Graphing exponential growth & decay

WEBVTT
Kind: captions
Language: en

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- [Voiceoer] This is from the graph basic

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exponential functions on Khan Academy.

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They asked us graph the
following exponential function.

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And they give us the function,

00:00:08.444 --> 00:00:12.043
h of x is equal to 27 times 1/3 to the x.

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So our initial value is 27
and 1/3 is our common ratio.

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It's written in a
standard exponential form.

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And it gives us some graphing tool

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where we can define these two points

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and we can also define

00:00:26.270 --> 00:00:28.400
a horizontal asymptote

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to construct our function.

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And these three things
are enough to define

00:00:32.703 --> 00:00:35.245
to graph an exponential if we know

00:00:35.245 --> 00:00:37.184
that it is an exponential function.

00:00:37.184 --> 00:00:38.739
So let's think about it a little bit.

00:00:38.739 --> 00:00:41.166
So the easiest thing
that I could think of is,

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well, let's think about its initial value.

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Its initial value is going
to be x equal to zero.

00:00:46.292 --> 00:00:49.775
X equal zeros, 1/3 to the
zero power is just one

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and so you're just left with
27 times one or just 27.

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That's what we call this number here

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when you've written in this form.

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You call this the initial value.

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So when x is equal to zero,

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h of x is equal to 27.

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Now we're graphing y=h(x).

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So now let's graph another point.

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So let's think about it a little bit.

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When x is equal to one,

00:01:12.430 --> 00:01:15.413
when x is equal to one, what is h of x?

00:01:15.413 --> 00:01:17.012
It's gonna be 1/3 to the first power

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which is just 1/3.

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And so 1/3 times 27 is gonna be nine.

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So when x is one, h of one is nine.

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And we can verify.

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And now let's just think about,

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let's think about the asymptote.

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So what's gonna happen here when x becomes

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really, really, really, really,
really, really, really big?

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If I take 1/3 to like a
really large exponent to,

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say to the 10th power
or to the 100th power

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or to the 1000th power, this
thing right over here is

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going to start approaching
zero as x becomes

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much, much, much, much larger.

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And so something that is
approaching zero times 27,

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well, that's going to
approach zero as well.

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So we're gonna have a
horizontal asymptote at zero.

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And you can verify that this works

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for more than just a two
points we talk about.

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When x is equal to, this is
telling us that the graph,

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y equals h of x goes to
the point two comma three.

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So h of two should be equal to three.

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And you can verify that
that is indeed the case.

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If x is two, 1/3 squared is nine.

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Oh sorry, 1/3 squared is 1/9

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times 27 is three.

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And we see that right over here.

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When x is two, h of two is three.

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So I feel pretty good about that.

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Let's do another one of this.

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So graph the following
exponential function.

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So same logic.

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When x is zero, the g
of zero is just gonna

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boiled out to that initial value.

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And so you scroll down.

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Initial value is negative 30.

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And so let's think about
when x is equal to one.

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When x is equal to one, two to
the first power is just two.

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And so two times negative
30 is negative 60.

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So when x is equal to one,

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the value of the graph is negative 60.

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Now let's think about this asymptote

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over that, it should sit.

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So let's think about what
happens when x becomes

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really, really, really, really,
really, really negative.

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So when x is really negative,

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so two to the negative one power is 1/2.

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Two to the negative two is 1/4.

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Two to the negative three is 1/8.

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As it gets larger and larger negative

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or higher magnitude negative values

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or in other words, as x becomes

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more and more and more negative,

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two to that power is
going to approach zero.

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And so negative 30 times
something approaching zero

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is going to approach zero.

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So this asymptote is in the right place,

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a horizontal asymptote as x
approaches negative infinity.

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As we move further and
further to the left,

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the value of a function
is going to approach zero.

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Now we can see it kind of
approaches zero from below.

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We can see they approaches below

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because we already looked
at the initial value

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and we used that common ratio
to find one other point.

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Hopefully, you found that interesting.


Worked example: rational vs. irrational expressions (unknowns)

WEBVTT
Kind: captions
Language: en

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- We're to let a and
b be rational numbers,

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and let b be non-zero, they
have to say let b be non-zero

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because we're about to divide by b.

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Is a over b rational or irrational?

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Well let's think about it,
they're both rational numbers

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so that means that a, since
it's rational, can be expressed

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as the ratio of two integers,
so I can write a is equal to m

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over n, and same thing about
b, I could write b as being

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equal to p over q, where

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where m, n, p,

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and q are integers,

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are integers, by definition of what a

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rational number is, they're
telling us these numbers

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are rational so I can express
them as these types of ratios.

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So what is a over b going to be?

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a over b is going to be

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m over n over p over q

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which is equal to m over n,
if I divide by a fraction,

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it's the same thing as
multiplying by the reciprocal.

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q over p, let me write that a little bit,

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q over p which is equal to

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mq over np.

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Well mq is going to be an
integer, if the product of

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two integers is going to be
an integer, and np is going

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to be another integer, the
product of two integers

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is an integer, so I've just shown that

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a over b can be

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expressed as the ratio of
two integers, so a over b

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is for sure, in fact I've just
proven it to you, a over b

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is for sure going to be rational.

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Let's do a few more of
these, this is interesting.

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All right, so now we're saying let a and b

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be irrational numbers.

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Is a over b, let a and
b be irrational numbers.

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Is a over b rational or irrational?

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And, like always, pause
the video and try to think

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this through, and you
might want do some examples

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of some irrational numbers
and see if you can get,

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when you divide them, you can get rational

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or irrational numbers.

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Well, let's just imagine
a world where, let's say,

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that a is equal to, I dunno,

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two square roots of two,

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and b is equal to the square root of two.

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Well, in that world, a
over b, a over b would be

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two square roots of two
over the square root of two

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which would be two, which is
very much a rational number,

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I can express that as a ratio
of integers, I can write that

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as two over one, there's actually
an infinite number of ways

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I can express that as a
ratio of two integers.

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So, in this case, I was able
to get a over b to be rational,

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based on a and b being irrational.

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But, what if, what if instead of,

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what if a was equal to

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the square root of two

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and b is equal to the square root

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of, let's say, b is equal
to the square root of seven.

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Well, then a over b would
be equal to the square root

00:02:54.190 --> 00:02:57.704
of two over the square root
of seven, which is still

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going to be irrational, I
mean another way to think

00:03:00.204 --> 00:03:02.377
about it, and I'm not proving
it here, but you could

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think about it, this is
going to be the square root

00:03:03.962 --> 00:03:07.427
of two sevens, so we
have something that's not

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a perfect square under the
radical, so we're going

00:03:10.243 --> 00:03:13.574
to end up with an irrational number.

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So, we can show one example where a over b

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is rational and we showed one example

00:03:18.796 --> 00:03:23.084
where it is irrational,
so it can be either way.

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Let's do a few more of these.

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All right, let a be a
non-zero rational number.

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Is a times the square root of
eight rational or irrational?

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Well, the key here is, if you
multiply an irrational number

00:03:38.043 --> 00:03:40.461
and why is this an irrational number?

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It has a perfect square
in it, but it's not

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a perfect square in and of itself.

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The square root of eight is,
the square root of eight is

00:03:47.165 --> 00:03:50.376
equal to the square
root of four times two,

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which is equal to the
square root of four times

00:03:52.467 --> 00:03:54.500
the square root of two, which is equal

00:03:54.500 --> 00:03:56.755
to two square roots of two.

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And, this is kind of
getting to the punchline

00:03:58.950 --> 00:04:02.566
of this problem, but if I
multiply a rational times

00:04:02.566 --> 00:04:05.918
an irrational, I am going
to get an irrational.

00:04:05.918 --> 00:04:07.909
So the square root of
eight is an irrational,

00:04:07.909 --> 00:04:11.973
and if I multiply that
times a rational number,

00:04:11.973 --> 00:04:16.107
I'm still going to get
an irrational number.

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So this is going to be,
for sure, irrational.

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Let's do one more of these.

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So we're to let a be an irrational number.

00:04:25.437 --> 00:04:29.296
Is negative 24 plus a
rational or irrational?

00:04:29.296 --> 00:04:32.771
And I won't give a formal
proof here, but to give you

00:04:32.771 --> 00:04:35.047
more of an intuitive feel,
it's nice to just try out

00:04:35.047 --> 00:04:37.017
some numbers, and I encourage
you to pause the video

00:04:37.017 --> 00:04:39.513
and try to think through it yourself.

00:04:39.513 --> 00:04:44.372
Let's just imagine some values,
imagine if a is irrational,

00:04:44.372 --> 00:04:48.977
a is irrational, so what
if a was equal to negative

00:04:48.977 --> 00:04:52.128
pi, which is approximately
equal to negative three point

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one four one five nine and it keeps going

00:04:54.911 --> 00:04:59.123
on and on forever, never repeating.

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Well, then we would have

00:05:02.468 --> 00:05:05.528
negative 24 plus a

00:05:05.528 --> 00:05:09.780
would be equal to negative 24 minus pi,

00:05:09.780 --> 00:05:12.539
which would be approximately negative 27

00:05:12.539 --> 00:05:15.979
point one for one five
nine, the decimal expansion,

00:05:15.979 --> 00:05:17.535
everything to the right of
the decimal, is going to be

00:05:17.535 --> 00:05:19.931
the exact same thing as pi.

00:05:19.931 --> 00:05:22.186
So this looks like, at least
for this example, is going

00:05:22.186 --> 00:05:25.620
to be irrational, and let's
see, if a was square root of two

00:05:25.620 --> 00:05:29.595
negative 24 plus the square root of two,

00:05:29.595 --> 00:05:32.855
well, once again, I'm
not doing a proof here,

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but intuitively, this is
going to be a decimal,

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it's going to have a decimal expansion

00:05:38.459 --> 00:05:40.836
that's going to go on
forever and never repeat,

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and so this would just
change what's to the left

00:05:43.294 --> 00:05:46.382
of the decimal, but not
really change what's,

00:05:46.382 --> 00:05:48.507
well it would change what's
to the right of the decimal

00:05:48.507 --> 00:05:50.681
because this is negative,
but it's still going to go on

00:05:50.681 --> 00:05:54.906
forever and never repeat,
and if, in fact, this was

00:05:54.906 --> 00:05:58.132
if this was this way, then
to the right of the decimal,

00:05:58.132 --> 00:06:00.326
you would have the same thing
as the square root of two

00:06:00.326 --> 00:06:02.012
to the left of the decimal,
you would just have

00:06:02.012 --> 00:06:05.974
a different value, you
would have negative 25 point

00:06:05.974 --> 00:06:09.306
whatever whatever whatever,
and so this is, when you add

00:06:09.306 --> 00:06:13.431
a rational number to an
irrational number, we've proven

00:06:13.431 --> 00:06:16.661
in other videos, a
rational plus an irrational

00:06:16.661 --> 00:06:20.217
is going to be irrational, irrational.

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If you want that proof,
we have other videos

00:06:22.269 --> 00:06:24.774
within this tutorial.


Graphing shifted functions

WEBVTT
Kind: captions
Language: en

00:00:00.156 --> 00:00:02.385
- [Voiceover] We’re told, "the
graph of the function f of x

00:00:02.385 --> 00:00:03.628
"is equal to x-squared."

00:00:03.628 --> 00:00:05.694
We see it right over here in grey.

00:00:05.694 --> 00:00:06.786
It’s shown in the grid below.

00:00:06.786 --> 00:00:08.527
"Graph the function g of x is equal to

00:00:08.527 --> 00:00:10.861
"x minus two-squared, minus four

00:00:10.861 --> 00:00:12.649
"in the interactive graph."

00:00:12.649 --> 00:00:15.052
This is from the shifting
functions exercise

00:00:15.052 --> 00:00:16.797
on Khan Academy, and we can see

00:00:20.507 --> 00:00:23.620
we can change the graph of g of x.

00:00:23.620 --> 00:00:25.687
But let’s see, we want
to graph it properly,

00:00:25.687 --> 00:00:27.649
so let’s see how they relate.

00:00:27.649 --> 00:00:28.707
Well, let’s think about a few things.

00:00:28.707 --> 00:00:31.963
Let’s first just make g
of x completely overlap.

00:00:34.163 --> 00:00:34.964
Okay, there you go.

00:00:34.964 --> 00:00:36.748
Now they're completely overlapping.

00:00:36.748 --> 00:00:38.191
And let’s see how they’re different.

00:00:38.191 --> 00:00:41.558
Well, g of x, if you look
at what's going on here,

00:00:41.558 --> 00:00:42.881
instead of having an x-squared,

00:00:42.881 --> 00:00:45.656
we have an x minus two-squared.

00:00:45.656 --> 00:00:49.359
So, one way to think about
it is, when x is zero,

00:00:49.359 --> 00:00:51.995
you have zero-squared is equal to zero.

00:00:51.995 --> 00:00:53.583
But how do you get zero here?

00:00:53.583 --> 00:00:55.091
Well, x has got to be equal to two.

00:00:55.091 --> 00:00:58.659
Two minus two-squared is zero-squared,

00:00:58.659 --> 00:01:00.842
if we don’t look at the
negative four just yet.

00:01:00.842 --> 00:01:03.582
And so, we would want to shift this graph

00:01:03.582 --> 00:01:05.788
over two to the right.

00:01:05.788 --> 00:01:08.551
This is essentially how much
do we shift to the right.

00:01:08.551 --> 00:01:10.490
It’s sometimes a little
bit counterintuitive

00:01:10.490 --> 00:01:13.055
that we have a negative there,

00:01:13.055 --> 00:01:14.518
because you might say, well, negative,

00:01:14.518 --> 00:01:17.084
that makes me think that I
want to shift to the left.

00:01:17.084 --> 00:01:18.686
But you have to remind yourself is like,

00:01:18.686 --> 00:01:21.078
well okay, for the original graph,

00:01:21.078 --> 00:01:23.005
when it was just x-squared,

00:01:23.005 --> 00:01:26.672
to get the zero-squared, I
just have to put x equals zero.

00:01:26.672 --> 00:01:30.261
Now to get a zero-squared,
I have to put in a two.

00:01:30.261 --> 00:01:33.756
So this is actually shifting
the graph to the right.

00:01:33.756 --> 00:01:36.624
And so, what do we do
with this negative four?

00:01:36.624 --> 00:01:38.319
Well, this is a little bit more intuitive,

00:01:38.319 --> 00:01:39.956
or at least for me when
I first learned it.

00:01:39.956 --> 00:01:42.452
This literally will just
shift the graph down.

00:01:42.452 --> 00:01:44.918
Whatever your value is
of x minus two-squared,

00:01:44.918 --> 00:01:47.328
it's gonna shift it down by four.

00:01:47.328 --> 00:01:49.333
So what we wanna do is just shift

00:01:49.333 --> 00:01:51.589
both of these points down by four.

00:01:51.589 --> 00:01:53.493
So this is gonna go from nine,

00:01:53.493 --> 00:01:56.523
and this is gonna go from the
coordinate five comma nine,

00:01:56.523 --> 00:02:01.051
to five comma, if we go
down four, five comma five.

00:02:01.051 --> 00:02:03.721
And this is gonna go from two comma zero,

00:02:03.721 --> 00:02:06.519
to two comma negative four.

00:02:06.519 --> 00:02:08.888
Two comma negative four.

00:02:08.888 --> 00:02:11.147
Did I do that right?

00:02:11.147 --> 00:02:14.043
I think that’s right.

00:02:14.043 --> 00:02:16.727
What, essentially, what
we have going on is,

00:02:16.727 --> 00:02:19.788
g of x is f of x shifted two

00:02:19.788 --> 00:02:23.148
to the right and four down.

00:02:23.148 --> 00:02:25.792
Two to the right and four down.

00:02:25.792 --> 00:02:27.626
And notice, if you look
at the vertex here,

00:02:27.626 --> 00:02:31.051
we shifted two to the right and four down.

00:02:31.051 --> 00:02:35.216
And I shifted this one also,
this one also I shifted

00:02:35.216 --> 00:02:38.596
two to the right, and four down.

00:02:38.596 --> 00:02:40.328
And, there you have it.

00:02:40.328 --> 00:02:43.880
We have graphed g of x,

00:02:43.880 --> 00:02:46.324
which is a shifted version of f of x.


Simplifying hairy exponent expressions

WEBVTT
Kind: captions
Language: en

00:00:00.414 --> 00:00:01.633
- [Voiceover] So let's get some practice

00:00:01.633 --> 00:00:04.825
simplifying hairy expressions
that have exponents in them.

00:00:04.825 --> 00:00:07.200
And so we have a hairy
expression right over here

00:00:07.200 --> 00:00:09.051
and I encourage you to pause the video

00:00:09.051 --> 00:00:11.880
and see if you can rewrite
this in a simpler way.

00:00:12.920 --> 00:00:14.150
All right, let's work
through this together

00:00:14.150 --> 00:00:15.602
and the first thing that jumps out at me

00:00:15.602 --> 00:00:16.647
is the numerator here.

00:00:16.647 --> 00:00:19.510
I have the number 125
raised to the 1/8 power

00:00:19.510 --> 00:00:22.914
times the same number, the same base, 125,

00:00:22.914 --> 00:00:24.957
raised to the 5/8 power.

00:00:24.957 --> 00:00:26.969
So I can rewrite this numerator.

00:00:26.969 --> 00:00:29.458
I can rewrite this
numerator using what I know

00:00:29.458 --> 00:00:31.075
of exponent properties

00:00:31.075 --> 00:00:34.659
as being equal to 125

00:00:34.659 --> 00:00:36.719
to the sum of these two exponents.

00:00:36.719 --> 00:00:40.586
To the negative 1/8 power plus 5/8 power.

00:00:41.696 --> 00:00:44.909
All of that is going to be
over the existing denominator

00:00:44.909 --> 00:00:47.429
we have, which is five to the 1/2.

00:00:47.429 --> 00:00:51.753
All of that is going to be
over five to the 1/2 power.

00:00:51.753 --> 00:00:53.005
So, these are equivalent.

00:00:53.005 --> 00:00:55.735
Notice, all I did is
I added the exponents,

00:00:55.735 --> 00:00:57.790
these two exponents
because I had the same base

00:00:57.790 --> 00:01:00.694
and we were taking the
product of both of these

00:01:00.694 --> 00:01:04.560
125 to the negative
1/8 and 125 to the 5/8.

00:01:04.560 --> 00:01:07.037
And so negative 1/8 plus 5/8,

00:01:07.037 --> 00:01:09.964
well that is 1/2.

00:01:09.964 --> 00:01:12.664
So, this is,

00:01:12.664 --> 00:01:15.824
this right over here is, 1/2.

00:01:15.824 --> 00:01:19.664
So, this is 125 to the
1/2, over five to the 1/2.

00:01:19.664 --> 00:01:21.892
Well, that's going to be the same thing.

00:01:21.892 --> 00:01:25.197
This is going to be equivalent to

00:01:25.197 --> 00:01:29.517
125 over five,

00:01:29.517 --> 00:01:33.612
over five, to the 1/2 power.

00:01:33.612 --> 00:01:36.610
To the 1/2 power.

00:01:36.610 --> 00:01:38.864
If I raise 125 to the 1/2 and I'm dividing

00:01:38.864 --> 00:01:40.174
by five to the 1/2,

00:01:40.174 --> 00:01:42.065
that's the same thing as
doing the division first

00:01:42.065 --> 00:01:44.372
and then raising that to the 1/2 power.

00:01:44.372 --> 00:01:47.063
Well, what's 125 divided by five?

00:01:47.063 --> 00:01:48.666
Well, that's just 25.

00:01:49.866 --> 00:01:51.473
Now what's 25 to the 1/2?

00:01:51.473 --> 00:01:53.419
Well, that's the same thing
as the principal square root

00:01:53.419 --> 00:01:57.272
of 25, which is equal to five.

00:01:57.272 --> 00:02:00.563
And we're all done, that
simplified quite nicely.

00:02:00.563 --> 00:02:01.721
Let's do another one of these.

00:02:01.721 --> 00:02:03.357
And this one is a little more interesting,

00:02:03.357 --> 00:02:05.380
'cause we are starting
to involve a variable,

00:02:05.380 --> 00:02:07.226
we have the variable w,

00:02:07.226 --> 00:02:10.834
but it's really going to be
somewhat the same process.

00:02:10.834 --> 00:02:14.439
And here the thing that jumps
out at me is the denominator.

00:02:14.439 --> 00:02:17.072
I have the same base, 3w-squared.

00:02:17.072 --> 00:02:20.739
3w-squared raised to
one power, one exponent,

00:02:20.739 --> 00:02:24.018
times the same base, 3w-squared,
raised to another power.

00:02:24.018 --> 00:02:26.690
So, this is going to be equal to,

00:02:26.690 --> 00:02:28.835
this is going to be equal to

00:02:28.835 --> 00:02:30.532
our numerator we can just rewrite it

00:02:30.532 --> 00:02:33.465
12w to the seventh power

00:02:33.465 --> 00:02:36.665
over negative 3/2

00:02:36.665 --> 00:02:41.390
over our denominator we
can write as this base,

00:02:41.390 --> 00:02:44.811
3w-squared,

00:02:44.811 --> 00:02:47.251
and we can add these two exponents.

00:02:47.251 --> 00:02:50.820
So, we could add negative
2/3 to negative 5/6.

00:02:50.820 --> 00:02:55.470
Negative 2/3 to negative 5/6.

00:02:55.470 --> 00:02:58.315
Well, what is that going to be?

00:02:58.315 --> 00:03:00.755
So, let's see, if I do negative,

00:03:00.755 --> 00:03:04.349
negative 2/3 is the same
thing as negative 4/6,

00:03:04.349 --> 00:03:06.671
minus 5/6,

00:03:06.671 --> 00:03:09.661
which is equal to negative 9/6,

00:03:09.661 --> 00:03:13.291
which is equal to negative 3/2.

00:03:13.291 --> 00:03:16.660
So, this right over here is
the same thing as negative 3/2.

00:03:18.020 --> 00:03:19.002
Let me just write that.

00:03:19.002 --> 00:03:21.038
Negative 3/2 power.

00:03:21.038 --> 00:03:25.806
Negative 2/3 plus negative
5/6 is negative 3/2.

00:03:25.806 --> 00:03:28.081
Now, what's interesting is I
have a negative 3/2 up here

00:03:28.081 --> 00:03:29.959
and I have a negative 3/2 over here,

00:03:29.959 --> 00:03:31.979
so, we can do the same thing
we did in the last problem.

00:03:31.979 --> 00:03:34.743
This could simplify to

00:03:34.743 --> 00:03:37.113
12w,

00:03:37.113 --> 00:03:40.177
12w to the seventh power,

00:03:40.177 --> 00:03:43.136
over 3w-squared.

00:03:43.136 --> 00:03:44.526
3w-squared.

00:03:44.526 --> 00:03:46.265
All of that,

00:03:46.265 --> 00:03:49.442
all of that to the,

00:03:49.442 --> 00:03:53.262
all of that to the negative 3/2 power.

00:03:53.262 --> 00:03:54.276
Notice what we did here.

00:03:54.276 --> 00:03:55.628
I had something to the negative 3/2

00:03:55.628 --> 00:03:58.137
divided by something
else to the negative 3/2.

00:03:58.137 --> 00:04:00.352
Well, that's the same thing
as doing the division first

00:04:00.352 --> 00:04:04.619
and then raising that
quotient to the negative 3/2.

00:04:04.619 --> 00:04:05.880
And what's nice about this,

00:04:05.880 --> 00:04:07.717
this is pretty
straightforward to simplify.

00:04:07.717 --> 00:04:10.371
12 divided by three is four,

00:04:10.371 --> 00:04:13.741
and w to the seventh divided by w-squared,

00:04:13.741 --> 00:04:16.128
well, we could divide both by w-squared

00:04:16.128 --> 00:04:17.438
or you could say this is the same thing

00:04:17.438 --> 00:04:20.108
as w to the seven-minus-two power.

00:04:20.108 --> 00:04:23.616
So this is going to be
w to the fifth power.

00:04:23.616 --> 00:04:25.931
W to the fifth power.

00:04:25.931 --> 00:04:29.315
And so it all simplified to 4w

00:04:29.315 --> 00:04:31.265
to the fifth power

00:04:31.265 --> 00:04:33.571
to the negative 3/2.

00:04:33.571 --> 00:04:35.622
To the negative 3/2.

00:04:35.622 --> 00:04:38.121
Now, if we want to, this
is already pretty simple

00:04:38.121 --> 00:04:40.410
and at some point it becomes somewhat

00:04:40.410 --> 00:04:41.972
someone's opinion on which is expression

00:04:41.972 --> 00:04:43.597
is simpler than another,
and it might depend

00:04:43.597 --> 00:04:45.415
on what you're using the expression for,

00:04:45.415 --> 00:04:47.760
but one could argue that
you could keep trying

00:04:47.760 --> 00:04:51.355
to simplify this, this
is the same thing as,

00:04:51.355 --> 00:04:55.991
this is the same thing as
four to the negative 3/2.

00:04:55.991 --> 00:04:58.700
To the negative 3/2.

00:04:58.700 --> 00:05:03.388
Times w to the fifth

00:05:03.388 --> 00:05:05.338
to the negative 3/2.

00:05:05.338 --> 00:05:06.959
And once again this is
just straight out of our

00:05:06.959 --> 00:05:08.798
exponent properties.

00:05:08.798 --> 00:05:11.209
Now, four to the negative 3/2,

00:05:11.209 --> 00:05:12.377
let's just think about that.

00:05:12.377 --> 00:05:15.333
Four to the negative 3/2.

00:05:15.333 --> 00:05:18.903
We're using a different color
just as a bit of an aside.

00:05:18.903 --> 00:05:22.873
So, four to the negative 3/2 is equal to,

00:05:22.873 --> 00:05:27.286
that's the same thing as
one over four to the 3/2,

00:05:27.286 --> 00:05:28.541
and, let's see.

00:05:30.101 --> 00:05:32.198
The square root of four is
two and then we raise that

00:05:32.198 --> 00:05:34.288
to the third power, it's gonna be eight.

00:05:34.288 --> 00:05:36.827
So, this is equal to 1/8.

00:05:36.827 --> 00:05:39.804
So, that's equal to
1/8, and so all of this

00:05:39.804 --> 00:05:41.674
is going to be equal to 1/8.

00:05:41.674 --> 00:05:43.667
And then w to the fifth, and then that

00:05:43.667 --> 00:05:47.128
to the negative 3/2, we can
multiply these exponents.

00:05:47.128 --> 00:05:51.354
That's going to be w to the
five times negative 3/2.

00:05:51.354 --> 00:05:55.112
Well, it's gonna be the
negative 15/2 power.

00:05:55.112 --> 00:05:57.673
So, I don't know which one
you would say is simpler.

00:05:57.673 --> 00:06:00.751
This one over here or this one over here

00:06:00.751 --> 00:06:03.377
but they are equivalent and
they both are a lot simpler

00:06:03.377 --> 00:06:05.077
than where we started.


Simplifying more involved radical expressions

WEBVTT
Kind: captions
Language: en

00:00:00.356 --> 00:00:02.245
- [Voiceover] We're asked to
simplify the expression by

00:00:02.245 --> 00:00:04.239
removing all factors
that are perfect squares

00:00:04.239 --> 00:00:07.826
from inside the radicals
and combining the terms.

00:00:07.826 --> 00:00:09.998
So, let's see if we can
do it and pause the video

00:00:09.998 --> 00:00:12.955
and give a-go at it
before we do it together.

00:00:12.955 --> 00:00:17.307
Alright, so let's see how we
can re-write these radicals.

00:00:17.307 --> 00:00:19.573
So, four times the square root of 20.

00:00:19.573 --> 00:00:22.399
Well, thats the same thing
as four times the square root

00:00:22.399 --> 00:00:25.623
of four, times the square root of five,

00:00:25.623 --> 00:00:28.764
because 20 is the same
thing as four times five.

00:00:29.844 --> 00:00:34.110
And 45, that's the same
thing as nine times five.

00:00:34.110 --> 00:00:35.826
And the reason why I'm
thinking about the four

00:00:35.826 --> 00:00:36.902
and I'm thinking about the nine

00:00:36.902 --> 00:00:38.711
is because those are perfect squares,

00:00:38.711 --> 00:00:40.241
so I could write this as,

00:00:40.241 --> 00:00:41.757
four times the square root of four times

00:00:41.757 --> 00:00:42.793
the square root of five.

00:00:42.793 --> 00:00:46.636
And then I could say, this
part right over here is

00:00:46.636 --> 00:00:50.385
minus three, times the
square root of nine,

00:00:51.225 --> 00:00:54.216
times the square root of five.

00:00:54.216 --> 00:00:56.064
The square root of 45
is the same thing as the

00:00:56.064 --> 00:00:57.616
square root of nine times five,
which is the same thing as

00:00:57.616 --> 00:01:00.519
square root of nine times
the square root of five,

00:01:00.519 --> 00:01:03.029
and then all of that is going to be over,

00:01:03.029 --> 00:01:06.043
all of that is over the square root of 35.

00:01:06.043 --> 00:01:11.043
Now, are there any perfect squares in 35?

00:01:11.062 --> 00:01:13.300
35 is seven times five.

00:01:13.300 --> 00:01:15.306
No, neither of those are perfect squares.

00:01:15.306 --> 00:01:20.306
So, I could just leave
that as square root of 35.

00:01:20.456 --> 00:01:22.134
And, let's see, the square root of four?

00:01:22.134 --> 00:01:23.660
Well, thats going to be two.

00:01:23.660 --> 00:01:25.328
This is the principal root,
so we're thinking about

00:01:25.328 --> 00:01:26.291
the positive square root.

00:01:26.291 --> 00:01:28.206
The square root of nine is three.

00:01:28.206 --> 00:01:31.770
And so, this part right over

00:01:31.770 --> 00:01:33.902
is going to be four times two,

00:01:33.902 --> 00:01:35.263
times the square root of
five, so it's going to be

00:01:35.263 --> 00:01:37.738
eight square roots of five.

00:01:37.738 --> 00:01:42.629
And then, this part
over here is going to be

00:01:42.629 --> 00:01:45.059
minus three, times three,
times the square root of five.

00:01:45.059 --> 00:01:49.889
So, minus nine square roots of five.

00:01:49.889 --> 00:01:53.812
And all of that is going to
be over the square root of 35.

00:01:53.812 --> 00:01:55.466
Square root of 35.

00:01:55.466 --> 00:01:56.970
And, so let's see, if I
have eight of something,

00:01:56.970 --> 00:01:58.779
and I subtract nine of that something,

00:01:58.779 --> 00:02:01.036
I'm gonna have negative
one of that something.

00:02:01.036 --> 00:02:03.883
So, I could say negative one
times the square root of five,

00:02:03.883 --> 00:02:06.839
or I could just say negative
square root of five.

00:02:06.839 --> 00:02:11.711
Negative square root of five
over the square root of 35.

00:02:11.711 --> 00:02:13.642
I actually think I could
simplify this even more,

00:02:13.642 --> 00:02:15.445
because this is the same thing.

00:02:15.445 --> 00:02:18.408
This is equal to the
negative of the square root

00:02:18.408 --> 00:02:22.733
of five over 35.

00:02:22.733 --> 00:02:24.273
Both the numerator and the denominator are

00:02:24.273 --> 00:02:25.828
divisible by five.

00:02:25.828 --> 00:02:27.550
So, we could divide them both by five,

00:02:27.550 --> 00:02:29.663
and we would get the square root of

00:02:29.663 --> 00:02:31.974
divide the numerator by five, you get one.

00:02:31.974 --> 00:02:34.465
Divide the denominator
by five, you get seven.

00:02:34.465 --> 00:02:37.350
So, we can view this as
the square root of 1/7th.

00:02:37.350 --> 00:02:40.948
Square root of 1/7th.

00:02:40.948 --> 00:02:42.674
And we are all done.

00:02:42.674 --> 00:02:43.999
Let's do another one of these.

00:02:43.999 --> 00:02:46.958
These are strangely, strangely fun.

00:02:46.958 --> 00:02:48.089
And once again, pause it,

00:02:48.089 --> 00:02:50.590
and see if you can work
it out on your own.

00:02:50.590 --> 00:02:52.815
Perform the indicated operations.

00:02:52.815 --> 00:02:56.735
Alright, so let's first multiply.

00:02:56.735 --> 00:02:58.234
So, this essentially is doing

00:02:58.234 --> 00:03:00.302
the distributive property twice.

00:03:00.302 --> 00:03:01.526
And, actually let me just do it that way,

00:03:01.526 --> 00:03:03.274
so let's distribute the
square root of five,

00:03:03.274 --> 00:03:04.499
plus the square root of six.

00:03:04.499 --> 00:03:08.339
Let's first multiply it times
the square root of five.

00:03:08.339 --> 00:03:11.234
So, the square root of five
times the square root of five

00:03:11.234 --> 00:03:13.564
is going to be five.

00:03:13.564 --> 00:03:15.639
Square root of five times
the square root of six is

00:03:15.639 --> 00:03:18.277
the square root of 30.

00:03:18.277 --> 00:03:21.741
So, five plus the square root of 30.

00:03:21.741 --> 00:03:26.486
And then, when I take this expression,

00:03:26.486 --> 00:03:30.113
and I multiply it times the second term,

00:03:30.113 --> 00:03:32.222
times the negative square root of six.

00:03:32.222 --> 00:03:33.728
Well, negative square root of six times

00:03:33.728 --> 00:03:34.947
the square root of five is going to be

00:03:34.947 --> 00:03:38.118
the negative of the square root of 30.

00:03:38.118 --> 00:03:40.821
And then the negative of
the square root of six

00:03:40.821 --> 00:03:42.814
times the square root
of six is going to be,

00:03:42.814 --> 00:03:44.778
we're gonna subtract six.

00:03:44.778 --> 00:03:46.389
The square root of six times
the square root of six is six,

00:03:46.389 --> 00:03:47.997
then we have the negative out there.

00:03:47.997 --> 00:03:51.084
And so, just like that, we are
left with, well, let's see.

00:03:51.084 --> 00:03:53.485
Square root of 30 minus
the square root of 30,

00:03:53.485 --> 00:03:55.387
well those cancel out, thats zero.

00:03:55.387 --> 00:03:57.637
And we're left with five minus six,

00:03:57.637 --> 00:04:02.421
which is going to be
equal to negative one.

00:04:02.421 --> 00:04:05.252
And we are, we're, we're all done.

00:04:05.252 --> 00:04:07.311
Now, another way that
you could've viewed this,

00:04:07.311 --> 00:04:08.929
is you could've seen a pattern here.

00:04:08.929 --> 00:04:09.590
You could've said,

00:04:09.590 --> 00:04:14.186
Well, this is the same thing
as A plus B, times A minus B,

00:04:14.186 --> 00:04:17.388
or A is square root of five
and B is square root of six,

00:04:17.388 --> 00:04:18.595
and we know that this will result in

00:04:18.595 --> 00:04:19.748
the difference of squares.

00:04:19.748 --> 00:04:22.445
This is will be A squared minus B squared.

00:04:22.445 --> 00:04:24.591
And, so this for this
particular case it would be

00:04:24.591 --> 00:04:28.180
square root of five squared,

00:04:28.180 --> 00:04:31.871
minus square root of six squared.

00:04:31.871 --> 00:04:35.005
Which is of course
equal to five minus six,

00:04:35.005 --> 00:04:36.605
which is equal to negative one.

00:04:36.605 --> 00:04:41.048
Either way, hopefully you found
that vaguely entertaining.


Simplifying radicals examples

WEBVTT
Kind: captions
Language: en

00:00:00.086 --> 00:00:02.001
- [Voiceover] Let's get
some practice rewriting

00:00:02.001 --> 00:00:03.825
and simplifying radical expressions.

00:00:03.825 --> 00:00:06.072
So in this first exercise,
and these are all from

00:00:06.072 --> 00:00:07.140
Khan Academy.

00:00:07.140 --> 00:00:09.550
It says simplify the expression
by removing all factors

00:00:09.550 --> 00:00:12.081
that are perfect squares
from inside the radicals,

00:00:12.081 --> 00:00:15.564
and combine the terms.

00:00:15.564 --> 00:00:19.558
If the expression cannot be
simplified, enter it as given.

00:00:19.558 --> 00:00:21.601
All right, let's see what we can do here.

00:00:21.601 --> 00:00:24.387
So, we have negative 40 (laughs),

00:00:24.387 --> 00:00:27.104
the negative square
root of 40 I should say.

00:00:27.104 --> 00:00:29.495
Let me write a little bit
bigger so you can see that.

00:00:29.495 --> 00:00:33.118
So the negative square root of 40

00:00:33.118 --> 00:00:36.856
plus the square root of 90.

00:00:36.856 --> 00:00:39.429
So let's see, what
perfect squares are in 40?

00:00:39.429 --> 00:00:40.659
So, what immediately jumps out at me

00:00:40.659 --> 00:00:42.489
is that this, it's divisible by four

00:00:42.489 --> 00:00:44.045
and four is a perfect square.

00:00:44.045 --> 00:00:47.043
So this is the negative square root of

00:00:47.043 --> 00:00:50.593
four times 10, plus

00:00:50.593 --> 00:00:53.356
the square root of, well
what jumps out at me

00:00:53.356 --> 00:00:54.675
is that this is divisible by nine.

00:00:54.675 --> 00:00:59.133
Nine is a perfect
square, so nine times 10.

00:00:59.133 --> 00:01:02.161
And if we look at the 10s
here, 10 does not have

00:01:02.161 --> 00:01:03.753
any perfect squares in it anymore.

00:01:03.753 --> 00:01:05.588
If you wanted to do a
full factorization of 10,

00:01:05.588 --> 00:01:09.672
a full prime factorization, it would be

00:01:09.672 --> 00:01:10.733
two times five.

00:01:10.733 --> 00:01:12.739
So there's no perfect squares in 10.

00:01:12.739 --> 00:01:14.667
And so we can work it out from here.

00:01:14.667 --> 00:01:16.775
This is the same thing
as the negative of the

00:01:16.775 --> 00:01:20.560
square root of four times
the square root of 10,

00:01:20.560 --> 00:01:22.998
plus the square root of nine,

00:01:22.998 --> 00:01:24.832
times the square root of 10.

00:01:24.832 --> 00:01:26.334
And when I say square
root, I'm really saying

00:01:26.334 --> 00:01:28.403
principal root, the positive square root.

00:01:28.403 --> 00:01:31.259
So it's the negative of the
positive square root of four,

00:01:31.259 --> 00:01:34.045
so that is, so let me do
this is in another color.

00:01:34.045 --> 00:01:35.762
so it can be clear.

00:01:35.762 --> 00:01:39.418
So, this right here is two.

00:01:39.418 --> 00:01:41.405
This right here is three.

00:01:41.405 --> 00:01:43.690
So it's going to be equal to

00:01:43.690 --> 00:01:46.049
negative two

00:01:46.049 --> 00:01:48.580
square roots of 10 plus

00:01:48.580 --> 00:01:51.552
three square roots of 10.

00:01:51.552 --> 00:01:54.199
So if I have negative two of something

00:01:54.199 --> 00:01:56.916
and I add three of that
same something to it,

00:01:56.916 --> 00:01:58.355
that's going to be what?

00:01:58.355 --> 00:02:02.790
Well that's going to be
one square root of 10.

00:02:02.790 --> 00:02:05.112
Now this last step
doesn't make full sense.

00:02:05.112 --> 00:02:07.550
Actually, let me slow
it down a little bit.

00:02:07.550 --> 00:02:10.542
I could rewrite it this way.

00:02:10.542 --> 00:02:13.633
I could write it as
three square roots of 10

00:02:13.633 --> 00:02:16.187
minus two square roots of 10.

00:02:16.187 --> 00:02:17.534
That might jump out at
you a little bit clear.

00:02:17.534 --> 00:02:19.438
If I have three of something
and I were to take away

00:02:19.438 --> 00:02:21.295
two of that something, and that case it's

00:02:21.295 --> 00:02:23.362
squares of 10s, well, I'm
going to be left with just

00:02:23.362 --> 00:02:24.662
one of that something.

00:02:24.662 --> 00:02:26.891
I'm just gonna be left
with one square root of 10.

00:02:26.891 --> 00:02:30.699
Which we could just write
as the square root of 10.

00:02:30.699 --> 00:02:34.498
Another way to think about it is,

00:02:34.498 --> 00:02:37.764
we could factor out a
square root of 10 here.

00:02:37.764 --> 00:02:39.676
So you undistribute it,

00:02:39.676 --> 00:02:41.892
do the distributive property in reverse.

00:02:41.892 --> 00:02:43.564
That would be the square root of 10

00:02:43.564 --> 00:02:47.053
times three minus two,

00:02:47.053 --> 00:02:48.765
which is of course, this is just one.

00:02:48.765 --> 00:02:50.460
So you're just left with
the square root of 10.

00:02:50.460 --> 00:02:54.152
So all of this simplifies
to square root of 10.

00:02:54.152 --> 00:02:56.311
Let's do a few more of these.

00:02:56.311 --> 00:02:59.314
So this says, simplify
the expression by removing

00:02:59.314 --> 00:03:01.048
all factors that are
perfect squares from inside

00:03:01.048 --> 00:03:02.441
the radicals, and combine the terms.

00:03:02.441 --> 00:03:06.163
So essentially the same idea.

00:03:06.163 --> 00:03:09.291
All right, let's see what we can do.

00:03:09.291 --> 00:03:10.707
So, this is interesting.

00:03:10.707 --> 00:03:14.561
We have a square root of 1/2.

00:03:14.561 --> 00:03:16.605
So can I, well actually,

00:03:16.605 --> 00:03:18.276
what could be interesting
is since if I have

00:03:18.276 --> 00:03:20.064
a square root of something
times the square root

00:03:20.064 --> 00:03:21.597
of something else.

00:03:21.597 --> 00:03:24.731
So the square root of 180 times

00:03:24.731 --> 00:03:26.914
the square root of 1/2,

00:03:26.914 --> 00:03:29.607
this is the same thing as the square root

00:03:29.607 --> 00:03:33.874
of 180 times 1/2.

00:03:33.874 --> 00:03:35.714
And this just comes straight
out of our exponent properties.

00:03:35.714 --> 00:03:38.152
It might look a little bit
more familiar if I wrote it as

00:03:38.152 --> 00:03:43.152
180 to the 1/2 power,
times 1/2 to the 1/2 power,

00:03:43.190 --> 00:03:47.974
is going to be equal to 180 times 1/2

00:03:47.974 --> 00:03:50.806
to the 1/2 power, taking the square root,

00:03:50.806 --> 00:03:53.476
the principal root is the same
thing as raising something

00:03:53.476 --> 00:03:55.450
to the 1/2 power.

00:03:55.450 --> 00:03:58.329
And so this is the square
root of 80 times 1/2

00:03:58.329 --> 00:04:02.207
which is going to be
the square root of 90,

00:04:02.207 --> 00:04:05.736
which is equal to the square root of nine

00:04:05.736 --> 00:04:09.358
times 10, and we just
simplified square root of 90

00:04:09.358 --> 00:04:11.936
in the last problem, that's
equal to the square root

00:04:11.936 --> 00:04:15.697
of nine times the square root
of, principle root of 10,

00:04:15.697 --> 00:04:19.714
which is equal to three
times the square root of 10.

00:04:19.714 --> 00:04:23.824
Three times the square root of 10.

00:04:23.824 --> 00:04:25.379
All right, let's keep going.

00:04:25.379 --> 00:04:26.772
So I have one more of these examples,

00:04:26.772 --> 00:04:28.142
and like always, pause the video

00:04:28.142 --> 00:04:29.698
and see if you can work
through these on your own

00:04:29.698 --> 00:04:32.391
before I work it out with you.

00:04:32.391 --> 00:04:34.458
Simplify the expression
by removing all factors

00:04:34.458 --> 00:04:35.456
that are perfect square,
okay, these are just

00:04:35.456 --> 00:04:38.034
same directions that we've
seen the last few times.

00:04:38.034 --> 00:04:39.195
And so let's see.

00:04:39.195 --> 00:04:42.376
If I wanted to do, if I
wanted to simplify this,

00:04:42.376 --> 00:04:47.043
this is equal to the square root of,

00:04:47.043 --> 00:04:50.363
well, 64 times two is 128,

00:04:50.363 --> 00:04:52.128
and 64 is a perfect square,
so I'm gonna write it as

00:04:52.128 --> 00:04:55.796
64 times two, over

00:04:55.796 --> 00:04:57.886
27 is nine times three.

00:04:57.886 --> 00:04:59.767
Nine is a perfect square.

00:04:59.767 --> 00:05:01.671
So this is going to be the same thing.

00:05:01.671 --> 00:05:03.528
And there's a couple of ways
that we can think about it.

00:05:03.528 --> 00:05:06.454
We could say this is the same thing as

00:05:06.454 --> 00:05:09.612
the square root of 64 times two,

00:05:09.612 --> 00:05:13.257
over the square root of nine times three,

00:05:13.257 --> 00:05:16.229
which is the same thing
as the square root of 64

00:05:16.229 --> 00:05:18.713
times the square root of two,

00:05:18.713 --> 00:05:21.871
over square root of nine
times the square root

00:05:21.871 --> 00:05:25.772
of three, which is
equal to, this is eight,

00:05:25.772 --> 00:05:29.835
this is three, so it would be
eight times the square root

00:05:29.835 --> 00:05:34.200
of two, over three times
the square root of three.

00:05:34.200 --> 00:05:35.710
That's one way to say it.

00:05:35.710 --> 00:05:37.590
Or we could even view the
square root of two over

00:05:37.590 --> 00:05:40.284
the square root of three
as a square root of 2/3.

00:05:40.284 --> 00:05:43.674
So we could say this is eight over three

00:05:43.674 --> 00:05:48.248
times the square root of 2/3.

00:05:48.248 --> 00:05:51.220
So these are all possible ways of trying

00:05:51.220 --> 00:05:52.427
to tackle this.

00:05:52.427 --> 00:05:54.053
So we could just write it, let's see.

00:05:54.053 --> 00:05:56.319
Have we removed all factors
that are perfect squares?

00:05:56.319 --> 00:05:58.037
Yes, from inside the radicals

00:05:58.037 --> 00:05:59.361
and we've combined terms.

00:05:59.361 --> 00:06:01.288
We weren't doing any
adding or subtracting here,

00:06:01.288 --> 00:06:03.865
so it's really just
removing the perfect squares

00:06:03.865 --> 00:06:06.141
from inside the radicals
and I think we've done that.

00:06:06.141 --> 00:06:09.322
So we could say this is
going to be 8/3 times

00:06:09.322 --> 00:06:11.597
the square root of 2/3.

00:06:11.597 --> 00:06:13.223
And there's other ways
that you could express this

00:06:13.223 --> 00:06:16.020
that would be equivalent but hopefully

00:06:16.020 --> 00:06:19.089
this makes some sense.


Rewriting roots as rational exponents

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:01.135
- [Voiceover] We're asked to determine

00:00:01.135 --> 00:00:04.036
whether each expression is equivalent to

00:00:04.036 --> 00:00:07.287
the seventh root of v to the third power.

00:00:07.287 --> 00:00:08.910
And, like always, pause the
video and see if you can

00:00:08.910 --> 00:00:11.054
figure out which of
these are equivalent to

00:00:11.054 --> 00:00:14.189
the seventh root of v to the third power.

00:00:14.189 --> 00:00:15.969
Well, a good way to figure
out if things are equivalent

00:00:15.969 --> 00:00:18.168
is to just try to get
them all in the same form.

00:00:18.168 --> 00:00:22.361
So, the seventh root of
v to the third power,

00:00:22.361 --> 00:00:23.353
v to the third power,

00:00:23.353 --> 00:00:24.994
the seventh root of
something is the same thing

00:00:24.994 --> 00:00:27.363
as raising it to the 1/7 power.

00:00:27.363 --> 00:00:30.411
So, this is equivalent
to v to the third power,

00:00:30.411 --> 00:00:33.640
raised to the 1/7 power.

00:00:33.640 --> 00:00:35.825
And if I raise something to
an exponent and then raise

00:00:35.825 --> 00:00:39.132
that to an exponent, well
then, that's the same thing as

00:00:39.132 --> 00:00:41.828
raising it to the product
of these two exponents.

00:00:41.828 --> 00:00:43.189
So, this is going to be the same thing as

00:00:43.189 --> 00:00:47.285
v to the three times 1/7 power,

00:00:47.285 --> 00:00:50.045
which, of course, is 3/7.

00:00:50.045 --> 00:00:52.754
3/7.

00:00:52.754 --> 00:00:54.870
So, we've written it
in multiple forms now.

00:00:54.870 --> 00:00:56.565
Let's see which of these match.

00:00:56.565 --> 00:00:58.957
So, v to the third to the 1/7 power,

00:00:58.957 --> 00:01:00.721
well, that was the form that
we have right over here,

00:01:00.721 --> 00:01:04.227
so that is equivalent.

00:01:04.227 --> 00:01:05.852
V to the 3/7.

00:01:05.852 --> 00:01:07.277
That's what we have right over here,

00:01:07.277 --> 00:01:10.560
so that one is definitely equivalent.

00:01:10.560 --> 00:01:11.645
Now, let's think about this one.

00:01:11.645 --> 00:01:14.672
This is the cube root of v to the seventh.

00:01:14.672 --> 00:01:17.333
Is this going to be equivalent?

00:01:17.333 --> 00:01:19.507
Well, one way to think about it,

00:01:19.507 --> 00:01:22.028
this is going to be the same thing as

00:01:22.028 --> 00:01:25.272
v to the 1/3 power ...

00:01:25.272 --> 00:01:26.877
actually, no, this wasn't the
cube root of v to the seventh,

00:01:26.877 --> 00:01:30.323
this was the cube root of v,
and that to the seventh power.

00:01:30.323 --> 00:01:33.158
So, that's the same thing
as v to the 1/3 power,

00:01:33.158 --> 00:01:35.712
and then, that to the seventh power.

00:01:35.712 --> 00:01:39.590
So, that is the same thing
as v to the 7/3 power,

00:01:39.590 --> 00:01:43.941
which is clearly different
to v to the 3/7 power.

00:01:43.941 --> 00:01:48.889
So, this is not going to
be equivalent for all v's,

00:01:48.889 --> 00:01:52.016
all v's for which this
expression is defined.

00:01:52.016 --> 00:01:55.565
Let's do a few more of these,
or similar types of problems

00:01:55.565 --> 00:01:59.976
dealing with roots and
fractional exponents.

00:01:59.976 --> 00:02:01.719
The following equation is true for g

00:02:01.719 --> 00:02:06.035
greater than or equal to
zero, and d is a constant.

00:02:06.035 --> 00:02:07.949
What is the value of d?

00:02:07.949 --> 00:02:09.512
Well, if I'm taking the
sixth root of something,

00:02:09.512 --> 00:02:12.448
that's the same thing as
raising it to the 1/6 power.

00:02:12.448 --> 00:02:16.014
So, the sixth root

00:02:16.014 --> 00:02:19.184
of g to the fifth, is the same thing as

00:02:19.184 --> 00:02:23.374
g to the fifth, raised to the 1/6 power.

00:02:23.374 --> 00:02:26.013
And, just like we just
saw in the last example,

00:02:26.013 --> 00:02:30.316
that's the same thing as g
to the five times 1/6 power.

00:02:30.316 --> 00:02:31.919
This is just our exponent properties.

00:02:31.919 --> 00:02:33.474
I raise something to an exponent and then

00:02:33.474 --> 00:02:35.401
raise that whole thing
to another exponent,

00:02:35.401 --> 00:02:37.233
I can just multiply the exponents.

00:02:37.233 --> 00:02:38.454
So, that's the same thing as

00:02:38.454 --> 00:02:42.514
g to the 5/6 power.

00:02:42.514 --> 00:02:46.460
And so d is 5/6.

00:02:46.460 --> 00:02:47.818
Five over six.

00:02:47.818 --> 00:02:50.243
The sixth root of g to the
fifth is the same thing

00:02:50.243 --> 00:02:53.583
as g to the 5/6 power.

00:02:53.583 --> 00:02:55.460
Let's do one more of these.

00:02:55.460 --> 00:02:58.249
The following equation is
true for x greater than zero,

00:02:58.249 --> 00:03:00.648
and d is a constant.

00:03:00.648 --> 00:03:02.142
What is the value of d?

00:03:02.142 --> 00:03:03.078
Alright, this is interesting.

00:03:03.078 --> 00:03:04.921
And I forgot to tell
you in the last one, but

00:03:04.921 --> 00:03:06.167
pause this video as
well and see if you can

00:03:06.167 --> 00:03:08.683
work it out on ...or pause
for this question as well

00:03:08.683 --> 00:03:10.736
and see if you can work it out.

00:03:10.736 --> 00:03:13.250
Well, here, let's just start rewriting

00:03:13.250 --> 00:03:14.952
the root as an exponent.

00:03:14.952 --> 00:03:16.761
So, I can rewrite the whole thing.

00:03:16.761 --> 00:03:18.287
This is the same thing as one over,

00:03:18.287 --> 00:03:20.225
instead of writing the seventh root of x,

00:03:20.225 --> 00:03:23.541
I'll write x to the 1/7 power

00:03:23.541 --> 00:03:26.241
is equal to x to the d.

00:03:26.241 --> 00:03:28.521
And if I have one over
something to a power,

00:03:28.521 --> 00:03:29.890
that's the same thing as that something

00:03:29.890 --> 00:03:31.860
raised to the negative of that power.

00:03:31.860 --> 00:03:33.588
So, that is the same thing as

00:03:33.588 --> 00:03:38.388
x to the negative 1/7 power.

00:03:38.388 --> 00:03:41.947
And so, that is going to
be equal to x to the d.

00:03:41.947 --> 00:03:44.678
And so, d must be equal to,

00:03:44.678 --> 00:03:48.472
d must be equal to negative 1/7.

00:03:48.472 --> 00:03:50.465
So, the key here is when
you're taking the reciprocal

00:03:50.465 --> 00:03:52.651
of something, that's the
same thing as raising it

00:03:52.651 --> 00:03:54.880
to the negative of that exponent.

00:03:54.880 --> 00:03:57.213
Another way of thinking about it is

00:03:57.213 --> 00:04:01.395
you could view this as,

00:04:01.395 --> 00:04:03.285
you could view it as,

00:04:03.285 --> 00:04:06.626
x to the 1/7

00:04:06.626 --> 00:04:08.666
to the negative one power.

00:04:08.666 --> 00:04:10.417
And then, if you multiply these exponents,

00:04:10.417 --> 00:04:12.747
you get what we have right over there.

00:04:12.747 --> 00:04:17.700
But, either way, d is
equal to negative 1/7.


Position vector valued functions

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.635
- [Voiceover] Let's say
I have some curve C,

00:00:02.635 --> 00:00:07.371
and it's described, it
can be parameterized,

00:00:07.371 --> 00:00:12.371
I can't say that word, as,
let's say x is equal to x of t,

00:00:13.417 --> 00:00:17.926
y is equal to some function y of t,

00:00:17.926 --> 00:00:22.757
and let's say that this is
valid for t is between a and b.

00:00:22.757 --> 00:00:25.642
So, t is greater than or equal to a

00:00:25.642 --> 00:00:28.466
and then less than or equal to b.

00:00:28.466 --> 00:00:32.158
So, if I were to just
draw this on, let me see,

00:00:32.158 --> 00:00:33.606
I could draw it like this.

00:00:33.606 --> 00:00:36.361
I'm staying very abstract right now.

00:00:36.361 --> 00:00:38.911
This is not a very specific example.

00:00:38.911 --> 00:00:41.517
This is the x axis, this is the y axis.

00:00:41.517 --> 00:00:46.206
My curve, let's say this
is when t is equal to a

00:00:46.206 --> 00:00:51.012
and then the curve might
do something like this.

00:00:51.012 --> 00:00:53.086
I don't know what it does,
let's say it's over there.

00:00:53.086 --> 00:00:55.633
This is t is equal to b.

00:00:55.633 --> 00:01:00.601
This actual point right
here will be x of b.

00:01:00.601 --> 00:01:01.809
That would be the x coordinate,

00:01:01.809 --> 00:01:06.514
you evaluate this
function at b, and y of b.

00:01:06.514 --> 00:01:08.914
And this is, of course,
when t is equal to a.

00:01:08.914 --> 00:01:13.372
The actual coordinate in R2
on the Cartesian coordinates

00:01:13.372 --> 00:01:17.404
will be x of a, which is this right here,

00:01:17.404 --> 00:01:22.130
and then y of a, which
is that right there.

00:01:22.130 --> 00:01:24.810
And we've seen that before,
that's just a standard way

00:01:24.810 --> 00:01:26.992
of describing a parametric
equation or curve

00:01:26.992 --> 00:01:29.675
using two parametric equations.

00:01:29.675 --> 00:01:33.687
What I want to do now is
describe this same exact curve

00:01:33.687 --> 00:01:36.474
using a vector valued function.

00:01:36.474 --> 00:01:39.052
So, if I define a vector valued function,

00:01:39.052 --> 00:01:40.724
and if you don't remember what those are

00:01:40.724 --> 00:01:41.954
we'll have a little bit of review here.

00:01:41.954 --> 00:01:45.261
Let me say I have a
vector valued function r,

00:01:45.261 --> 00:01:48.224
and I'll put a little
vector arrow on top of it.

00:01:48.224 --> 00:01:50.379
In a lot of textbooks,
they'll just bold it

00:01:50.379 --> 00:01:53.053
and they'll leave scalar
valued functions unbolded,

00:01:53.053 --> 00:01:54.655
but it's hard to draw bold

00:01:54.655 --> 00:01:56.777
so I'll put a little vector on top.

00:01:56.777 --> 00:01:59.427
And let's say that r is a function of t.

00:01:59.427 --> 00:02:02.236
And these are going to
be position vectors.

00:02:02.236 --> 00:02:04.037
Position vectors.

00:02:04.037 --> 00:02:07.286
Position vectors.

00:02:07.286 --> 00:02:09.813
And I'm specifying that
because in general,

00:02:09.813 --> 00:02:11.303
when someone talks about a vector,

00:02:11.303 --> 00:02:15.390
this vector and this vector
are considered equivalent

00:02:15.390 --> 00:02:17.665
as long as they have the
same magnitude and direction.

00:02:17.665 --> 00:02:19.616
No one really cares about what their

00:02:19.616 --> 00:02:21.501
start and end points are as long as their

00:02:21.501 --> 00:02:23.781
direction is the same and
their length is the same.

00:02:23.781 --> 00:02:25.304
But, when you talk about position vectors,

00:02:25.304 --> 00:02:27.819
you're saying, "No, these
vectors are all going to

00:02:27.819 --> 00:02:30.755
start at zero, at the origin."

00:02:30.755 --> 00:02:32.015
And when you state the position vector

00:02:32.015 --> 00:02:33.361
you're implicitly saying,

00:02:33.361 --> 00:02:36.302
"This is specifying a unique position."

00:02:36.302 --> 00:02:38.052
In this case, it's going to
be in two dimensional space,

00:02:38.052 --> 00:02:39.538
but it could be in
three dimensional space,

00:02:39.538 --> 00:02:42.435
or really four, five,
whatever, n dimensional space.

00:02:42.435 --> 00:02:43.671
So, when you state the position vector,

00:02:43.671 --> 00:02:46.364
you're literally saying,
"Okay, this vector

00:02:46.364 --> 00:02:49.817
literally specifies that point in space."

00:02:49.817 --> 00:02:52.896
So, let's see if we
can describe this curve

00:02:52.896 --> 00:02:57.896
as a vector, a position
vector valued function.

00:02:58.226 --> 00:02:59.235
So, we can say r of t,

00:02:59.235 --> 00:03:01.633
let me switch back to that pink color,

00:03:01.633 --> 00:03:03.192
oh, this keeps staying green,

00:03:03.192 --> 00:03:06.421
is equal to x of t

00:03:06.421 --> 00:03:08.807
times the unit vector in the x direction,

00:03:08.807 --> 00:03:11.133
the unit vector gets
a little caret on top,

00:03:11.133 --> 00:03:12.887
a little hat that's like the arrow for it.

00:03:12.887 --> 00:03:14.639
That just says it's a unit vector.

00:03:14.639 --> 00:03:16.324
Plus,

00:03:16.324 --> 00:03:20.204
y of t times j.

00:03:20.204 --> 00:03:22.835
If I was dealing with a
curve in three dimensions,

00:03:22.835 --> 00:03:24.902
I would have, plus z of t times k,

00:03:24.902 --> 00:03:27.316
but we're dealing with
two dimensions right here.

00:03:27.316 --> 00:03:29.610
And so, the way this works is,
you're just taking your ...

00:03:29.610 --> 00:03:32.030
for any t, and still we're gonna have

00:03:32.030 --> 00:03:36.000
t is greater than or equal to a

00:03:36.000 --> 00:03:37.840
and then less than or equal to b.

00:03:37.840 --> 00:03:39.994
And this is the exact same thing as that.

00:03:39.994 --> 00:03:41.536
Let me just redraw it.

00:03:41.536 --> 00:03:44.227
So, let me draw our coordinates.

00:03:44.227 --> 00:03:46.286
Our coordinates right here.

00:03:46.286 --> 00:03:48.571
Our axes.

00:03:48.571 --> 00:03:51.621
So, that's the y axis
and this is the x axis.

00:03:51.621 --> 00:03:53.809
So, when you evaluate r of a, right?

00:03:53.809 --> 00:03:56.383
That's our starting
point, so let me do that.

00:03:56.383 --> 00:04:00.440
So, r of a, maybe I'll
do it right over here.

00:04:00.440 --> 00:04:03.539
Our position vector valued
function evaluated at

00:04:03.539 --> 00:04:07.183
t is equal to a, is going to be equal to

00:04:07.183 --> 00:04:12.106
x of a times our unit
vector in the x direction,

00:04:12.106 --> 00:04:16.068
plus y of a times our unit vector in the

00:04:16.068 --> 00:04:18.145
vertical direction, or in the y direction.

00:04:18.145 --> 00:04:19.629
And what's that going to look like?

00:04:19.629 --> 00:04:24.342
Well, x of a is this thing right here.

00:04:24.342 --> 00:04:26.595
So, it's x of a times the unit vector.

00:04:26.595 --> 00:04:30.147
So, it's really, maybe the
unit vector is this long,

00:04:30.147 --> 00:04:31.821
it has length one, so
now we're just gonna have

00:04:31.821 --> 00:04:34.187
a length of x of a in that direction,

00:04:34.187 --> 00:04:35.650
and then same thing in y of a.

00:04:35.650 --> 00:04:38.332
It's going to be y of a
length in that direction.

00:04:38.332 --> 00:04:40.152
But the bottom line,
this vector right here,

00:04:40.152 --> 00:04:44.311
if you add these scaled values
of these two unit vectors,

00:04:44.311 --> 00:04:48.742
you're going to get r of a
looking something like this.

00:04:48.742 --> 00:04:52.518
It's going to be a vector that
looks something like that.

00:04:52.518 --> 00:04:53.525
Just like that.

00:04:53.525 --> 00:04:54.968
It's a vector, it's a position vector,

00:04:54.968 --> 00:04:56.803
that's why we're nailing it at the origin

00:04:56.803 --> 00:04:58.939
but drawing it in standard position.

00:04:58.939 --> 00:05:02.465
And that right there, is

00:05:02.465 --> 00:05:04.725
r of a.

00:05:04.725 --> 00:05:07.112
Now, what happens if a
increases a little bit?

00:05:07.112 --> 00:05:09.774
What is r of a plus a little bit?

00:05:09.774 --> 00:05:12.343
And, I don't know, we could
call that r of a plus delta

00:05:12.343 --> 00:05:14.784
or r of a plus h.

00:05:14.784 --> 00:05:15.662
We'll do a different color.

00:05:15.662 --> 00:05:18.977
So, let's say r, let's say
we increase a a little bit,

00:05:18.977 --> 00:05:22.947
r of a plus some small h,
well that's just going to be

00:05:22.947 --> 00:05:27.947
x of a plus h, times the unit vector i,

00:05:27.957 --> 00:05:31.990
plus y times a plus h,
times the unit vector j.

00:05:31.990 --> 00:05:33.415
And, what's that going to look like?

00:05:33.415 --> 00:05:35.509
Well, we're gonna go a little
bit further down the curve.

00:05:35.509 --> 00:05:36.613
That's like saying the coordinate

00:05:36.613 --> 00:05:39.108
x of a plus h and y of a plus h.

00:05:39.108 --> 00:05:40.733
It might be that point right there,

00:05:40.733 --> 00:05:44.019
so it'll be a new unit vector.

00:05:44.019 --> 00:05:45.887
It'll be a new unit vector.

00:05:45.887 --> 00:05:48.418
Sorry, it'll be a new
vector, position vector,

00:05:48.418 --> 00:05:50.996
not a unit vector, these won't
necessarily have length one,

00:05:50.996 --> 00:05:53.682
that might be right here.

00:05:53.682 --> 00:05:55.918
Let me do that same color as this.

00:05:55.918 --> 00:05:58.298
So, it might be just like,

00:05:58.298 --> 00:06:00.000
just like that.

00:06:00.000 --> 00:06:05.000
So, that right here, is r of a plus h.

00:06:05.131 --> 00:06:07.778
So, you see as you keep
increasing your value

00:06:07.778 --> 00:06:11.261
of t until you get to b,
these position vectors

00:06:11.261 --> 00:06:13.856
are going to keep specifying, we're gonna

00:06:13.856 --> 00:06:16.034
keep specifying points along this curve.

00:06:16.034 --> 00:06:19.364
So, the curve, let me draw the
curve in a different color.

00:06:19.364 --> 00:06:21.376
The curve looks something like this.

00:06:21.376 --> 00:06:22.537
It's meant to look exactly like

00:06:22.537 --> 00:06:24.325
the curve that I have up here.

00:06:24.325 --> 00:06:27.504
And, for example, r of b is going to be

00:06:27.504 --> 00:06:29.596
a vector that looks like this.

00:06:29.596 --> 00:06:33.125
It's going to be a vector
that looks like that.

00:06:33.125 --> 00:06:36.366
I want to draw it relatively straight.

00:06:36.366 --> 00:06:39.078
That vector, right there,

00:06:39.078 --> 00:06:42.508
is r of b.

00:06:42.508 --> 00:06:43.852
So, hopefully you realize that, look,

00:06:43.852 --> 00:06:46.174
these position vectors
really are specifying the

00:06:46.174 --> 00:06:49.192
same points on this curve
as this original, I guess

00:06:49.192 --> 00:06:53.233
straight up parameterization
that we did for this curve.

00:06:53.233 --> 00:06:54.835
And I just wanted to do that
as a little bit of review

00:06:54.835 --> 00:06:57.586
cause we're now gonna
break in into the idea of

00:06:57.586 --> 00:07:01.922
actually taking a derivative
of this vector valued function.

00:07:01.922 --> 00:07:04.484
And I'll do that in the next video.


Comparing features of quadratic functions

WEBVTT
Kind: captions
Language: en

00:00:00.017 --> 00:00:01.611
- [Voiceover] So we're
asked which function

00:00:01.611 --> 00:00:04.021
has the greater y-intercept?

00:00:05.115 --> 00:00:07.157
So, the y-intercept is the y-coordinate

00:00:07.157 --> 00:00:09.328
when x is equal to zero.

00:00:09.328 --> 00:00:12.289
So, f of zero, when x is equal to zero,

00:00:12.289 --> 00:00:14.433
the function is equal to, let's see,

00:00:14.433 --> 00:00:17.103
f of zero is going to be equal to

00:00:17.103 --> 00:00:20.923
zero minus zero, plus four,
is going to be equal to four.

00:00:20.923 --> 00:00:22.389
So this function right over here,

00:00:22.389 --> 00:00:25.025
it has a y-intercept of four.

00:00:25.025 --> 00:00:29.518
So it would intersect the
y-axis right over there.

00:00:29.518 --> 00:00:31.689
While the function that
we're comparing it to,

00:00:31.689 --> 00:00:33.454
g of x, we're looking at its graph,

00:00:33.454 --> 00:00:35.323
y is equal to g of x,

00:00:35.323 --> 00:00:37.958
its y-intercept is right over here,

00:00:37.958 --> 00:00:40.791
at y is equal to three.

00:00:40.791 --> 00:00:43.787
So which function has
a greater y-intercept?

00:00:43.787 --> 00:00:45.640
Well, it's going to be f of x.

00:00:45.640 --> 00:00:50.451
F of x has a greater
y-intercept than g of x does.

00:00:50.451 --> 00:00:51.798
Let's do a few more of these

00:00:51.798 --> 00:00:53.260
where we're comparing different functions.

00:00:53.260 --> 00:00:56.987
One of them that has a visual depiction,

00:00:56.987 --> 00:00:59.054
and one of them where we're
just given the equation.

00:00:59.054 --> 00:01:01.991
How many roots do the
functions have in common?

00:01:01.991 --> 00:01:04.758
Well, g of x, we can see their roots.

00:01:04.758 --> 00:01:07.837
The roots are, x equals negative one

00:01:07.837 --> 00:01:11.017
and x is equal to two.

00:01:11.017 --> 00:01:13.427
So these two functions, at most,

00:01:13.427 --> 00:01:14.925
are going to have two roots in common,

00:01:14.925 --> 00:01:17.920
because this g of x only has two roots.

00:01:17.920 --> 00:01:19.290
There's a couple of
ways we could tackle it.

00:01:19.290 --> 00:01:21.983
We could just try to find f's roots,

00:01:21.983 --> 00:01:25.156
or we could plug in
either one of these values

00:01:25.156 --> 00:01:29.054
and see if it makes the
function equal to a zero.

00:01:29.054 --> 00:01:31.225
I'll do the first way,
I'll try to factor this.

00:01:31.225 --> 00:01:35.393
So let's see, what two numbers,
if I add them I get one,

00:01:35.393 --> 00:01:38.191
'cause that's the coefficient
here, or implicitly there.

00:01:38.191 --> 00:01:40.722
And if I take the product,
I get negative six.

00:01:40.722 --> 00:01:42.289
Well, their gonna have
to have different signs

00:01:42.289 --> 00:01:43.555
since their product is negative.

00:01:43.555 --> 00:01:46.817
So, let's see, negative
three and positive two.

00:01:46.817 --> 00:01:48.953
No, actually, the other way
around 'cause it's positive one.

00:01:48.953 --> 00:01:50.590
So positive three, and negative two.

00:01:50.590 --> 00:01:55.590
So this is equal to x plus
three, times x minus two.

00:01:55.722 --> 00:01:58.427
So f of x is going to have zeros when

00:01:58.427 --> 00:02:01.550
x is equal to negative three.

00:02:01.550 --> 00:02:03.385
X is equal to negative three.

00:02:03.385 --> 00:02:06.159
Or, x is equal to two.

00:02:06.159 --> 00:02:07.193
These are the two zeros.

00:02:07.193 --> 00:02:08.493
If x is equal to negative three,

00:02:08.493 --> 00:02:11.024
this expression becomes zero.

00:02:11.024 --> 00:02:12.990
Zero times anything is zero.

00:02:12.990 --> 00:02:15.389
If x equals two, this
expression becomes zero,

00:02:15.389 --> 00:02:16.887
and zero times anything is zero.

00:02:16.887 --> 00:02:19.615
So f of negative three is
zero, and f of two is zero.

00:02:19.615 --> 00:02:21.891
These are the zeros of that function.

00:02:21.891 --> 00:02:24.085
So let's see, which of
these are in common?

00:02:24.085 --> 00:02:26.593
Well, negative three is out
here, that's not in common.

00:02:26.593 --> 00:02:28.509
X equals two is in common,

00:02:28.509 --> 00:02:32.189
so they only have one common
zero right over there.

00:02:32.189 --> 00:02:36.136
So how many roots do the
functions have in common?

00:02:36.136 --> 00:02:37.321
One.

00:02:37.321 --> 00:02:38.522
All right.

00:02:38.522 --> 00:02:40.623
Let's do one more of these.

00:02:40.623 --> 00:02:44.588
And they ask us, "Do the functions
have the same concavity?"

00:02:44.588 --> 00:02:48.846
And one way to think about concavity is

00:02:48.846 --> 00:02:51.717
whether it's opening upwards
or opening downwards.

00:02:51.717 --> 00:02:55.631
So this is often viewed
as concave upwards,

00:02:55.631 --> 00:02:58.708
and this is viewed as concave downwards.

00:02:58.708 --> 00:03:02.387
Concave downwards.

00:03:02.387 --> 00:03:04.825
And the key realization is,

00:03:04.825 --> 00:03:07.787
well, if you just look at this
blue, if you look at g of x

00:03:07.787 --> 00:03:09.956
right over here, it is concave downwards.

00:03:09.956 --> 00:03:11.791
So the question is, "Would this be concave

00:03:11.791 --> 00:03:12.952
"downwards or upwards?"

00:03:12.952 --> 00:03:14.894
And the key here is the coefficient on

00:03:14.894 --> 00:03:19.894
the second degree term,
on the x squared term.

00:03:19.894 --> 00:03:22.019
If the coefficient is positive,

00:03:22.019 --> 00:03:24.318
you're going to be concave upwards,

00:03:24.318 --> 00:03:27.452
because as x gets suitably
far away from zero,

00:03:27.452 --> 00:03:30.355
this term is going to
overpower everything else,

00:03:30.355 --> 00:03:32.387
and it's going to become positive.

00:03:32.387 --> 00:03:35.289
So as x gets further and further away,

00:03:35.289 --> 00:03:36.983
we're not even further away from zero,

00:03:36.983 --> 00:03:38.853
as x gets further and
further away from the vertex,

00:03:38.853 --> 00:03:41.873
as x gets further and
further away from the vertex,

00:03:41.873 --> 00:03:43.892
this term dominates everything else,

00:03:43.892 --> 00:03:46.458
and we get more and more positive values.

00:03:46.458 --> 00:03:49.419
And so that's why if your
coefficient is positive,

00:03:49.419 --> 00:03:51.195
you're going to have concave upwards,

00:03:51.195 --> 00:03:53.087
a concave upwards graph.

00:03:53.087 --> 00:03:54.759
And so if this is concave upwards,

00:03:54.759 --> 00:03:56.791
this one is clearly concave downwards.

00:03:56.791 --> 00:04:00.959
They do not have the
same concavity, so no.

00:04:00.959 --> 00:04:03.989
If this was negative
four x squared minus 108,

00:04:03.989 --> 00:04:06.789
then it would be concave
downwards and we would say yes.

00:04:06.789 --> 00:04:09.356
Anyway, hopefully you
found that interesting.


Finding features of quadratic functions

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:02.299
- [Voiceover] So, I have three
different functions here.

00:00:02.299 --> 00:00:03.204
I know they're all called f,

00:00:03.204 --> 00:00:06.035
but we're gonna just assume
they are different functions.

00:00:06.035 --> 00:00:10.472
And, for each of these, I
want to do three things.

00:00:10.472 --> 00:00:12.770
I want to find the zeros.

00:00:12.770 --> 00:00:14.860
And so, the zeros are the input values

00:00:14.860 --> 00:00:16.927
that make the value of the
function equal to zero.

00:00:16.927 --> 00:00:21.099
So, here it'd be the t values
that make f of t equal zero.

00:00:21.099 --> 00:00:22.231
Here, it'd be the x values

00:00:22.231 --> 00:00:23.736
that make the function equal zero.

00:00:23.736 --> 00:00:25.447
So, I want to find the zeros.

00:00:25.447 --> 00:00:29.386
I also want to find the
coordinates of the vertex.

00:00:29.386 --> 00:00:31.126
And finally, I want to find the equation

00:00:31.126 --> 00:00:33.348
of the line of symmetry.

00:00:33.348 --> 00:00:35.368
Line of symmetry.

00:00:35.368 --> 00:00:37.681
And in particular, to make it
a little bit more specific,

00:00:37.681 --> 00:00:39.733
the vertical line of symmetry,
which will actually be

00:00:39.733 --> 00:00:42.448
the only line of symmetry for these three.

00:00:42.448 --> 00:00:44.098
So, pause the video and
see if you can figure out

00:00:44.098 --> 00:00:46.898
the zeros, the vertex,
and the line of symmetry.

00:00:46.898 --> 00:00:48.649
And I'm assuming you just did that

00:00:48.649 --> 00:00:49.973
and now I am going to attempt to do it.

00:00:49.973 --> 00:00:52.451
And if, at any point, you get inspired

00:00:52.451 --> 00:00:54.554
pause the video again and
keep on working on it.

00:00:54.554 --> 00:00:57.165
The best way to learn this
stuff is to do it yourself.

00:00:57.165 --> 00:00:59.220
So, let's see, so let's
first find the zeros.

00:00:59.220 --> 00:01:01.334
So, to find the zeros, we can set

00:01:01.334 --> 00:01:05.505
t minus five squared,
minus nine, equal to zero.

00:01:05.505 --> 00:01:08.970
So, we could say t ... for what t's does

00:01:08.970 --> 00:01:12.909
t minus five squared,
minus nine, equal zero?

00:01:12.909 --> 00:01:15.955
Let's see, to solve this, we
could add nine to both sides.

00:01:15.955 --> 00:01:18.927
And so we could say, if
we add nine to both sides

00:01:18.927 --> 00:01:22.503
the lefthand side's just
t minus five squared.

00:01:22.503 --> 00:01:24.641
The righthand side is going to be nine.

00:01:24.641 --> 00:01:27.532
And so, if t minus five squared
is nine, that means that

00:01:27.532 --> 00:01:30.163
t minus five could be equal to

00:01:30.163 --> 00:01:31.883
the positive square root of nine

00:01:31.883 --> 00:01:34.902
or t minus five could equal

00:01:34.902 --> 00:01:38.280
the negative square root of nine.

00:01:38.280 --> 00:01:41.246
And to solve for t, we could
add five to both sides,

00:01:41.246 --> 00:01:44.830
so we get t is equal to
eight or t is equal to,

00:01:44.830 --> 00:01:46.528
if we add five to both sides here,

00:01:46.528 --> 00:01:48.638
t is equal to two.

00:01:48.638 --> 00:01:51.068
And just like that, we
have found the zeros

00:01:51.068 --> 00:01:54.200
for this function because if
t is equal to eight or two,

00:01:54.200 --> 00:01:55.831
the function is going to be zero.

00:01:55.831 --> 00:02:00.831
F of eight is zero and f
of two is going to be zero.

00:02:01.046 --> 00:02:05.708
Now, let's find the vertex,
the coordinates of the vertex.

00:02:05.708 --> 00:02:08.484
The coordinates of the vertex.

00:02:08.484 --> 00:02:10.118
So, the x coordinate of the vertex,

00:02:10.118 --> 00:02:12.642
or sorry, I should say the
t coordinate of the vertex

00:02:12.642 --> 00:02:15.241
since the input variable here is t,

00:02:15.241 --> 00:02:17.752
the t coordinate of the
vertex is going to be

00:02:17.752 --> 00:02:20.682
halfway in between the zeros.

00:02:20.682 --> 00:02:23.505
It's going to be halfway in between where

00:02:23.505 --> 00:02:26.595
the parabola, in this case,

00:02:26.595 --> 00:02:28.573
is going to intersect the x axis,

00:02:28.573 --> 00:02:30.358
or the t axis, I keep saying x axis,

00:02:30.358 --> 00:02:31.937
the t axis for this case.

00:02:31.937 --> 00:02:33.725
So, halfway between eight and two.

00:02:33.725 --> 00:02:35.466
Well, it's gonna be the average of them.

00:02:35.466 --> 00:02:37.219
Eight plus two, over two.

00:02:37.219 --> 00:02:39.844
That's ten over two, that's five.

00:02:41.534 --> 00:02:43.593
The t coordinate is five and five is

00:02:43.593 --> 00:02:46.492
three away from eight
and three away from two.

00:02:46.492 --> 00:02:49.612
And, when t is equal to
five, what is f of t?

00:02:49.612 --> 00:02:50.930
What is f of five?

00:02:50.930 --> 00:02:52.207
Well, when t is equal to five,

00:02:52.207 --> 00:02:54.825
five minus five squared is just zero.

00:02:54.825 --> 00:02:59.416
And then, f of five is just
going to be negative nine.

00:02:59.416 --> 00:03:02.811
And this form of a function,
this is actually called

00:03:02.811 --> 00:03:06.066
vertex form because it's very
easy to pick out the vertex.

00:03:06.066 --> 00:03:08.140
It's very easy to
realize, like, okay look,

00:03:08.140 --> 00:03:10.824
for this particular one, we're
going to hit a minimum point

00:03:10.824 --> 00:03:14.428
when this part of the
expression is equal to zero

00:03:14.428 --> 00:03:17.539
because this thing, the
lowest value it can take on

00:03:17.539 --> 00:03:19.443
is zero cause you're squaring it,

00:03:19.443 --> 00:03:21.301
it can never take on a negative value.

00:03:21.301 --> 00:03:23.971
And it takes on zero
when t is equal to five.

00:03:23.971 --> 00:03:25.902
And when that's, if this part is zero,

00:03:25.902 --> 00:03:29.015
then the f of five is
going to be negative nine.

00:03:29.015 --> 00:03:31.903
So, just like that, we have
established the vertex.

00:03:31.903 --> 00:03:33.682
Now, we actually have a lot of information

00:03:33.682 --> 00:03:35.534
if we wanted to draw it.

00:03:35.534 --> 00:03:38.260
So, if we want to draw this function.

00:03:38.260 --> 00:03:41.270
I'll just do a very quick sketch of it.

00:03:41.270 --> 00:03:42.546
Whoops.

00:03:42.546 --> 00:03:44.473
So, a very quick sketch of it.

00:03:44.473 --> 00:03:47.933
So, that is our t axis, not our x axis,

00:03:47.933 --> 00:03:50.198
I have to keep reminding myself.

00:03:50.198 --> 00:03:53.970
And that is my,

00:03:53.970 --> 00:03:56.740
let's call that my y axis.

00:03:56.740 --> 00:04:00.960
And we're going to graph
y is equal to f of t.

00:04:00.960 --> 00:04:02.213
Well, we know the vertex is at the point

00:04:02.213 --> 00:04:04.019
five comma negative nine.

00:04:04.019 --> 00:04:07.530
So, this is, t is equal to five

00:04:07.530 --> 00:04:11.732
and y is equal to negative nine,

00:04:11.732 --> 00:04:14.704
so that's the vertex right over there.

00:04:14.704 --> 00:04:15.760
And then, we know we have zeros

00:04:15.760 --> 00:04:18.001
at t equals eight and t equals two.

00:04:18.001 --> 00:04:22.413
So, t equals eight and t equals two.

00:04:22.413 --> 00:04:25.594
Let me make that a little
bit in, t equals two.

00:04:25.594 --> 00:04:27.139
Those are the two zeros.

00:04:27.139 --> 00:04:29.240
So, eight and two.

00:04:29.240 --> 00:04:32.211
And so, just like that,
we can graph f of t,

00:04:32.211 --> 00:04:35.183
or we can graph y is equal to f of t.

00:04:35.183 --> 00:04:40.183
So, y is equal to f of t is
going to look something like

00:04:42.507 --> 00:04:45.754
let me draw it a little
... something like that

00:04:45.754 --> 00:04:49.020
That's the graph of y is equal to f of t.

00:04:49.020 --> 00:04:51.837
Now, the last thing that I
said is the line of symmetry.

00:04:51.837 --> 00:04:55.262
Well, the line of symmetry is going to be

00:04:55.262 --> 00:04:58.526
the vertical line that
goes through the vertex.

00:04:58.526 --> 00:05:00.897
So, the equation of that line of symmetry

00:05:00.897 --> 00:05:04.354
is going to be t is equal to five.

00:05:04.354 --> 00:05:07.309
That's really just the t
coordinate of the vertex.

00:05:07.309 --> 00:05:09.834
That defines the line of symmetry.

00:05:09.834 --> 00:05:12.206
Let's do the other two right over here.

00:05:12.206 --> 00:05:14.228
So, what are the zeros?

00:05:14.228 --> 00:05:18.666
Well, if you set this equal
to zero, if we say x plus two

00:05:18.666 --> 00:05:21.843
times x plus four is equal to zero,

00:05:21.843 --> 00:05:26.296
well that's going to happen
if x plus two is equal to zero

00:05:26.296 --> 00:05:29.825
or x plus four is equal to zero.

00:05:29.825 --> 00:05:32.193
This is going to happen if we
subtract two from both sides,

00:05:32.193 --> 00:05:34.903
when x is negative two,
and if we subtract four

00:05:34.903 --> 00:05:38.555
from both sides, or when x
is equal to negative four.

00:05:38.555 --> 00:05:42.682
As we said, the vertex, the
x coordinate of the vertex,

00:05:42.682 --> 00:05:44.831
is going to be halfway in between these.

00:05:44.831 --> 00:05:47.953
So, it's going to be negative
two plus negative four,

00:05:47.953 --> 00:05:51.618
over two, so that would
be negative six, over two,

00:05:51.618 --> 00:05:54.135
which is just negative three.

00:05:54.135 --> 00:05:55.832
Negative three.

00:05:55.832 --> 00:05:59.365
And when x is negative
three, f of x is going to be,

00:05:59.365 --> 00:06:02.338
let's see, it's going to be negative one

00:06:02.338 --> 00:06:05.368
times one, right?

00:06:05.368 --> 00:06:10.257
Negative three plus two is
negative one, and so times one.

00:06:10.257 --> 00:06:12.970
So, it's just going to be negative one.

00:06:12.970 --> 00:06:14.003
There you have it.

00:06:14.003 --> 00:06:16.857
And the line of symmetry is
going to be the vertical line

00:06:16.857 --> 00:06:19.197
x is equal to negative three.

00:06:19.197 --> 00:06:23.121
And once again, we can
graph that really fast.

00:06:23.121 --> 00:06:25.486
So, let me,

00:06:25.486 --> 00:06:28.866
this is my y axis.

00:06:28.866 --> 00:06:31.780
See everything is
happening for negative x's,

00:06:31.780 --> 00:06:36.216
so I'll draw it a little
bit more skewed this way.

00:06:36.216 --> 00:06:38.615
This is my x axis.

00:06:38.615 --> 00:06:42.160
And we see that we have zeros
at x equals negative two

00:06:42.160 --> 00:06:45.388
and x is equal to negative four.

00:06:45.388 --> 00:06:49.605
So, negative one, two, three, four.

00:06:49.605 --> 00:06:54.142
So, we have zeros

00:06:54.142 --> 00:06:56.912
there, negative two, be careful.

00:06:56.912 --> 00:06:59.682
Negative two and negative four.

00:06:59.682 --> 00:07:03.382
And the vertex is at negative
three comma negaitve one.

00:07:03.382 --> 00:07:06.531
So, negative three comma negative one.

00:07:06.531 --> 00:07:08.049
Make sure we see that, so this is

00:07:08.049 --> 00:07:10.708
negative one right over
here, negative one.

00:07:10.708 --> 00:07:13.978
This is negative two,
this is negative four.

00:07:13.978 --> 00:07:18.212
And so we can sketch out what the graph of

00:07:18.212 --> 00:07:20.704
y is equal to f of x
is going to look like.

00:07:20.704 --> 00:07:24.226
It's gonna look something like that.

00:07:24.226 --> 00:07:27.295
That is y is equal to f of x.

00:07:27.295 --> 00:07:28.456
Let's do one more.

00:07:28.456 --> 00:07:30.621
Hopefully, we're getting the hang of this.

00:07:30.621 --> 00:07:34.817
So, here, to solve x squared,
plus six x, plus eight

00:07:34.817 --> 00:07:38.239
is equal to zero, it will
be useful to factor this.

00:07:38.239 --> 00:07:40.221
And so, this can be written as, and if you

00:07:40.221 --> 00:07:41.831
have trouble doing this,
I encourage you to watch

00:07:41.831 --> 00:07:43.618
videos on factoring polynomials,

00:07:43.618 --> 00:07:45.830
What adds up to six and when you

00:07:45.830 --> 00:07:47.492
take their product is eight?

00:07:47.492 --> 00:07:48.889
Well, four and two.

00:07:48.889 --> 00:07:53.679
Four plus two is six and
four times two is eight.

00:07:53.679 --> 00:07:56.817
So, is equal to zero.

00:07:56.817 --> 00:07:58.664
And then, this is actually
the exact same thing

00:07:58.664 --> 00:08:00.591
as what we have in blue right over here.

00:08:00.591 --> 00:08:03.456
These are actually the
exact same function.

00:08:03.456 --> 00:08:05.628
They're just written in different forms.

00:08:05.628 --> 00:08:08.321
And so, the solutions are
gonna be the exact solutions

00:08:08.321 --> 00:08:11.568
that we just saw right
over here and the graph

00:08:11.568 --> 00:08:14.610
is going to be the same thing
that we have right over there.

00:08:14.610 --> 00:08:17.982
So, same vertex, same line
of symmetry, same zeros.

00:08:17.982 --> 00:08:21.815
These functions were just
written in different ways.


Worked example: Solving equations by completing the square

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:01.192
- [Voiceover] So let's see if we can solve

00:00:01.192 --> 00:00:03.374
this quadratic equation right over here:

00:00:03.374 --> 00:00:06.286
x squared minus two x minus eight

00:00:06.286 --> 00:00:08.855
is equal to zero.

00:00:08.855 --> 00:00:10.810
And actually, they're cutting
down some trees outside,

00:00:10.810 --> 00:00:14.327
so my apologies if you hear
some chopping of trees.

00:00:14.327 --> 00:00:15.619
I'll try to ignore it myself.

00:00:15.619 --> 00:00:19.586
All right. So back to the problem at hand,

00:00:19.586 --> 00:00:20.808
and there's actually several ways

00:00:20.808 --> 00:00:22.130
that you could attack this problem.

00:00:22.130 --> 00:00:24.752
We could just try to
factor the left-hand side

00:00:24.752 --> 00:00:27.284
and go that way, but the
way we're going to tackle it

00:00:27.284 --> 00:00:29.392
is by completing the square.

00:00:29.392 --> 00:00:30.577
Now what does that mean?

00:00:30.577 --> 00:00:32.799
Well that means that I wanna write,

00:00:32.799 --> 00:00:36.169
I wanna write the left-hand
side of this equation

00:00:36.169 --> 00:00:37.588
into the form

00:00:37.588 --> 00:00:42.148
x plus a squared plus b,

00:00:42.148 --> 00:00:45.936
and we'll see if we can write
the left-hand in this form

00:00:45.936 --> 00:00:47.640
that we can actually solve it

00:00:47.640 --> 00:00:49.072
in a pretty straightforward way.

00:00:49.072 --> 00:00:50.744
So let's see if we could do that.

00:00:50.744 --> 00:00:52.736
Well let's just remind ourselves

00:00:52.736 --> 00:00:54.818
how we need to rearrange
the left-hand side

00:00:54.818 --> 00:00:56.419
in order to get to this form.

00:00:56.419 --> 00:00:59.075
If I were to expand out x plus a squared,

00:00:59.075 --> 00:01:00.717
let me do that in a different color.

00:01:00.717 --> 00:01:03.140
So if I were to expand
out x plus a squared,

00:01:03.140 --> 00:01:07.160
that is x squared plus two a x,

00:01:07.160 --> 00:01:08.722
I'll make that plus sign you can see,

00:01:08.722 --> 00:01:10.183
plus two a x,

00:01:10.183 --> 00:01:11.845
plus a squared,

00:01:11.845 --> 00:01:16.069
and of course you still have
that plus b there, plus b.

00:01:16.069 --> 00:01:18.648
So let's see if we can
write this in that form.

00:01:18.648 --> 00:01:20.493
So, what I'm going to do,

00:01:20.493 --> 00:01:21.551
this is what you typically do

00:01:21.551 --> 00:01:22.763
when you try to complete the square.

00:01:22.763 --> 00:01:25.155
All right. The x squared minus two x.

00:01:25.155 --> 00:01:27.387
Now I'm gonna have a little bit of a gap

00:01:27.387 --> 00:01:29.989
and I'm gonna have minus eight,

00:01:29.989 --> 00:01:31.601
and I have another a little bit of a gap

00:01:31.601 --> 00:01:33.416
and I'm gonna say equals zero.

00:01:33.416 --> 00:01:34.958
So I just rewrote this equation,

00:01:34.958 --> 00:01:36.629
but I gave myself some space

00:01:36.629 --> 00:01:38.271
so I can add or subtract some things

00:01:38.271 --> 00:01:39.933
that might make it a little bit easier

00:01:39.933 --> 00:01:41.468
to get into this form.

00:01:41.468 --> 00:01:44.616
So, if we just match our terms,

00:01:44.616 --> 00:01:47.388
x squared, x squared,

00:01:47.388 --> 00:01:51.544
two a x, negative two x.

00:01:51.544 --> 00:01:54.180
So, if this is two a x,

00:01:54.180 --> 00:01:56.813
that means that two a is negative two,

00:01:56.813 --> 00:01:58.828
two a is equal to negative two

00:01:58.828 --> 00:02:01.648
or a is equal to negative one.

00:02:01.648 --> 00:02:02.559
Another way to think about it,

00:02:02.559 --> 00:02:07.549
your a is going to be half of
your first degree coefficient

00:02:07.549 --> 00:02:09.341
or the coefficient on the x term.

00:02:09.341 --> 00:02:11.013
So the coefficient of the
x term is a negative two,

00:02:11.013 --> 00:02:13.214
half of that is a negative one.

00:02:13.214 --> 00:02:14.925
And then we wanna have,

00:02:14.925 --> 00:02:19.443
and then we want to have an a squared.

00:02:19.443 --> 00:02:22.425
So if a is negative one, a
squared would be plus one.

00:02:22.425 --> 00:02:24.758
So let's throw a plus one there.

00:02:24.758 --> 00:02:25.990
But like we've done and said before,

00:02:25.990 --> 00:02:28.382
we can't just willy-nilly add something

00:02:28.382 --> 00:02:30.817
on one side of equation
without adding it to the other

00:02:30.817 --> 00:02:33.730
or without subtracting it
again on that same side.

00:02:33.730 --> 00:02:37.372
Otherwise, you're fundamentally changing

00:02:37.372 --> 00:02:39.137
the truth of the equation.

00:02:39.137 --> 00:02:41.989
So if I add one on that side,
I even have to add one on the,

00:02:41.989 --> 00:02:43.050
if I add one of the left side,

00:02:43.050 --> 00:02:44.652
I even have to add one on the right side

00:02:44.652 --> 00:02:46.384
to make the equation still hold true

00:02:46.384 --> 00:02:50.518
or I could add one and subtract
one from the left-hand side,

00:02:50.518 --> 00:02:53.429
so I'm not really changing the
value of the left-hand side.

00:02:53.429 --> 00:02:54.717
All I've done is added one

00:02:54.717 --> 00:02:57.320
and subtracted one from
the left-hand side.

00:02:57.320 --> 00:02:59.412
Now why did I do this again?

00:02:59.412 --> 00:03:00.467
Well now, I've been able,

00:03:00.467 --> 00:03:02.379
I haven't changed its value.

00:03:02.379 --> 00:03:04.621
I just added and
subtracted the same thing,

00:03:04.621 --> 00:03:07.113
but this part of the left-hand side

00:03:07.113 --> 00:03:10.650
now matches this pattern right over here,

00:03:10.650 --> 00:03:13.329
x squared plus two a x,

00:03:13.329 --> 00:03:16.471
where a is negative one,
so it's minus two x,

00:03:16.471 --> 00:03:19.165
plus a squared, plus negative one squared

00:03:19.165 --> 00:03:21.413
and then this,

00:03:21.413 --> 00:03:24.384
this part right over here is the plus b.

00:03:24.384 --> 00:03:28.471
So we already know that b
is equal to negative nine.

00:03:28.471 --> 00:03:31.758
Negative eight minus one is negative nine,

00:03:31.758 --> 00:03:35.148
and so that's going to be
our b right over there.

00:03:35.148 --> 00:03:37.937
And so we can rewrite this as,

00:03:37.937 --> 00:03:39.632
what I squared off in green,

00:03:39.632 --> 00:03:42.842
that's gonna be x plus a squared.

00:03:42.842 --> 00:03:45.358
So we could write it as x plus

00:03:45.358 --> 00:03:48.061
and I could write a is negative one.

00:03:48.061 --> 00:03:50.847
Actually, let me, I could
write it like that first.

00:03:50.847 --> 00:03:54.344
x plus a squared or x plus negative one.

00:03:54.344 --> 00:03:56.886
Well, let's just x minus one,
so I'm just gonna write it

00:03:56.886 --> 00:04:01.808
as x minus negative one squared

00:04:01.808 --> 00:04:04.644
and then we have minus nine,

00:04:04.644 --> 00:04:08.625
minus nine is equal to zero,

00:04:08.625 --> 00:04:11.398
is equal to zero.

00:04:11.398 --> 00:04:13.400
And then I can add nine to both side,

00:04:13.400 --> 00:04:15.745
so I just have this squared expression

00:04:15.745 --> 00:04:18.658
on the left-hand side, so let's do that.

00:04:18.658 --> 00:04:22.215
Let me add nine to both sides.

00:04:22.215 --> 00:04:27.215
And what I am going to be
left with, so let me just,

00:04:27.634 --> 00:04:30.447
on the left-hand side, those cancel out.

00:04:30.447 --> 00:04:31.988
That's why added the nine.

00:04:31.988 --> 00:04:36.893
I'm just gonna be left with
the x minus one squared.

00:04:36.893 --> 00:04:41.614
It's going to be equal to, on this side,

00:04:41.614 --> 00:04:44.758
zero plus nine is nine.

00:04:44.758 --> 00:04:47.920
So if x minus one, let
me do that in blue color.

00:04:47.920 --> 00:04:50.323
So, it's gonna be nine.

00:04:50.323 --> 00:04:52.622
And so if x minus one squared is nine,

00:04:52.622 --> 00:04:54.344
if I have something
square is equal to nine,

00:04:54.344 --> 00:04:56.843
that means that that something
is either going to be

00:04:56.843 --> 00:04:59.592
the positive or the negative
square root of nine.

00:04:59.592 --> 00:05:01.981
So it's either gonna be
positive or negative three.

00:05:01.981 --> 00:05:06.981
So we can say x minus one
is equal to positive three

00:05:07.063 --> 00:05:10.510
or x minus one is equal to negative three

00:05:10.510 --> 00:05:11.501
and you could see that here.

00:05:11.501 --> 00:05:14.013
If x minus one is three,
three squared is nine.

00:05:14.013 --> 00:05:15.875
If x minus one is negative three,

00:05:15.875 --> 00:05:18.341
negative three squared is nine.

00:05:18.341 --> 00:05:20.283
And so here, we can just add one

00:05:20.283 --> 00:05:22.405
to both sides of this equation,

00:05:22.405 --> 00:05:25.835
add one to both sides of this equation,

00:05:25.835 --> 00:05:30.063
and you get x is equal to four

00:05:30.063 --> 00:05:34.467
or x is, if we add one to
both sides of this equation,

00:05:34.467 --> 00:05:37.684
we get, my digital ink is
acting up, I don't know.

00:05:37.684 --> 00:05:42.684
All right. Then we get x is
equal to negative three plus one

00:05:42.729 --> 00:05:45.144
is negative two.

00:05:45.144 --> 00:05:46.446
So, x could be equal to four

00:05:46.446 --> 00:05:48.858
or x could be equal to
negative two, and we're done.

00:05:48.858 --> 00:05:49.830
Now, some of you might be saying,

00:05:49.830 --> 00:05:51.102
"Well, why did we go through the trouble

00:05:51.102 --> 00:05:52.043
"of completing the square?

00:05:52.043 --> 00:05:54.135
"I might have been able
to just factor this

00:05:54.135 --> 00:05:55.793
"and then solve it that way."

00:05:55.793 --> 00:05:58.125
And you could have, actually,
for this particular problem.

00:05:58.125 --> 00:05:59.497
Completing the square is very powerful

00:05:59.497 --> 00:06:02.249
because you could actually
always apply this,

00:06:02.249 --> 00:06:04.161
and in the future, what you will learn

00:06:04.161 --> 00:06:05.211
in the quadratic formula

00:06:05.211 --> 00:06:07.664
and the quadratic formula
actually comes directly

00:06:07.664 --> 00:06:09.892
out of completing the square.

00:06:09.892 --> 00:06:11.314
In fact, when you're applying
he quadratic formula,

00:06:11.314 --> 00:06:13.683
you're essentially applying the result

00:06:13.683 --> 00:06:15.675
of completing the square.

00:06:15.675 --> 00:06:18.091
So hopefully you found that fun.