.650 --> 00:11:53.320
negative 3.