Algebra II: Circles and Logarithms

WEBVTT
Kind: captions
Language: en

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We're on problem 43.

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Which statement describes
the graph of

00:00:06.440 --> 00:00:07.330
the following equation?

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They write x squared plus y
squared plus 4x minus 6y minus

00:00:17.710 --> 00:00:20.250
3 is equal to 0.

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Essentially you want to get
this into standard conic

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square format-- I don't know
what it's officially called--

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and you do that by completing
the square on both the x and

00:00:28.560 --> 00:00:32.330
the y I guess you could call
it sub-polynomial.

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Then you should be able
recognize what kind of conic

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section it is.

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So let's separate out the x
and the y terms. So the x

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terms-- I'll do that in blue--
you get x squared plus 4x.

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Then we're going to add
something that makes it a

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perfect square.

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You can already guess what
we're going to add.

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Let's say we're going to add 4,
because x squared plus 4x

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plus 4 is x plus 2 squared.

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So I'm going to write that in
magenta just so we remember

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that we added it.

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This came out of nowhere
right now.

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So if I added plus 4 here,
I'm going to have to add

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plus 4 on this side.

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This is the equals sign.

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All right, then let me do the
y terms. I'll do it in this

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brown color.

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Plus y squared minus 6y.

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Then what would add here?

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Half of minus 6 is mine 3.

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Then if I were to square that,
minus 3 squared-- I'll do it

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in this magenta color
again-- is plus 9.

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So I want to add plus 9 to the
right-hand side as well.

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Of course this is y minus 3
squared, this whole thing

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right here.

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Then I have that minus
3 right over here.

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All right now let's rewrite
this term right here.

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I added this plus 4 here for a
reason so that I could make

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this into x plus 2 squared, then
plus-- and I added this

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plus 9 here for a reason,
so I could make

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this y minus 3 squared.

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Let me just keep the minus
3 there for now.

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Minus-- let me do it in the
other color-- minus 3 is equal

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to 4 plus 9.

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4 plus 9 is equal to 13.

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Let's add the 3 to both sides
of that equation.

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So you get x plus 2 squared
plus y minus 3 squared is

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equal to 16.

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Now if you think about it, this
is the form of a circle.

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x minus a squared plus y minus
b squared is equal to r

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squared, where the center of
the circle is at where?

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Whatever the x-value is what
makes this equal to 0.

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So it's x, it's minus 2.

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Then the y-value of its center
is what makes this equal to 0.

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Minus 2 comma 3, that's
its center.

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It's radius is the square
root of 16.

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So the radius is equal to 4.

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It can't be negative 4.

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You can't have a negative
radius.

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So the answer is a circle with
radius at minus 2 comma

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3 and radius 4.

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That's choice D.

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Problem 44.

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What is the solution to
the equation 5 to the

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x is equal to 17?

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OK so that means that we if we
take the log base, so five to

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the x power is equal to 17.

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We could rewrite this
in log format.

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That means that log base 5 of
seventeen is equal to x.

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You could read this exactly the
same way you read that.

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5 to the x is equal to 17.

00:04:00.680 --> 00:04:02.810
Of course that's not one of
our choices, and this is

00:04:02.810 --> 00:04:04.510
fairly useless on your
calculator because you don't

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have a log base 5 button.

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Let me write it this way, x is
equal to-- I'm just switching

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it-- log base 5 of 17, just
because it's easier

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for me to deal with.

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I prove it in previous Khan
Academy logarithm videos, but

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there's just a general
property.

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I'll write the property here.

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If x is equal to log base, I
don't know, base b of a, that

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is equal to log base 10 of a,
lowercase a, divided by log

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base 10 of b.

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You just have to pick
the same base.

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You could have the base, you
could have the natural log of

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a divided by the natural
log of b.

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You could do whatever log you
want, but log base 10 is on

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your calculator.

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So that's why this
is convenient.

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So we could say this is equal
to the log base 10 of 17

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divided by the log
base 10 of 5.

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That should be a choice.

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Yep, that's choice D.

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Next question.

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This problem 45.

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I'll write it down, 45.

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They say if log base 10 of x is
equal to minus 2, what is

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the value of x?

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So that essentially --
10 to the minus 2

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power is equal to x.

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All a logarithm is, it tells you
what exponent do I have to

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raise the base to to get
whatever I'm inputting into

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the logarithm function.

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Right So if I raise 10 to
the minus 2, I get x.

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10 to the minus 2, that's 1 over
10 squared is equal to x,

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which is 1 over 100
is equal to x.

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That's choice C.

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I think I could do problem
46 right here.

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46.

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Which equation is equivalent
to log base 3 of

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1/9 is equal to x?

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OK, read it the exact same
way we read question 45.

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This is just saying that 3 to
the x power is equal to 1/9.

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Let's write that.

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3 to the x power is
equal to 1/9.

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Is that one of the choices?

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Yeah.

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That's choice C.

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That was suspiciously
easy, I think.

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You just have to know
the definition of

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what a logarithm is.

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OK problem 47.

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I'll clear the screen.

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Let me switch colors.

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47.

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Oh I'll cut and paste this one
because this is interesting.

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Let me pull it up a little bit
so you can definitely read it.

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Because they want us to identify
an incorrect step.

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OK let me go here.

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OK they ask us, which is the
first incorrect step in

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simplifying log base
4 of 4 over 64.

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All right step 1, they said log
base 4 4/64 is equal to

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log base 4 4 minus
log base 4 64.

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That's right.

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When you divide within the
logarithm function or when

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you're inputting into the
logarithm function, that's

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equivalent of taking the log of
the top one and subtracting

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from that the log
of the bottom.

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Actually when you divide two
numbers of the same base, you

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subtract their exponents.

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We've proved that in
previous videos.

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But that's just good enough
that you need to know that

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that's the correct
step right now.

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So what's the log base 4 of 4?

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4 to what power is equal to 4?

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Yeah sure, that's equal to 1.

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4 to what power is
equal to 64?

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Let's see, 4 squared is 16.

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4 to the 1 is equal to 4.

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4 squared is equal to 16.

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4 to the third is 4 times
16, which is 64.

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So 4 to the third power is 64.

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So this is 3.

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So this should be-- let's see,
log of 4 minus 16-- right,

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this should be a 3 here.

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This should be a minus 3.

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So this step 2 was the
first incorrect step.

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They mis-evaluated
log base 4 of 64.

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They through it was 16
for some reason.

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I guess they divided 64 by 4.

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But it's actually 4 to the third
power of 16, sorry, 4 to

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the third power of 64.

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So the answer there is 3.

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So step 2 is the first
incorrect step.

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Next question.

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All right let me see.

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All right, clear image,
invert colors.

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Let me copy and paste this
one too because this is

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interesting.

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I'll try to copy and paste
as many of the

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word problems as possible.

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This is problem 48.

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Let me write that down
in the corner.

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48, right in that corner.

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Jeremy, Michael, Shanan, and
Brenda each worked the same

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math problem of the
chalkboard.

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Each student's work
is shown below.

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Their teacher said that while
two of them had the correct

00:09:51.460 --> 00:09:54.210
answer, only one of them had
arrived at the correct

00:09:54.210 --> 00:09:56.560
conclusion using
correct steps.

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OK they want to which is the
completely correct solution.

00:09:58.640 --> 00:10:02.100
OK let me just copy
and paste those.

00:10:02.100 --> 00:10:09.500
So the completely correct
solution, OK so let's just

00:10:09.500 --> 00:10:12.410
think about what everyone
did here.

00:10:12.410 --> 00:10:15.690
It looks like this is it
dealing with exponents.

00:10:15.690 --> 00:10:18.840
All right he had x to third
times x to the minus 7.

00:10:18.840 --> 00:10:21.050
OK this is already wrong.

00:10:21.050 --> 00:10:24.610
Because x to the minus 7
is 1 over x to the 7.

00:10:24.610 --> 00:10:25.880
This is one over x to the 7.

00:10:25.880 --> 00:10:27.760
So this should be a
plus right here.

00:10:27.760 --> 00:10:29.920
So he's wrong.

00:10:29.920 --> 00:10:32.960
OK Shanan, x to the third
times x to the minus 7.

00:10:32.960 --> 00:10:37.830
Yep that equals x to the third
over x to the positive 7.

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Then let's see, 1
over x to the 4.

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That is right, right?

00:10:41.940 --> 00:10:45.380
Because this is equal to
x to the 3 minus 7.

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So that's x to the minus 4.

00:10:48.420 --> 00:10:51.800
x to the minus 4 is
1 over x to the 4.

00:10:51.800 --> 00:10:53.470
So as far as I can tell,
Shanan's work

00:10:53.470 --> 00:10:55.730
looks pretty good.

00:10:55.730 --> 00:10:58.020
Although this was a little weird
how she went straight to

00:10:58.020 --> 00:10:58.740
1 over x to the 4.

00:10:58.740 --> 00:11:00.310
She may just skipped a step.

00:11:00.310 --> 00:11:03.090
Michael's work, x to the
over x to the minus 7.

00:11:03.090 --> 00:11:05.165
OK, he made the same
mistake as Jeremy.

00:11:05.165 --> 00:11:07.710
X to the minus 7 when you put it
in the denominator becomes

00:11:07.710 --> 00:11:09.460
x to the positive 7.

00:11:09.460 --> 00:11:11.790
Not right.

00:11:11.790 --> 00:11:15.560
Then Brenda's work got
this step right.

00:11:15.560 --> 00:11:18.150
But then this should be
x to the minus 4, x

00:11:18.150 --> 00:11:18.960
to the minus 4 power.

00:11:18.960 --> 00:11:23.140
So this is not correct.

00:11:23.140 --> 00:11:26.320
So the only person who got
it right was Shanan.

00:11:26.320 --> 00:11:27.920
What choices is that?

00:11:27.920 --> 00:11:29.970
Shanan is choice C.

00:11:32.510 --> 00:11:33.045
Out of time.

00:11:33.045 --> 00:11:35.240
See you in the next video.


Hyperbola and parabola examples

WEBVTT
Kind: captions
Language: en

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We're on problem 39.

00:00:03.940 --> 00:00:09.060
And it says which ordered pairs
the vertex of f of x is

00:00:09.060 --> 00:00:14.280
equal to x squared
plus 6x plus 5?

00:00:14.280 --> 00:00:16.660
Now, finding vertexes of
parabolas, there's a formula,

00:00:16.660 --> 00:00:19.830
and negative b over 2a, but you
might forget that when you

00:00:19.830 --> 00:00:21.280
turn 30 years old.

00:00:21.280 --> 00:00:23.515
The whole goal of this isn't
just to pass an algebra exam.

00:00:23.515 --> 00:00:25.400
It's to learn algebra.

00:00:25.400 --> 00:00:27.190
Then when you learn calculus,
you'll learn actually a much

00:00:27.190 --> 00:00:30.160
easier way to figure out minimum
and maximum points.

00:00:30.160 --> 00:00:32.330
But there's actually an
intuitive way of doing it in

00:00:32.330 --> 00:00:33.440
algebra as well.

00:00:33.440 --> 00:00:37.530
What we have to do is just
write this in that form.

00:00:37.530 --> 00:00:40.510
If we could write this function
in the form like

00:00:40.510 --> 00:00:46.060
this, f of x is equal to A,
times x plus or minus

00:00:46.060 --> 00:00:51.550
something, I don't know,
b, squared, plus C.

00:00:51.550 --> 00:00:56.440
Then we know the vertex happens
when x is equal to b.

00:00:56.440 --> 00:00:59.530
Because if you look at this form
of a parabola-- and you

00:00:59.530 --> 00:01:03.490
can write any of these in this
form-- then you say OK, this

00:01:03.490 --> 00:01:06.380
term right here is always
going to be positive.

00:01:06.380 --> 00:01:09.320
So the minimum point of this
parabola is when this term is

00:01:09.320 --> 00:01:10.560
equal to 0.

00:01:10.560 --> 00:01:13.090
Then the minimum point of this
term happens when x is equal

00:01:13.090 --> 00:01:16.030
to b, when this term
is equal to 0.

00:01:16.030 --> 00:01:19.720
So the vertex becomes
xb comma C.

00:01:19.720 --> 00:01:21.400
So let's do that with this.

00:01:21.400 --> 00:01:22.530
You could memorize it.

00:01:22.530 --> 00:01:24.200
It ends up being minus
b over 2A.

00:01:24.200 --> 00:01:26.110
But let's just actually solve
it, because I want you to have

00:01:26.110 --> 00:01:27.440
the intuition.

00:01:27.440 --> 00:01:32.370
f of x is equal to x squared
plus 6x, plus 5.

00:01:32.370 --> 00:01:33.800
I wrote a gap there for a
reason, because we're

00:01:33.800 --> 00:01:35.430
essentially going to
complete a square.

00:01:35.430 --> 00:01:39.210
We want to turn this x squared
plus 6x, plus something, into

00:01:39.210 --> 00:01:40.890
a perfect square.

00:01:40.890 --> 00:01:42.020
Hopefully you have
some practice

00:01:42.020 --> 00:01:43.860
completing the square.

00:01:43.860 --> 00:01:47.130
What number is 1/2 of 6?

00:01:47.130 --> 00:01:52.000
Because when we square it, if
we do x plus a squared, that

00:01:52.000 --> 00:01:57.320
equals what? x squared plus
2ax plus a squared.

00:01:57.320 --> 00:02:01.390
So if we map that to that,
x squared is x squared.

00:02:01.390 --> 00:02:04.050
2ax is equal to 6.

00:02:04.050 --> 00:02:06.140
So a would be equal to 3.

00:02:06.140 --> 00:02:12.240
And if a is equal to 3, a
squared would be equal to 9.

00:02:12.240 --> 00:02:14.650
You can't just add a 9 to one
side of the equation.

00:02:14.650 --> 00:02:16.770
You either have to do it to both
sides, or you have to do

00:02:16.770 --> 00:02:17.940
something to offset that.

00:02:17.940 --> 00:02:23.920
So if we added a 9 here, let's
just subtract a 9 as well.

00:02:23.920 --> 00:02:29.500
So then we get x squared plus
6x, plus 9, plus 5, minus 9.

00:02:29.500 --> 00:02:35.150
So this right here, this part of
it, that's equal to x plus

00:02:35.150 --> 00:02:38.530
3, squared.

00:02:38.530 --> 00:02:41.150
And then this part, 5
minus 9, minus 4.

00:02:41.150 --> 00:02:44.350
There, we got it into our
format that we needed.

00:02:44.350 --> 00:02:49.190
So when does this function
achieve a minimum point?

00:02:49.190 --> 00:02:51.620
Well, it achieves a minimum
point when this part is equal

00:02:51.620 --> 00:02:55.700
to 0, or when x is
equal to minus 3.

00:02:55.700 --> 00:02:59.120
When x is equal to minus
3, this is 0.

00:02:59.120 --> 00:03:01.690
So f of x is going to
be equal to minus 4.

00:03:01.690 --> 00:03:04.820
So that's the vertex,
and that's choice A.

00:03:04.820 --> 00:03:07.400
You could have done
minus b over 2A.

00:03:07.400 --> 00:03:13.010
You'd get minus 12 over
2-- no minus 6 over 2,

00:03:13.010 --> 00:03:14.620
which is minus 3.

00:03:14.620 --> 00:03:15.700
That's something you
could memorize.

00:03:15.700 --> 00:03:18.670
But it's better, I think,
to have the intuition.

00:03:18.670 --> 00:03:24.650
Problem 40.

00:03:24.650 --> 00:03:36.630
The graph of x over 2, squared,
minus y over 2,

00:03:36.630 --> 00:03:41.020
squared equals 1
is a hyperbola.

00:03:41.020 --> 00:03:42.670
That makes sense.

00:03:42.670 --> 00:03:46.050
This is y over 3. x over 2,
squared, minus y over 3,

00:03:46.050 --> 00:03:46.770
squared is equal to 1.

00:03:46.770 --> 00:03:48.530
If this is a plus, we would
have an ellipse.

00:03:48.530 --> 00:03:49.250
Fair enough.

00:03:49.250 --> 00:03:51.590
Which set of equations
represents the asymptotes of

00:03:51.590 --> 00:03:53.140
the hyperbola's graph?

00:03:53.140 --> 00:03:56.250
Once again, there's some
equations that people memorize

00:03:56.250 --> 00:03:58.310
and formulas people memorize,
but you can actually have an

00:03:58.310 --> 00:03:59.670
intuition on this.

00:03:59.670 --> 00:04:03.570
So let's think about what
happens as x approaches

00:04:03.570 --> 00:04:04.990
positive or negative infinity.

00:04:04.990 --> 00:04:07.640
Because that's what we care
about as an asymptote, right?

00:04:07.640 --> 00:04:10.290
What does the graph approach
when we get to really, really

00:04:10.290 --> 00:04:12.720
large values of x, and
it turns out, really

00:04:12.720 --> 00:04:14.430
large values of y?

00:04:14.430 --> 00:04:16.640
So let's just rearrange
this a little bit.

00:04:16.640 --> 00:04:19.425
Let's put y on the other side.

00:04:22.610 --> 00:04:23.470
Let's square this.

00:04:23.470 --> 00:04:28.160
So this is x squared over 4.

00:04:28.160 --> 00:04:29.590
Let me keep it actually
in that format.

00:04:29.590 --> 00:04:36.110
Let's keep that as x over 2,
squared is equal to 1, plus y

00:04:36.110 --> 00:04:38.360
over 3, squared.

00:04:38.360 --> 00:04:41.700
Let's subtract 1 from
both sides.

00:04:41.700 --> 00:04:46.550
So you get x over 2, squared,
minus 1 is equal

00:04:46.550 --> 00:04:51.800
to y over 3, squared.

00:04:51.800 --> 00:04:56.740
And now, let's multiply both
sides by 3 squared.

00:04:56.740 --> 00:05:00.230
Let's multiply both
sides by 9.

00:05:00.230 --> 00:05:01.450
Let's do that.

00:05:01.450 --> 00:05:04.350
Actually, to simplify this,
let me rewrite this.

00:05:04.350 --> 00:05:06.440
I wanted to keep in that form.

00:05:06.440 --> 00:05:08.790
That's the same thing as x
squared over 4, minus 1, is

00:05:08.790 --> 00:05:11.950
equal to y squared over 9.

00:05:11.950 --> 00:05:14.370
Multiply both sides by 9.

00:05:14.370 --> 00:05:16.140
I'm going to switch the sides
so we get y here.

00:05:16.140 --> 00:05:20.700
So you get y squared is
equal to 9 over 4 x

00:05:20.700 --> 00:05:23.600
squared, minus 9.

00:05:23.600 --> 00:05:25.850
All right, and that's about as
far as we can get just with

00:05:25.850 --> 00:05:27.210
straight up algebra.

00:05:27.210 --> 00:05:29.330
When we think about asymptotes,
those are points

00:05:29.330 --> 00:05:32.590
that the graphs will never reach
but it'll get close to.

00:05:32.590 --> 00:05:34.580
So we have to think about what
happens with x gets really

00:05:34.580 --> 00:05:38.020
large and correspondingly
y gets really large.

00:05:38.020 --> 00:05:41.350
Well, as x gets really large, so
as x approaches infinity--

00:05:41.350 --> 00:05:43.120
and you'll learn this is
precalculus and calculus,

00:05:43.120 --> 00:05:45.120
these are limits, but it's
really a very intuitive

00:05:45.120 --> 00:05:49.050
concept-- as x gets really
large, this minus 9 doesn't

00:05:49.050 --> 00:05:50.000
matter much.

00:05:50.000 --> 00:05:52.150
This term starts
to dominate it.

00:05:52.150 --> 00:05:54.600
These terms matter a lot
more than this minus 9.

00:05:54.600 --> 00:05:59.580
So as x approaches infinity, y
squared is approximately equal

00:05:59.580 --> 00:06:03.730
to 9/4 x squared.

00:06:03.730 --> 00:06:05.800
Now we could take the square
root of both sides of this.

00:06:05.800 --> 00:06:12.850
So you get y is approximately
equal to plus or minus 3/2 x.

00:06:12.850 --> 00:06:17.420
Those are the actual asymptotes,
y equals 3/2 x and

00:06:17.420 --> 00:06:20.670
y is equal to minus 3/2 x.

00:06:20.670 --> 00:06:23.480
That's actually choice A.

00:06:23.480 --> 00:06:25.400
I won't go into the actual
instruction of how you would

00:06:25.400 --> 00:06:26.160
think about graphing.

00:06:26.160 --> 00:06:26.870
Actually, why not?

00:06:26.870 --> 00:06:27.730
Why don't we just do that.

00:06:27.730 --> 00:06:28.940
We know what the
asymptotes are.

00:06:28.940 --> 00:06:31.730
Let's actually think about
how you would graph this.

00:06:31.730 --> 00:06:35.000
So if you were to draw the--
it's all about learning, not

00:06:35.000 --> 00:06:40.100
just about assessing-- that's
the x- and y-axis.

00:06:40.100 --> 00:06:43.010
And they tell us their
asymptotes, 3/2 x

00:06:43.010 --> 00:06:44.550
and minus 3/2 x.

00:06:44.550 --> 00:06:48.350
So 3/2 x, the slope
of 3/2 is 1.5.

00:06:48.350 --> 00:06:52.265
So it's a slope that will look
something like-- let me get

00:06:52.265 --> 00:06:58.220
the line tool out-- so x might
look like, so plus 3/2 x would

00:06:58.220 --> 00:07:00.760
look something like that.

00:07:00.760 --> 00:07:05.800
Minus 3/2 x would be something
like that.

00:07:05.800 --> 00:07:12.520
Now we have to figure out does
this hyperbola go from there

00:07:12.520 --> 00:07:14.780
and there, or does it go
from here and here?

00:07:14.780 --> 00:07:21.950
There were just say, OK, when y
is equal to 0 we get x over

00:07:21.950 --> 00:07:24.200
2 is equal to plus or minus 1.

00:07:32.950 --> 00:07:35.325
You actually get x
would be there.

00:07:39.470 --> 00:07:41.970
So you know that this thing
has x-intercepts.

00:07:41.970 --> 00:07:43.740
You can set y is equal to 0.

00:07:43.740 --> 00:07:45.250
You can't set x is equal to 0.

00:07:45.250 --> 00:07:49.410
If you get x is equal to 0, then
you get minus y squared

00:07:49.410 --> 00:07:50.300
is equal to something.

00:07:50.300 --> 00:07:51.530
You get y squared is equal
to negative 1.

00:07:51.530 --> 00:07:53.090
You get some imaginary
solution.

00:07:53.090 --> 00:07:55.370
So you know that it
has x-intercepts.

00:07:55.370 --> 00:07:57.160
You know that these are
the asymptotes.

00:07:57.160 --> 00:07:58.750
So if you want to draw the
graph, it would look

00:07:58.750 --> 00:08:00.970
something like that.

00:08:00.970 --> 00:08:03.350
I know that's not a great
drawing, but hopefully it

00:08:03.350 --> 00:08:05.000
serves the purpose.

00:08:05.000 --> 00:08:06.710
Anyway, the choice was A.

00:08:06.710 --> 00:08:08.880
And hopefully you learned a
little bit about asymptotes.

00:08:08.880 --> 00:08:12.310
Next problem, 41.

00:08:12.310 --> 00:08:14.540
Which of the following
represents a hyperbola?

00:08:14.540 --> 00:08:15.940
Let me copy those.

00:08:15.940 --> 00:08:19.210
No, which of the following
represents a parabola, they

00:08:19.210 --> 00:08:20.460
want to know.

00:08:23.010 --> 00:08:23.990
That's too small.

00:08:23.990 --> 00:08:25.250
I don't think you
can see that.

00:08:25.250 --> 00:08:28.600
I'll get a bigger
version of it.

00:08:28.600 --> 00:08:29.975
All right, that's the
bigger version.

00:08:33.200 --> 00:08:34.260
There you go.

00:08:34.260 --> 00:08:36.870
Which of these represents
a parabola?

00:08:36.870 --> 00:08:38.710
Well, we could just
go one by one.

00:08:38.710 --> 00:08:40.980
This is a circle.

00:08:40.980 --> 00:08:42.230
That's a circle.

00:08:44.330 --> 00:08:45.230
Then you could graph
if you like.

00:08:45.230 --> 00:08:47.790
This is an ellipse, where you
just kind of weight the y's

00:08:47.790 --> 00:08:49.840
and the x squareds.

00:08:49.840 --> 00:08:50.470
This is an ellipse.

00:08:50.470 --> 00:08:55.250
If this was a negative, this
would be a hyperbola, ellipse.

00:08:55.250 --> 00:08:56.320
This one is a hyperbola
because you have

00:08:56.320 --> 00:08:57.570
this negative here.

00:09:00.170 --> 00:09:02.780
This one might not look like
a traditional parabola,

00:09:02.780 --> 00:09:03.950
but it really is.

00:09:03.950 --> 00:09:06.660
Because if I were to divide both
sides of this by 4p, I'd

00:09:06.660 --> 00:09:12.070
get x is equal to 1 over
4p times y squared.

00:09:12.070 --> 00:09:14.690
So it's a sideways parabola.

00:09:14.690 --> 00:09:15.860
It looks something like this.

00:09:15.860 --> 00:09:18.440
If that's the x- and y-axis,
the graph will look

00:09:18.440 --> 00:09:19.680
something like this.

00:09:19.680 --> 00:09:21.210
It's going to be a sideways
parabola.

00:09:21.210 --> 00:09:23.040
But it's a parabola
nonetheless.

00:09:23.040 --> 00:09:24.395
So the choice is C.

00:09:29.150 --> 00:09:33.020
We have time for at least
one more, problem 42.

00:09:33.020 --> 00:09:38.202
They give the following
equation.

00:09:38.202 --> 00:09:39.560
Let me paste it here.

00:09:42.530 --> 00:09:44.960
All right, and they say what
is the standard form of the

00:09:44.960 --> 00:09:48.040
equation of the conic
section above?

00:09:48.040 --> 00:09:50.530
And essentially they want us
to complete the square with

00:09:50.530 --> 00:09:53.590
the x's and the y's so we get
that x minus something

00:09:53.590 --> 00:09:58.300
squared, over some m plus or
minus y, minus something

00:09:58.300 --> 00:10:00.050
squared, over some constant.

00:10:00.050 --> 00:10:02.260
So let's do that.

00:10:02.260 --> 00:10:07.140
So let's group the x and the y
terms. So you get 4x squared,

00:10:07.140 --> 00:10:11.670
minus 16x, plus something.

00:10:11.670 --> 00:10:14.460
That's how we're going to
complete the square.

00:10:14.460 --> 00:10:19.350
Then you get plus minus
5y squared.

00:10:19.350 --> 00:10:23.470
Let me just minus, and I'll
do plus 5y squared.

00:10:23.470 --> 00:10:28.630
We have a minus here, so
it becomes plus 30y.

00:10:28.630 --> 00:10:36.680
And then out here you have
a minus 9 is equal to 0.

00:10:36.680 --> 00:10:39.380
Just to simplify it, let's
factor a 4 out of this part.

00:10:39.380 --> 00:10:49.260
So you get 4 times x squared,
minus 4x, plus something,

00:10:49.260 --> 00:10:55.340
minus-- let's factor a 5 out--
5, times y squared, plus 6y,

00:10:55.340 --> 00:10:58.550
probably plus something-- and
let's add the 9 to both

00:10:58.550 --> 00:11:02.030
sides-- is equal to 9.

00:11:02.030 --> 00:11:03.670
All right, so let's
think about it.

00:11:03.670 --> 00:11:06.920
If we wanted to complete the
square here, what do we put

00:11:06.920 --> 00:11:08.300
here to make this a
perfect square?

00:11:08.300 --> 00:11:09.740
If this was a plus 4,
then this would

00:11:09.740 --> 00:11:12.120
be x minus 2 squared.

00:11:12.120 --> 00:11:14.890
So we're adding a 4 here.

00:11:14.890 --> 00:11:17.000
But we're really adding
a 4 times a 4.

00:11:17.000 --> 00:11:19.140
We're really adding a 16
if you think about it.

00:11:19.140 --> 00:11:22.740
If I were to redistribute
this, I just added 16.

00:11:22.740 --> 00:11:25.150
So if I add 16 to one side of
the equation, I have to add 16

00:11:25.150 --> 00:11:26.260
to the other side
of the equation.

00:11:26.260 --> 00:11:27.970
So I'll add 16 there.

00:11:27.970 --> 00:11:30.890
To make this a perfect square,
let's see, 1/2 of 6 is 3, so 3

00:11:30.890 --> 00:11:33.490
squared is 9.

00:11:33.490 --> 00:11:35.290
But I'm really adding
a 5 times a 9.

00:11:35.290 --> 00:11:36.420
I could redistribute the 5.

00:11:36.420 --> 00:11:39.470
I really added a 45.

00:11:39.470 --> 00:11:40.000
Oh, no, no, no.

00:11:40.000 --> 00:11:42.610
I added a minus 5 times 9.

00:11:42.610 --> 00:11:43.610
I caught myself.

00:11:43.610 --> 00:11:45.220
I added a minus 5 times 9.

00:11:45.220 --> 00:11:47.210
So I'm really subtracting
45 from that side.

00:11:47.210 --> 00:11:50.530
So I have to subtract
45 from that side.

00:11:50.530 --> 00:11:56.900
And so this simplifies to 4
times x minus 2 squared--

00:11:56.900 --> 00:12:01.110
right, that's x minus two
squared-- minus 5 times-- this

00:12:01.110 --> 00:12:08.440
is y plus 3 squared-- is equal
to-- let's see, 9 plus 16 is

00:12:08.440 --> 00:12:14.600
25, 25 minus 45.

00:12:14.600 --> 00:12:16.670
Let's see, have I made
a mistake someplace?

00:12:16.670 --> 00:12:20.890
25 minus 45, no, that's right.

00:12:20.890 --> 00:12:26.000
25 minus 45 is minus 20.

00:12:26.000 --> 00:12:29.440
And now we can divide both sides
by minus 20, if I did

00:12:29.440 --> 00:12:31.750
this correctly.

00:12:31.750 --> 00:12:33.930
We'll divide both sides
by minus 20.

00:12:33.930 --> 00:12:38.310
So you get x minus 2 squared.

00:12:38.310 --> 00:12:40.920
What's 4 divided by minus 20?

00:12:40.920 --> 00:12:42.800
It's minus 1/5.

00:12:42.800 --> 00:12:43.840
Let me just do that.

00:12:43.840 --> 00:12:55.000
So 4 over minus 20, minus 5,
times y plus 3 squared, over

00:12:55.000 --> 00:12:57.910
minus 20 is equal to 1.

00:12:57.910 --> 00:13:01.410
So this is equal to, let's see
this becomes a positive, and

00:13:01.410 --> 00:13:03.060
this one becomes a negative.

00:13:03.060 --> 00:13:06.140
So let's put this one first. So
5 over 20 is the same thing

00:13:06.140 --> 00:13:11.900
as y plus 3 squared, over
4-- right, that's this

00:13:11.900 --> 00:13:14.490
one-- plus, no, sorry.

00:13:14.490 --> 00:13:15.190
We have a minus.

00:13:15.190 --> 00:13:21.430
Minus-- and this becomes 1/5--
x minus 2 squared, over 5.

00:13:21.430 --> 00:13:24.830
That is equal to 1.

00:13:24.830 --> 00:13:26.240
We haven't made a careless
mistake.

00:13:26.240 --> 00:13:28.100
That is choice B.

00:13:28.100 --> 00:13:30.040
See you in the the next video.


Algebra II: Quadratics and shifts

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.680
We're on problem 32.

00:00:04.680 --> 00:00:10.650
What are the solutions to the
equation 1 plus 1 over x

00:00:10.650 --> 00:00:15.080
squared is equal to 3 over x?

00:00:15.080 --> 00:00:17.310
So at first. This looks like
a pretty daunting equation.

00:00:17.310 --> 00:00:19.340
You have these x's in the
denominator and x squared in

00:00:19.340 --> 00:00:20.240
the denominator.

00:00:20.240 --> 00:00:23.020
But I think we can simplify it
if we can just get rid of

00:00:23.020 --> 00:00:24.250
these x squares in
the denominator.

00:00:24.250 --> 00:00:25.770
The easiest way to do
that is to multiply

00:00:25.770 --> 00:00:27.530
everything by x squared.

00:00:27.530 --> 00:00:30.050
So let's multiply both sides of
this equation by x squared.

00:00:30.050 --> 00:00:34.340
Then we'll get x squared times
1 is x squared, x squared

00:00:34.340 --> 00:00:38.780
times 1 over x squared,
that's just 1.

00:00:38.780 --> 00:00:42.890
Then x squared times 3 over x,
that's 3x squared over x.

00:00:42.890 --> 00:00:47.350
x squared divided by x is just
x, so that is equal to 3x.

00:00:47.350 --> 00:00:51.700
We can subtract 3x from both
sides and you get x squared

00:00:51.700 --> 00:00:55.450
minus 3x plus 1 is equal to 0.

00:00:55.450 --> 00:00:57.810
This is a simple quadratic.

00:00:57.810 --> 00:01:00.250
It's not obvious that
you can factor it.

00:01:00.250 --> 00:01:03.690
In fact, two numbers when you
multiply them equal 1, and

00:01:03.690 --> 00:01:07.040
then when you add them
equal minus 3.

00:01:07.040 --> 00:01:08.670
I'm guessing it might
even be imaginary.

00:01:08.670 --> 00:01:10.110
It will probably not be
imaginary, but it's just a

00:01:10.110 --> 00:01:10.900
strange number.

00:01:10.900 --> 00:01:12.230
So let's use the quadratic
equation.

00:01:12.230 --> 00:01:14.260
When in doubt, use the
quadratic equation.

00:01:14.260 --> 00:01:17.750
So minus B, this is B right?

00:01:17.750 --> 00:01:21.900
B is this 3 right there,
negative 3.

00:01:21.900 --> 00:01:22.690
B is negative 3.

00:01:22.690 --> 00:01:26.480
So minus B is going to be plus
3, plus or minus the square

00:01:26.480 --> 00:01:27.240
root of B squared.

00:01:27.240 --> 00:01:33.940
Minus 3 squared is 9, minus 4
times A, which is 1, times C

00:01:33.940 --> 00:01:34.480
which is 1.

00:01:34.480 --> 00:01:38.060
So it's minus 4, all
of that over 2A.

00:01:38.060 --> 00:01:40.740
A is 1 so it's just over 2.

00:01:40.740 --> 00:01:47.080
That is equal to 3/2 plus
or minus the square

00:01:47.080 --> 00:01:49.130
root of 5 over 2.

00:01:49.130 --> 00:01:50.990
I just separated these out
because I'm looking at the

00:01:50.990 --> 00:01:52.390
choices and it seems
like they did that.

00:01:52.390 --> 00:01:55.770
So you could've said that's 3/2
plus square root of 5 over

00:01:55.770 --> 00:02:00.390
2 or 3/2 minus the square
root of 5 over 2.

00:02:00.390 --> 00:02:01.860
I just did that because
it seems like that's

00:02:01.860 --> 00:02:03.250
how they write it.

00:02:03.250 --> 00:02:05.840
That is choice A.

00:02:05.840 --> 00:02:10.870
Next problem, 33.

00:02:10.870 --> 00:02:13.840
I think this one actually might
be good to copy and

00:02:13.840 --> 00:02:15.510
paste the problem.

00:02:15.510 --> 00:02:18.230
Let me see if I can do this.

00:02:18.230 --> 00:02:21.330
OK, there are two numbers with
the following properties.

00:02:21.330 --> 00:02:23.450
Let me write down the properties
and let me copy and

00:02:23.450 --> 00:02:25.720
paste it for you.

00:02:25.720 --> 00:02:27.410
OK, I've copied it.

00:02:27.410 --> 00:02:29.350
Let me go here.

00:02:29.350 --> 00:02:32.010
Then I've pasted it for you.

00:02:32.010 --> 00:02:33.260
All right.

00:02:36.950 --> 00:02:38.370
So there's two numbers with
the following properties.

00:02:38.370 --> 00:02:40.740
The second number is 3 more
than the first number.

00:02:40.740 --> 00:02:44.180
So let's say S for second number
and F for first number.

00:02:44.180 --> 00:02:48.490
So the second number is 3 more
than the first number.

00:02:48.490 --> 00:02:52.680
So the second number is equal
to the first number plus 3.

00:02:52.680 --> 00:02:54.570
That's from statement 1.

00:02:54.570 --> 00:02:57.950
The product of the two numbers
is 9 more than the sum.

00:02:57.950 --> 00:03:07.450
So the product of the two
numbers, that's S times F is 9

00:03:07.450 --> 00:03:15.290
more, 9 plus their sum,
plus S plus F.

00:03:15.290 --> 00:03:17.180
So let's see, we have two
equations and two unknowns.

00:03:17.180 --> 00:03:18.380
This is nonlinear because I'm

00:03:18.380 --> 00:03:20.000
multiplying these two variables.

00:03:20.000 --> 00:03:23.070
But I think we should be
able to solve them

00:03:23.070 --> 00:03:24.690
one way or the other.

00:03:24.690 --> 00:03:27.730
So let's see, we have
what S is equal to.

00:03:27.730 --> 00:03:31.440
So let's just substitute that
back into this equation.

00:03:31.440 --> 00:03:34.350
So let's say that S is
equal to F plus 3.

00:03:34.350 --> 00:03:41.340
So if we substitute for these
S's, we get F plus 3 times F

00:03:41.340 --> 00:03:46.990
is equal to 9 plus F plus 3,
right, instead of an S, F plus

00:03:46.990 --> 00:03:49.772
3 and then plus F.

00:03:49.772 --> 00:03:51.450
Let's see if we can
simplify this.

00:03:51.450 --> 00:03:58.460
F times F is F squared plus
3F is equal to 9 plus

00:03:58.460 --> 00:04:05.730
3 is 12 plus 2F.

00:04:05.730 --> 00:04:09.740
Subtract 2F from both sides,
you get F squared plus F is

00:04:09.740 --> 00:04:11.180
equal to 12.

00:04:11.180 --> 00:04:18.000
Subtract 12, you get F squared
plus F minus 12 is equal to 0.

00:04:18.000 --> 00:04:19.380
This one looks factorable.

00:04:19.380 --> 00:04:21.510
I don't have to take out
the quadratic equation.

00:04:21.510 --> 00:04:30.170
Let's see, this is F plus 4
times F minus 3, right?

00:04:30.170 --> 00:04:31.510
Because when you multiply those,
you get negative 12.

00:04:31.510 --> 00:04:35.400
When you add those, you get plus
1, so that is equal to 0.

00:04:35.400 --> 00:04:38.420
So in order for this to be true,
one or both of these

00:04:38.420 --> 00:04:39.500
have to be equal to 0.

00:04:39.500 --> 00:04:45.870
So if F plus 4 is equal to
0, that means F could be

00:04:45.870 --> 00:04:47.520
equal to minus 4.

00:04:47.520 --> 00:04:50.500
If F minus 3 is equal
to 0, then that says

00:04:50.500 --> 00:04:51.690
that F could be 3.

00:04:51.690 --> 00:04:54.810
So F could be minus 4 or 3.

00:04:54.810 --> 00:04:59.710
Now, S is F plus 3, so if we're
dealing with the minus 4

00:04:59.710 --> 00:05:03.730
scenario, if F is equal to
minus 4, then what is S?

00:05:03.730 --> 00:05:07.940
Then S is going to be minus 4
plus 3, and S is going to be

00:05:07.940 --> 00:05:09.390
equal to minus 1.

00:05:09.390 --> 00:05:13.090
Then if F is equal to 3,
than S is equal to 6.

00:05:13.090 --> 00:05:16.010
So let's see if we see either
of these combinations.

00:05:16.010 --> 00:05:19.900
Minus 4, minus 1,
that's choice B.

00:05:19.900 --> 00:05:21.040
Excellent.

00:05:21.040 --> 00:05:27.890
All right, problem 34.

00:05:27.890 --> 00:05:31.400
Let me see, maybe I should
copy and paste these word

00:05:31.400 --> 00:05:35.460
problems so we can see how we
parse the problems. So I've

00:05:35.460 --> 00:05:41.320
copied it, then I'll go here,
and then pasted it.

00:05:41.320 --> 00:05:45.020
Jenny is solving the equation x
squared minus 8x equals 9 by

00:05:45.020 --> 00:05:46.470
completing the square.

00:05:46.470 --> 00:05:48.910
What number should be added to
both sides of the equation to

00:05:48.910 --> 00:05:49.660
complete the square?

00:05:49.660 --> 00:05:55.280
So x squared minus
8x is equal to 9.

00:05:55.280 --> 00:05:57.600
I wrote it with space
for a reason.

00:05:57.600 --> 00:05:59.420
When you're completing the
square, you're trying to turn

00:05:59.420 --> 00:06:02.700
the left-hand side of this
equation into some type of a

00:06:02.700 --> 00:06:04.650
perfect square.

00:06:04.650 --> 00:06:07.590
So if it's a perfect square,
I have two numbers.

00:06:07.590 --> 00:06:11.880
It's the same number that when
you add them together, you get

00:06:11.880 --> 00:06:14.810
minus 8, and when you square
them, you should get something

00:06:14.810 --> 00:06:15.470
else, right?

00:06:15.470 --> 00:06:17.410
So what's half of minus 8?

00:06:17.410 --> 00:06:21.270
Half of minus 8 is minus 4.

00:06:21.270 --> 00:06:24.430
So minus 4 squared is 16.

00:06:24.430 --> 00:06:27.830
So if I add 16 to both
sides, I'm all set.

00:06:27.830 --> 00:06:28.470
Why did that work?

00:06:28.470 --> 00:06:29.460
Well, now it's a
perfect square.

00:06:29.460 --> 00:06:34.940
This is now x minus 4 squared
is equal to 9 plus 16 is 25.

00:06:34.940 --> 00:06:36.400
They're not even asking
us to solve it.

00:06:36.400 --> 00:06:39.070
They just want to know what we
had to add to both sides.

00:06:39.070 --> 00:06:40.990
So it's 16, D.

00:06:40.990 --> 00:06:43.310
Remember the whole logic here,
and I've done a few videos on

00:06:43.310 --> 00:06:46.030
completing the squares, is what
number do I add here to

00:06:46.030 --> 00:06:47.980
make this a perfect square?

00:06:47.980 --> 00:06:51.470
You say, OK, I have a minus
8x, so I take half of this

00:06:51.470 --> 00:06:54.540
number, because the same number
added to itself twice

00:06:54.540 --> 00:06:56.590
is going to become minus 8.

00:06:56.590 --> 00:06:58.375
I take half of that number,
then I squared it.

00:06:58.375 --> 00:07:00.120
So half of minus 8 is minus 4.

00:07:00.120 --> 00:07:01.990
If you square it,
you get the 16.

00:07:01.990 --> 00:07:03.420
So add 16 to both sides,
you get this.

00:07:03.420 --> 00:07:04.420
You can actually
solve for this.

00:07:04.420 --> 00:07:06.070
x minus 4 is plus or minus 5.

00:07:06.070 --> 00:07:07.110
You keep going.

00:07:07.110 --> 00:07:10.070
That's actually where the
quadratic equation comes from.

00:07:10.070 --> 00:07:13.140
Anyway, next problem.

00:07:13.140 --> 00:07:20.980
16 was choice number D.

00:07:20.980 --> 00:07:25.950
I'm going to copy and paste
this entire problem here.

00:07:25.950 --> 00:07:29.810
Let's go here and
paste it here.

00:07:29.810 --> 00:07:32.660
OK, which of the following most
accurately describes the

00:07:32.660 --> 00:07:36.810
translation of the graph y is
equal to x plus 3 squared

00:07:36.810 --> 00:07:41.555
minus 2 to the graph y equals
x minus 2 squared plus 2?

00:07:44.070 --> 00:07:50.090
So the y translation tends to be
pretty easy to figure out.

00:07:50.090 --> 00:07:52.610
Let me just draw some
example graph.

00:07:55.650 --> 00:07:59.190
So if I had the graph x squared,
the graph x squared

00:07:59.190 --> 00:08:00.540
looks something like this.

00:08:00.540 --> 00:08:01.840
Let's see if I can draw it.

00:08:01.840 --> 00:08:04.980
The graph x squared looks
something like this, right?

00:08:04.980 --> 00:08:05.790
And it intersects.

00:08:05.790 --> 00:08:08.330
When x is equal to 0, we're
at our minimum point.

00:08:08.330 --> 00:08:11.210
And any other value increases
in both directions.

00:08:11.210 --> 00:08:16.150
The graph of x squared plus
2, you're shifting up.

00:08:16.150 --> 00:08:17.870
This is the graph of
x squared plus 2.

00:08:17.870 --> 00:08:20.030
You would shift it up by 2.

00:08:20.030 --> 00:08:24.420
The graph of x squared minus 2,
you would shift down by 2.

00:08:24.420 --> 00:08:27.210
This would be x squared
plus 2.

00:08:27.210 --> 00:08:30.500
This would be x squared
minus 2.

00:08:30.500 --> 00:08:34.110
So the shift in the y direction
is very easy to see.

00:08:34.110 --> 00:08:44.760
So if we're going from something
minus 2 to plus 2

00:08:44.760 --> 00:08:48.110
we're going to be shifting
it up 4, right?

00:08:48.110 --> 00:08:49.710
So that's always the
easy one to just

00:08:49.710 --> 00:08:51.120
eyeball and figure out.

00:08:51.120 --> 00:08:53.920
So we're definitely going to be
shifting from minus 2 to 2.

00:08:53.920 --> 00:08:54.860
So it's up 4.

00:08:54.860 --> 00:08:58.520
It's either going to be
choice A or choice D.

00:08:58.520 --> 00:09:02.270
The left/right shift is often
a little bit more hard for

00:09:02.270 --> 00:09:06.400
people to visualize or to at
least internalize, but I'll

00:09:06.400 --> 00:09:09.240
give you an attempt.

00:09:09.240 --> 00:09:10.100
Let's just go back to this.

00:09:10.100 --> 00:09:12.300
This is the graph of x squared,
this yellow line

00:09:12.300 --> 00:09:14.920
right there.

00:09:14.920 --> 00:09:16.280
That's the graph of x squared.

00:09:16.280 --> 00:09:17.120
Let me ask you a question.

00:09:17.120 --> 00:09:23.820
What is the graph of
x minus 3 squared?

00:09:23.820 --> 00:09:28.040
So does this shift it down 3 to
the negative direction or 3

00:09:28.040 --> 00:09:28.660
to the positive direction?

00:09:28.660 --> 00:09:31.290
Your intuition might say,
oh, I'm subtracting 3.

00:09:31.290 --> 00:09:33.130
When I did minus 2,
I shifted down.

00:09:33.130 --> 00:09:34.840
But it's actually the
opposite here.

00:09:34.840 --> 00:09:37.230
Because you have to think about
for what value of x am I

00:09:37.230 --> 00:09:40.540
going to have a 0
squared here?

00:09:40.540 --> 00:09:43.100
That happens with
x is equal to 3.

00:09:43.100 --> 00:09:44.200
So you can think
of it this way.

00:09:44.200 --> 00:09:48.190
Now, when we're at this point,
when x is equal to 3, it's the

00:09:48.190 --> 00:09:51.650
same thing as this point when
we have just x squared.

00:09:51.650 --> 00:09:55.480
When you put 3 in here, this
whole expression becomes zero.

00:09:55.480 --> 00:09:58.610
As you get above 3, that's
like going above zero.

00:09:58.610 --> 00:10:01.370
As you go below 3, that's
like going below zero.

00:10:01.370 --> 00:10:07.090
So this graph will just get
shifted to the right by 3.

00:10:07.090 --> 00:10:10.242
That's x minus 3 shifts
to the right by 3.

00:10:10.242 --> 00:10:14.410
x plus three would go in the
other direction, because when

00:10:14.410 --> 00:10:16.870
x is minus 3, that's when
it would equal to zero.

00:10:16.870 --> 00:10:17.580
I haven't written that down.

00:10:17.580 --> 00:10:18.400
So let's think about this.

00:10:18.400 --> 00:10:25.640
We're going from x plus 3, so if
this is x squared, x plus 3

00:10:25.640 --> 00:10:28.100
would look something-- let me do
it in a different color. x

00:10:28.100 --> 00:10:30.410
plus 3 is actually shifted
to the left.

00:10:30.410 --> 00:10:31.630
The way I always think about
it, there's two ways

00:10:31.630 --> 00:10:32.990
to think about it.

00:10:32.990 --> 00:10:36.390
The y shift is intuitive and
the x shift might not be.

00:10:36.390 --> 00:10:38.510
If you have a plus 3 here,
you're actually shifting in

00:10:38.510 --> 00:10:40.070
the downward direction.

00:10:40.070 --> 00:10:43.430
The way to actually think about
the intuition is when

00:10:43.430 --> 00:10:45.570
will this whole expression
equal 0?

00:10:45.570 --> 00:10:49.740
This whole expression equals 0
when x is equal to minus 3.

00:10:49.740 --> 00:10:53.630
So that's the point at which
you're getting 0 squared.

00:10:53.630 --> 00:10:55.080
When I'm drawing these
graphs, I'm not

00:10:55.080 --> 00:10:56.770
doing the y shift here.

00:10:56.770 --> 00:11:00.350
So this is going to be shifted
to the left 3.

00:11:00.350 --> 00:11:03.250
This is going to be shifted
to the right by 2.

00:11:03.250 --> 00:11:05.810
So if this is shifted to the
left 3 and this is shifted to

00:11:05.810 --> 00:11:08.700
the right by 2, to go from this
to this, you're shifting

00:11:08.700 --> 00:11:09.950
to the right by 5.

00:11:14.180 --> 00:11:17.000
So the actual graph x plus
3 squared minus 2

00:11:17.000 --> 00:11:20.500
is going to be here.

00:11:20.500 --> 00:11:23.710
Then to go here, you have a plus
2, so you're shifting the

00:11:23.710 --> 00:11:27.970
graph up by 4, and then you're
going to x minus 2.

00:11:27.970 --> 00:11:31.100
So this graph right here
is going to be up here.

00:11:31.100 --> 00:11:35.550
So you're shifting up by 4 and
then you're shifting to the

00:11:35.550 --> 00:11:38.640
right by 5.

00:11:38.640 --> 00:11:41.180
Actually, even if you're
confused with your shifting

00:11:41.180 --> 00:11:42.740
left or right, you can just
say the difference between

00:11:42.740 --> 00:11:45.110
plus 3 and minus 2 is 5,
and 5 is only there.

00:11:45.110 --> 00:11:47.440
But you should hopefully
understand the problems a

00:11:47.440 --> 00:11:48.800
little bit deeper than that.

00:11:48.800 --> 00:11:51.400
Anyway, I'll see you
in the next video.


Examples: Graphing and interpreting quadratics

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:02.200
We're on problem 36.

00:00:02.200 --> 00:00:04.130
It says which of the following
sentences is true about the

00:00:04.130 --> 00:00:08.230
graphs of y is equal to 3 times
x minus 5 squared plus 1

00:00:08.230 --> 00:00:12.270
and y equals 3 times x plus
5 squared plus 1?

00:00:12.270 --> 00:00:15.120
So let's do something very
similar to what we did in the

00:00:15.120 --> 00:00:18.430
past. If you think about it,
both of these equations, y is

00:00:18.430 --> 00:00:20.090
going to be 1 or greater.

00:00:20.090 --> 00:00:23.110
Let's just analyze this
a little bit.

00:00:23.110 --> 00:00:28.140
This term right here, since
we're squaring, is always

00:00:28.140 --> 00:00:30.170
going to be positive.

00:00:30.170 --> 00:00:33.470
Even if what's inside the
parentheses becomes negative,

00:00:33.470 --> 00:00:36.030
if we have x is minus 10, inside
the parenthesis becomes

00:00:36.030 --> 00:00:38.180
negative, but when
you square it, it

00:00:38.180 --> 00:00:38.870
always becomes positive.

00:00:38.870 --> 00:00:40.810
You're going to multiply 3 times
a positive number, so

00:00:40.810 --> 00:00:42.250
you're going to get
a positive number.

00:00:42.250 --> 00:00:44.560
So the lowest value that
this could be is 0.

00:00:44.560 --> 00:00:48.420
The lowest value that y could
be is actually 1.

00:00:48.420 --> 00:00:54.470
The same thing here, this
number can become very

00:00:54.470 --> 00:00:55.790
negative, but when you
square it, it's

00:00:55.790 --> 00:00:57.070
going to become positive.

00:00:57.070 --> 00:01:00.250
So this expression with the
squared here is going to be

00:01:00.250 --> 00:01:01.320
positive, and you multiply
it by 3, and

00:01:01.320 --> 00:01:02.430
it's going to be positive.

00:01:02.430 --> 00:01:05.126
So the lowest value here is
always going to be zero when

00:01:05.126 --> 00:01:08.210
you include this whole term.

00:01:08.210 --> 00:01:10.160
So similarly, the lowest
value y can be is 1.

00:01:10.160 --> 00:01:11.760
I just want to think about it
a little bit just give you a

00:01:11.760 --> 00:01:13.380
little bit of an intuition.

00:01:13.380 --> 00:01:15.750
Let's think of this in the
context of what we learned

00:01:15.750 --> 00:01:17.282
last time with the shifting.

00:01:17.282 --> 00:01:21.390
So let me draw it in a color
that you can see.

00:01:21.390 --> 00:01:26.220
So if that is the y-axis, and
I'll just draw mainly the

00:01:26.220 --> 00:01:34.240
positive area, so if I were to
just draw y is equal to x

00:01:34.240 --> 00:01:39.100
squared plus 1, it would look
like this where this is 1.

00:01:39.100 --> 00:01:40.120
That's y is equal to 1.

00:01:40.120 --> 00:01:43.330
The graph would look something
along this.

00:01:43.330 --> 00:01:45.600
y is equal to x squared
plus 1.

00:01:45.600 --> 00:01:47.030
That's a horrible drawing.

00:01:47.030 --> 00:01:49.790
Normally, I wouldn't redo it,
but that was just atrocious.

00:01:49.790 --> 00:01:52.070
y is equal to x squared plus 1
looks something like that.

00:01:52.070 --> 00:01:53.540
It's symmetric.

00:01:53.540 --> 00:01:54.180
You get the idea.

00:01:54.180 --> 00:01:55.890
You've seen these parabolas
before.

00:01:55.890 --> 00:01:57.970
This is y is equal to
x squared plus 1.

00:02:00.720 --> 00:02:07.720
Now, if we were to do x minus
5 squared plus 1,

00:02:07.720 --> 00:02:08.520
what happens to it?

00:02:08.520 --> 00:02:09.190
Well, let me think about it.

00:02:09.190 --> 00:02:10.910
What is 3x squared plus 1?

00:02:10.910 --> 00:02:12.940
Well, then it just increases
a little bit faster.

00:02:12.940 --> 00:02:15.340
So if I were say y equals 3x
squared plus 1, it might look

00:02:15.340 --> 00:02:16.860
something like this.

00:02:16.860 --> 00:02:19.670
It'll just increase a little
bit faster, three times as

00:02:19.670 --> 00:02:21.550
fast actually.

00:02:21.550 --> 00:02:24.630
So that would be 3x
squared plus 1.

00:02:24.630 --> 00:02:26.590
The rate of increase in both
directions just goes faster

00:02:26.590 --> 00:02:29.680
because you have this constant
term 3 out there multiplying

00:02:29.680 --> 00:02:30.140
the numbers.

00:02:30.140 --> 00:02:33.730
OK, now what happens
when you shift it?

00:02:33.730 --> 00:02:35.980
So let's do x minus 5.

00:02:35.980 --> 00:02:40.830
So where x equals 0 was the
minimum point before, now if

00:02:40.830 --> 00:02:44.490
we substitute a 5 here, that'll
be our minimum point.

00:02:44.490 --> 00:02:47.470
Because then that whole
term becomes zero.

00:02:47.470 --> 00:02:50.785
So this vertex will now be
shifted to the right.

00:02:50.785 --> 00:02:53.370
Let me do it in another color.

00:02:53.370 --> 00:02:56.560
So if this is the point 5, now
this would be the graph.

00:02:56.560 --> 00:02:58.870
If you just took this graph and
you shifted it over to the

00:02:58.870 --> 00:03:04.240
right by 5-- I won't draw the
whole thing-- that graph right

00:03:04.240 --> 00:03:10.230
there would be 3 times x
minus 5 squared plus 1.

00:03:10.230 --> 00:03:12.990
Remember, the y shift
is always intuitive.

00:03:12.990 --> 00:03:14.400
If you add 1, you're
shifting it up.

00:03:14.400 --> 00:03:16.250
If you subtract 1, you're
shifting it down.

00:03:16.250 --> 00:03:17.100
The x shift isn't.

00:03:17.100 --> 00:03:19.800
We subtracted 5, x minus 5.

00:03:19.800 --> 00:03:23.080
We replaced x with x minus 5,
but we shifted to the right.

00:03:23.080 --> 00:03:25.900
The intuition is there, because
now plus 5 makes this

00:03:25.900 --> 00:03:27.980
expression zero.

00:03:27.980 --> 00:03:30.590
So that's 3x minus 5 squared.

00:03:30.590 --> 00:03:34.430
In the same logic, 3 times x
plus 5 squared is going to be

00:03:34.430 --> 00:03:37.830
to here, plus 1.

00:03:37.830 --> 00:03:40.720
That's going to be shifted
to-- let me pick a good

00:03:40.720 --> 00:03:43.740
color-- to the left.

00:03:43.740 --> 00:03:44.840
This is going to look
something like this.

00:03:44.840 --> 00:03:46.890
It's going to be this blue graph
shifted to the left.

00:03:46.890 --> 00:03:48.370
This is minus 5.

00:03:48.370 --> 00:03:53.670
So this is the graph right
here of 3 times x plus 5

00:03:53.670 --> 00:03:55.570
squared plus 1

00:03:55.570 --> 00:03:56.810
Now, hopefully, you
have an intuition.

00:03:56.810 --> 00:03:59.190
So let's read their statements
and see which one makes sense.

00:03:59.190 --> 00:04:00.580
Which of the following
is true?

00:04:00.580 --> 00:04:02.500
Their vertices are maximums.
No, that's not

00:04:02.500 --> 00:04:03.545
true of any of these.

00:04:03.545 --> 00:04:06.610
Because the vertices is that
point right there.

00:04:06.610 --> 00:04:09.120
It's actually the
minimum point.

00:04:09.120 --> 00:04:11.900
A maximum point would look
something like that.

00:04:11.900 --> 00:04:13.540
We know that, because you
just go positive.

00:04:13.540 --> 00:04:15.510
This term can only
be positive.

00:04:15.510 --> 00:04:18.399
If this was a negative 3, then
it would flip it over.

00:04:18.399 --> 00:04:20.220
So it's not choice A.

00:04:20.220 --> 00:04:22.820
The graphs have the same shape
with different vertices.

00:04:22.820 --> 00:04:27.330
Yeah, both of these graphs have
the shape of 3x squared,

00:04:27.330 --> 00:04:30.370
but 1 vertices is 10 to the
left of the other one.

00:04:30.370 --> 00:04:31.660
So I think B is our choice.

00:04:31.660 --> 00:04:32.860
Let's read the other ones.

00:04:32.860 --> 00:04:33.960
The graphs have different
shapes

00:04:33.960 --> 00:04:34.610
with different vertices.

00:04:34.610 --> 00:04:35.880
No, they have the same shape.

00:04:35.880 --> 00:04:37.100
They definitely have
the same shape.

00:04:37.100 --> 00:04:40.040
They both have this
3x squared shape.

00:04:40.040 --> 00:04:42.580
One graph has a vertex that is
a maximum, while the other

00:04:42.580 --> 00:04:43.570
has-- no, that's not right.

00:04:43.570 --> 00:04:45.410
They both are upward facing,
so they both

00:04:45.410 --> 00:04:46.270
have minimum points.

00:04:46.270 --> 00:04:49.710
So it's choice B.

00:04:49.710 --> 00:04:59.420
Next problem, problem 37.

00:04:59.420 --> 00:05:02.530
Let me see what it says.

00:05:02.530 --> 00:05:03.965
What are the x-intercepts?

00:05:03.965 --> 00:05:05.580
Let me copy and paste that.

00:05:08.620 --> 00:05:12.530
OK, I'll paste it there.

00:05:12.530 --> 00:05:15.080
What are the x-intercepts
of the graph of that?

00:05:15.080 --> 00:05:17.190
Well, the x-intercepts, whatever
this graph looks

00:05:17.190 --> 00:05:18.850
like, I don't know exactly
what it looks like.

00:05:22.730 --> 00:05:26.470
This graph is going to look
something like this.

00:05:26.470 --> 00:05:27.800
I actually have no idea
what it looks like

00:05:27.800 --> 00:05:28.540
until I solve it.

00:05:28.540 --> 00:05:29.550
It's going to look something
like this.

00:05:29.550 --> 00:05:31.820
When they say x-intercepts,
they're like, where does it

00:05:31.820 --> 00:05:33.120
intersect the x-axis?

00:05:33.120 --> 00:05:35.080
So that's like there
and there.

00:05:35.080 --> 00:05:38.180
I don't know if those are the
actual points, right?

00:05:38.180 --> 00:05:41.680
To do that, we set the function
equal to zero,

00:05:41.680 --> 00:05:43.630
because this is the point
y is equal to 0.

00:05:43.630 --> 00:05:47.440
You're essentially saying when
does this function equal zero

00:05:47.440 --> 00:05:50.230
because that's the x-axis
when y is equal to 0.

00:05:50.230 --> 00:05:55.080
So you set y is equal to 0, and
you get 0 is equal to 12x

00:05:55.080 --> 00:06:00.030
squared minus 5x minus 2.

00:06:00.030 --> 00:06:03.260
Whenever I have a coefficient
larger than 1 in front of the

00:06:03.260 --> 00:06:06.540
x squared term, I find that very
hard to just eyeball and

00:06:06.540 --> 00:06:08.920
factor, so I use the
quadratic equation.

00:06:08.920 --> 00:06:12.200
So negative B, this is the B.

00:06:12.200 --> 00:06:13.650
B is minus 5.

00:06:13.650 --> 00:06:16.310
So negative negative
5 is plus 5.

00:06:16.310 --> 00:06:20.720
Negative B plus or minus the
square root of B squared,

00:06:20.720 --> 00:06:27.530
negative 5 squared is 25, minus
4 times A, which is 12,

00:06:27.530 --> 00:06:29.775
times C, which is minus 2.

00:06:32.440 --> 00:06:34.730
So let's just make that
times plus 2 and put

00:06:34.730 --> 00:06:35.600
the plus out there.

00:06:35.600 --> 00:06:37.370
A minus times a minus
is a plus.

00:06:37.370 --> 00:06:42.650
All of that over 2A, all of
that over 24, 2 times A.

00:06:42.650 --> 00:06:48.440
So that is equal to 5 plus or
minus the square root-- let's

00:06:48.440 --> 00:06:58.130
see, it was 25 plus 4
times 12 times 2.

00:06:58.130 --> 00:07:00.080
Because that was a minus 2,
but we had a minus there

00:07:00.080 --> 00:07:05.920
before, so 8 times 12, so
96, all of that over 24.

00:07:05.920 --> 00:07:07.830
What's 25 plus 96?

00:07:07.830 --> 00:07:13.490
It's 121, which is 11 squared.

00:07:13.490 --> 00:07:21.820
So this becomes 5 plus
or minus 11 over 24.

00:07:21.820 --> 00:07:26.250
Remember, these are the x-values
where that original

00:07:26.250 --> 00:07:28.050
function will equal zero.

00:07:28.050 --> 00:07:29.050
It's always important
to remember

00:07:29.050 --> 00:07:31.320
what we're even doing.

00:07:31.320 --> 00:07:36.750
So let's see, if x is equal to
5 plus 11 over 24, that is

00:07:36.750 --> 00:07:41.890
equal to 16/24, which
is equal to 2/3.

00:07:41.890 --> 00:07:43.850
That's one potential
intercept.

00:07:43.850 --> 00:07:46.610
So maybe that's right here.

00:07:46.610 --> 00:07:50.500
That's x is equal to 2/3
and y is equal to 0.

00:07:50.500 --> 00:07:55.460
The other value is x is equal
to 5 minus 11 over 24.

00:07:55.460 --> 00:08:00.860
That's minus 6/24, which is
equal to minus 1/4, which

00:08:00.860 --> 00:08:01.650
could be this point.

00:08:01.650 --> 00:08:03.940
I actually drew the graph
not that far off of

00:08:03.940 --> 00:08:04.470
what it could be.

00:08:04.470 --> 00:08:06.480
So this would be x is
equal to minus 1/4.

00:08:06.480 --> 00:08:09.370
Those are the x-intercepts
of that graph.

00:08:09.370 --> 00:08:17.240
So 2/3 and minus 1/4 is
choice C on the test.

00:08:17.240 --> 00:08:18.700
We have time for at
least one more.

00:08:22.290 --> 00:08:24.900
Oh boy, they drew us all
these this graphs.

00:08:24.900 --> 00:08:26.250
Let me shrink it.

00:08:29.790 --> 00:08:32.030
I want to be able to
fit all the graphs.

00:08:32.030 --> 00:08:35.919
So let me copy and paste
their graphs.

00:08:35.919 --> 00:08:40.289
So this is one where the
clipboard is definitely going

00:08:40.289 --> 00:08:41.539
to come in useful.

00:08:52.450 --> 00:08:53.700
OK that's good enough.

00:09:02.030 --> 00:09:05.400
I've never done something
this graphical.

00:09:05.400 --> 00:09:10.740
So the graph they say is y is
equal to minus 2 times x minus

00:09:10.740 --> 00:09:13.640
1 squared plus 1.

00:09:13.640 --> 00:09:15.380
So that's what we have
to find the graph of.

00:09:18.670 --> 00:09:21.220
So immediately when you look
at it, you say, OK, this is

00:09:21.220 --> 00:09:26.640
like the same thing as y is
equal to minus 2x squared plus

00:09:26.640 --> 00:09:28.970
1, but they shifted the x.

00:09:28.970 --> 00:09:31.780
They shifted the x to
the right by 1.

00:09:31.780 --> 00:09:33.300
I know it says a minus 1,
but think about it.

00:09:33.300 --> 00:09:38.470
When x is equal to positive
1, this is equal to 0.

00:09:38.470 --> 00:09:43.080
So it's going to be shifted
to the right by 1, plus 1.

00:09:43.080 --> 00:09:43.870
We know that.

00:09:43.870 --> 00:09:50.600
We know that it's going to be
shifted up by 1, so up plus 1.

00:09:50.600 --> 00:09:51.610
Then we have to think
is it going to be

00:09:51.610 --> 00:09:53.110
opening upwards or downwards?

00:09:53.110 --> 00:09:57.970
Think of it this way: If this
was y is equal to 2x squared

00:09:57.970 --> 00:10:00.830
plus 1, then this term would
always be positive.

00:10:00.830 --> 00:10:02.780
It'll just become more and
more positive as you get

00:10:02.780 --> 00:10:05.680
further and further away from
zero, so it would open up.

00:10:05.680 --> 00:10:07.760
But if you put a negative number
there, if you say y is

00:10:07.760 --> 00:10:10.630
equal to minus 2x squared
plus 1, then you're

00:10:10.630 --> 00:10:11.470
going to open downward.

00:10:11.470 --> 00:10:13.760
You're just going to get more
and more negative as you get

00:10:13.760 --> 00:10:16.280
away from your vertex.

00:10:16.280 --> 00:10:20.565
So we're shifted to the right
by 1, we're shifted up by 1,

00:10:20.565 --> 00:10:22.640
and we're going to be
opening downwards.

00:10:22.640 --> 00:10:25.050
So if we look at our
choices, only these

00:10:25.050 --> 00:10:26.690
two are opening downwards.

00:10:29.410 --> 00:10:32.540
Both of them are shifted
up by 1.

00:10:32.540 --> 00:10:36.520
Their vertex is at
y is equal to 1.

00:10:36.520 --> 00:10:38.740
But this is shifted 1 to
the right and this is

00:10:38.740 --> 00:10:40.380
shifted 1 to the left.

00:10:40.380 --> 00:10:43.350
Remember, we said it was
x minus 1 squared.

00:10:43.350 --> 00:10:45.370
So the vertex happens
when this whole

00:10:45.370 --> 00:10:48.070
expression is equal to zero.

00:10:48.070 --> 00:10:52.370
This whole expression is equal
to zero when x is equal to

00:10:52.370 --> 00:10:55.370
positive 1.

00:10:55.370 --> 00:10:57.340
So that's right here.

00:10:57.340 --> 00:10:59.555
So it's actually choice C.

00:11:02.580 --> 00:11:04.710
When your shifting graphs, that
can be one of the hardest

00:11:04.710 --> 00:11:05.430
things to ingrain.

00:11:05.430 --> 00:11:09.760
But I just really encourage
you to explore graphs,

00:11:09.760 --> 00:11:11.960
practice with graphs with your
graphing calculator and really

00:11:11.960 --> 00:11:15.590
try to plot points and try to
get a really good grasp of why

00:11:15.590 --> 00:11:19.190
when you go from minus 2x
squared plus 1 to minus 2

00:11:19.190 --> 00:11:22.830
times x minus 1 squared, why
when you replace an x with an

00:11:22.830 --> 00:11:26.870
minus 1, why this shifts the
graph to the right by 1.

00:11:26.870 --> 00:11:29.470
Anyway, I'll see you
in the next video.


Algebra II: Imaginary and complex numbers

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.560
We're on problem 21.

00:00:02.560 --> 00:00:06.310
Which product is equivalent
to 4x squared minus 16

00:00:06.310 --> 00:00:08.510
over 2 minus x?

00:00:08.510 --> 00:00:10.780
So let's see if we can simplify
this a little bit.

00:00:10.780 --> 00:00:14.300
4x squared minus 16, if we were
to factor out a 4, that's

00:00:14.300 --> 00:00:21.980
the same thing as 4 times
x squared minus 4.

00:00:21.980 --> 00:00:25.880
That's 4x squared minus 16, all
of that over 2 minus x.

00:00:25.880 --> 00:00:28.630
Then x squared minus 4, that's
that same pattern that keeps

00:00:28.630 --> 00:00:31.550
showing up, a squared minus
b squared, right?

00:00:31.550 --> 00:00:37.750
So that numerator becomes 4
times x plus 2 times x minus

00:00:37.750 --> 00:00:41.280
2, right? a plus b
times a minus b.

00:00:41.280 --> 00:00:43.570
Then the denominator, let's
think about it.

00:00:43.570 --> 00:00:47.620
This has 2 minus x, and we
have an x minus 2 here.

00:00:47.620 --> 00:00:52.300
But what if we were to factor
out a negative 1 from this?

00:00:52.300 --> 00:00:53.470
What happens?

00:00:53.470 --> 00:00:57.930
If you factor out a negative 1,
then the negative x becomes

00:00:57.930 --> 00:01:02.130
a positive x, and the positive
2 becomes a minus 2.

00:01:02.130 --> 00:01:05.920
I did that just to show you that
a negative, if you divide

00:01:05.920 --> 00:01:08.430
this by or multiply by
a negative 1, you

00:01:08.430 --> 00:01:09.320
just switch the order.

00:01:09.320 --> 00:01:12.070
2 minus x becomes x minus 2,
which is a good thing to

00:01:12.070 --> 00:01:15.110
recognize because we have an x
minus 2 in the numerator, too.

00:01:15.110 --> 00:01:19.150
So this will cancel
out with that.

00:01:19.150 --> 00:01:25.800
If we assume that x does not
ever equal 2 because x

00:01:25.800 --> 00:01:29.200
equaling 2 would make this an
undefined statement because

00:01:29.200 --> 00:01:32.000
you can't divide by zero, but if
we make that assumption, we

00:01:32.000 --> 00:01:33.060
can cancel those out.

00:01:33.060 --> 00:01:37.360
Then we have 4 divided by
negative 1 is minus 4

00:01:37.360 --> 00:01:40.270
times x plus 2.

00:01:40.270 --> 00:01:44.480
That is choice D.

00:01:44.480 --> 00:01:47.960
Next question, 22.

00:01:47.960 --> 00:01:49.220
I like this cutting and pasting

00:01:49.220 --> 00:01:51.260
of the actual question.

00:01:51.260 --> 00:01:52.510
It makes it fun.

00:01:56.610 --> 00:01:58.850
I don't want it to slow down
my computer too much.

00:01:58.850 --> 00:02:03.950
I sound like I'm melting
when that happens.

00:02:03.950 --> 00:02:05.030
So that's our problem.

00:02:05.030 --> 00:02:07.560
I didn't write the
choices down.

00:02:07.560 --> 00:02:09.400
All right, let's see
what we can do.

00:02:09.400 --> 00:02:12.390
So they have x squared
plus 4x.

00:02:12.390 --> 00:02:14.250
Let me write this.

00:02:14.250 --> 00:02:23.590
x squared plus 4x over x plus 3
times x squared minus 9, all

00:02:23.590 --> 00:02:26.940
of that over x squared
plus x minus 12.

00:02:26.940 --> 00:02:29.990
You might say, oh, my God, to
multiply these using algebraic

00:02:29.990 --> 00:02:31.890
multiplication would take
forever, then I'd have to

00:02:31.890 --> 00:02:33.810
factor it and simplify it.

00:02:33.810 --> 00:02:36.470
The trick here is to factor it
and simplify it before you do

00:02:36.470 --> 00:02:37.370
any multiplication.

00:02:37.370 --> 00:02:39.690
So let's simplify each of these
expressions and cancel

00:02:39.690 --> 00:02:41.910
out as much as possible.

00:02:41.910 --> 00:02:45.330
So here we can just
factor out an x.

00:02:45.330 --> 00:02:51.640
So this just becomes
x times x plus 4.

00:02:51.640 --> 00:02:55.090
This is a squared minus b
squared, so this becomes x

00:02:55.090 --> 00:03:00.050
plus 3 times x minus 3.

00:03:00.050 --> 00:03:01.360
With that you can't do much.

00:03:01.360 --> 00:03:03.260
Let's think about this one.

00:03:03.260 --> 00:03:08.430
What two numbers add up to a 1,
a 1x, and when you multiply

00:03:08.430 --> 00:03:09.840
them, you get a minus 12?

00:03:09.840 --> 00:03:16.100
So let's see, if you have an x
plus 4 times x minus 3, that

00:03:16.100 --> 00:03:16.510
should work.

00:03:16.510 --> 00:03:18.970
4 minus 3 is 1.

00:03:18.970 --> 00:03:21.900
4 times minus 3 is minus 12.

00:03:21.900 --> 00:03:23.540
So that works.

00:03:23.540 --> 00:03:25.190
All right now we're
ready to cancel.

00:03:28.600 --> 00:03:28.910
Normally.

00:03:28.910 --> 00:03:29.980
I would just start canceling,
but I just

00:03:29.980 --> 00:03:31.070
want to make the point.

00:03:31.070 --> 00:03:39.140
So this is equal to x times x
plus 4 times these, x plus 3

00:03:39.140 --> 00:03:52.680
times x minus 3, all of that
over x plus 3 times x plus 4

00:03:52.680 --> 00:03:55.970
times x minus 3.

00:03:55.970 --> 00:03:57.890
Now we can cancel.

00:03:57.890 --> 00:04:04.640
We have an x minus 3, x minus
3, x plus 4 x plus 4, x plus

00:04:04.640 --> 00:04:07.370
3, x plus 3, and all we're
left with is an x.

00:04:07.370 --> 00:04:09.250
So all of this fancy expression,
all that

00:04:09.250 --> 00:04:10.740
equals is an x.

00:04:10.740 --> 00:04:13.280
That's choice B.

00:04:13.280 --> 00:04:13.870
Where am I?

00:04:13.870 --> 00:04:15.940
Next problem.

00:04:15.940 --> 00:04:20.630
Image, clear image, invert
colors, there you go.

00:04:20.630 --> 00:04:21.990
All right, let me see
if I have to copy

00:04:21.990 --> 00:04:24.510
and paste this one.

00:04:24.510 --> 00:04:25.340
Where was I?

00:04:25.340 --> 00:04:27.430
Oh, why not?

00:04:27.430 --> 00:04:29.570
Another one where they're
trying to get

00:04:29.570 --> 00:04:30.520
the simplest form.

00:04:30.520 --> 00:04:31.770
So I copied it.

00:04:34.810 --> 00:04:36.910
Let me paste it.

00:04:36.910 --> 00:04:39.170
All right, I hope you
can read this.

00:04:39.170 --> 00:04:41.640
So what is the simplest form
of this expression here?

00:04:41.640 --> 00:04:42.850
I'll rewrite it.

00:04:42.850 --> 00:04:52.290
5x to the third y plus 20x
squared y squared plus 20x y

00:04:52.290 --> 00:04:57.690
to the third, everything
divided by 5xy.

00:04:57.690 --> 00:05:00.560
So 5xy actually goes
into every one of

00:05:00.560 --> 00:05:01.810
these terms, right?

00:05:01.810 --> 00:05:03.060
Let's just get rid of the 5's.

00:05:05.570 --> 00:05:07.150
Well, actually, let's just
divide the numerator and the

00:05:07.150 --> 00:05:09.310
denominator by 5xy.

00:05:09.310 --> 00:05:12.830
So what's 5x to the third
y divided by 5xy?

00:05:12.830 --> 00:05:15.600
Well, 5 divided by 5-- I'll
do it in magenta.

00:05:15.600 --> 00:05:18.220
5 divided by 5 is 1 so we don't
have to write that.

00:05:18.220 --> 00:05:21.950
x to the third divided
by x is x squared.

00:05:21.950 --> 00:05:23.590
y divided by y is 1.

00:05:23.590 --> 00:05:26.390
So this term divided by this
term reduced to that.

00:05:26.390 --> 00:05:27.550
Easy enough.

00:05:27.550 --> 00:05:28.600
Next one.

00:05:28.600 --> 00:05:31.620
Plus 20 divided by 5 is 4.

00:05:31.620 --> 00:05:35.090
x squared divided by x is x.

00:05:35.090 --> 00:05:38.356
y squared divided by y is y.

00:05:38.356 --> 00:05:40.200
We did the second term.

00:05:40.200 --> 00:05:42.500
Let's do the third term.

00:05:42.500 --> 00:05:46.010
20 divided by 5 is 4.

00:05:46.010 --> 00:05:50.465
x divided by x is 1 so we don't
have to write it. y to

00:05:50.465 --> 00:05:55.150
the third divided by
y is y squared.

00:05:55.150 --> 00:05:57.230
What we have here is this
simplified, but that doesn't

00:05:57.230 --> 00:05:58.420
look like anything here.

00:05:58.420 --> 00:06:00.650
So my guess is that this is
probably one of these.

00:06:00.650 --> 00:06:02.760
This is the square
of something.

00:06:02.760 --> 00:06:04.010
Let's see if we can
factor this.

00:06:04.010 --> 00:06:06.520
This would be x.

00:06:06.520 --> 00:06:13.170
So what two numbers when I add
them become 4xy or what number

00:06:13.170 --> 00:06:17.330
when I multiply it times x is
equal to half of this-- that's

00:06:17.330 --> 00:06:20.250
another way to think about it--
and when I square it is

00:06:20.250 --> 00:06:21.690
equal to 4y squared?

00:06:21.690 --> 00:06:26.220
That's just plus 2y squared.

00:06:26.220 --> 00:06:28.850
If that didn't make a lot of
sense, multiply this out and

00:06:28.850 --> 00:06:30.540
you'll get this answer.

00:06:30.540 --> 00:06:33.960
When you look at the choices,
you should just say, OK, I

00:06:33.960 --> 00:06:37.740
know that this choice and
this choice, those look

00:06:37.740 --> 00:06:38.970
nothing like that.

00:06:38.970 --> 00:06:40.730
Then you say, oh, well,
maybe this is one of

00:06:40.730 --> 00:06:42.310
these choices, right?

00:06:42.310 --> 00:06:44.460
But you have a 4y
squared here.

00:06:44.460 --> 00:06:46.440
This one has no y
at all in it.

00:06:46.440 --> 00:06:48.620
So you can even just use
deductive reasoning to say,

00:06:48.620 --> 00:06:52.410
oh, the choice must be
x plus 2y squared.

00:06:52.410 --> 00:06:54.480
Next problem.

00:06:54.480 --> 00:06:57.810
OK, this one I'm definitely
going to copy and paste

00:06:57.810 --> 00:07:01.155
because they have a big image.

00:07:04.580 --> 00:07:06.340
All right, let me see
if I can do it.

00:07:10.140 --> 00:07:12.280
This makes it a lot easier than
having to draw it myself.

00:07:15.930 --> 00:07:17.730
All right.

00:07:17.730 --> 00:07:20.840
If i equals the square root of
negative 1, which, if you

00:07:20.840 --> 00:07:23.960
didn't know it, that's what i
is defined as, and I go on

00:07:23.960 --> 00:07:27.240
many videos about the magical
of i and e and pi.

00:07:27.240 --> 00:07:31.100
Which point shows the location
of 5 minus 2i on the plane?

00:07:33.990 --> 00:07:35.630
The x-axis is the real axis.

00:07:35.630 --> 00:07:38.240
You take the real part,
which is 5.

00:07:38.240 --> 00:07:40.840
So you go 1, 2, 3, 4, 5.

00:07:40.840 --> 00:07:42.170
So that's the real part.

00:07:42.170 --> 00:07:44.400
Then the y-axis is the
imaginary axis.

00:07:44.400 --> 00:07:49.950
You take the imaginary part,
minus 2i, minus 2, right?

00:07:49.950 --> 00:07:53.440
So you go down 2 in the i
direction and you're at point

00:07:53.440 --> 00:07:55.610
D, which is choice D.

00:07:55.610 --> 00:07:56.490
Fair enough.

00:07:56.490 --> 00:07:58.300
That problem had a lot
of graphics but

00:07:58.300 --> 00:08:01.540
was unusually simple.

00:08:01.540 --> 00:08:02.790
Next problem.

00:08:05.140 --> 00:08:07.800
OK, now they're giving us a
whole parade of imaginary

00:08:07.800 --> 00:08:09.820
number questions.

00:08:09.820 --> 00:08:11.640
So this one I don't have to
cut, copy, and paste.

00:08:11.640 --> 00:08:14.810
They say if i is equal to the
square root of negative 1,

00:08:14.810 --> 00:08:17.770
which is a definition that we've
all learned, what is the

00:08:17.770 --> 00:08:21.270
value of i to the fourth?

00:08:23.820 --> 00:08:25.250
Well let's just go back.

00:08:25.250 --> 00:08:27.250
Let's say you've never seen
i before in your life.

00:08:27.250 --> 00:08:27.700
Let's think about it.

00:08:27.700 --> 00:08:30.915
If i is equal to negative 1--
sorry, this is negative 1.

00:08:30.915 --> 00:08:33.010
I know that negative looks
like it got messed up.

00:08:33.010 --> 00:08:35.549
Let's square both sides
of this equation.

00:08:35.549 --> 00:08:40.710
i squared will be equal
to negative 1.

00:08:40.710 --> 00:08:44.059
Now, let's square both sides
of this equation.

00:08:44.059 --> 00:08:49.100
i squared squared is equal
to negative 1 squared.

00:08:49.100 --> 00:08:51.900
Well, i squared squared is just
i to the fourth, right?

00:08:51.900 --> 00:08:54.790
2 times 2 is equal to negative
1 squared, which is

00:08:54.790 --> 00:08:55.890
just equal to 1.

00:08:55.890 --> 00:08:56.545
And we're done.

00:08:56.545 --> 00:08:57.795
That's choice C.

00:09:01.040 --> 00:09:04.580
All right, now they have another
imaginary graph.

00:09:04.580 --> 00:09:09.085
So let me just copy and
paste this graph.

00:09:11.760 --> 00:09:14.190
I'll go here.

00:09:14.190 --> 00:09:14.720
There you go.

00:09:14.720 --> 00:09:17.070
I didn't have to erase
what I had before.

00:09:17.070 --> 00:09:20.660
The question they ask us: Which
of the following complex

00:09:20.660 --> 00:09:23.330
numbers is represented by the
point on the graph below?

00:09:23.330 --> 00:09:24.226
So we just have to figure
out the real and

00:09:24.226 --> 00:09:26.060
the imaginary parts.

00:09:26.060 --> 00:09:27.020
So what's the real part?

00:09:27.020 --> 00:09:29.790
This coordinate, its
real part is 3.

00:09:33.440 --> 00:09:36.640
Its imaginary part is
minus 4 right there.

00:09:39.500 --> 00:09:43.640
So it's 3, and its imaginary
part, minus 4 in

00:09:43.640 --> 00:09:44.620
the imaginary direction.

00:09:44.620 --> 00:09:45.090
And we're done.

00:09:45.090 --> 00:09:45.840
3 minus 4i.

00:09:45.840 --> 00:09:47.090
That's choice C.

00:09:50.660 --> 00:09:51.850
I'm almost out of time.

00:09:51.850 --> 00:09:54.080
I'll see you in the
next video.


Algebra II: Complex numbers and conjugates

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.110
We're on problem 27.

00:00:03.110 --> 00:00:08.080
They tell us if i is equal to
the square root of minus 1,

00:00:08.080 --> 00:00:16.750
then what is 4 times i-- that's
an i-- times 6 times i?

00:00:16.750 --> 00:00:18.330
Well, multiplication
is associative.

00:00:18.330 --> 00:00:19.780
You can switch the
order around.

00:00:19.780 --> 00:00:25.720
So this is the same thing as 4
times 6 times i times i, or

00:00:25.720 --> 00:00:30.670
the same thing as 24 times
i squared, right?

00:00:30.670 --> 00:00:33.010
If we square both sides of this,
you get i squared is

00:00:33.010 --> 00:00:35.940
equal to negative 1.

00:00:35.940 --> 00:00:39.160
So this is 24 times
negative 1.

00:00:39.160 --> 00:00:43.220
So this is minus 24,
which is choice C.

00:00:43.220 --> 00:00:47.550
Next problem, 28.

00:00:47.550 --> 00:00:49.120
They want to know what
an equivalent

00:00:49.120 --> 00:00:53.660
form of 3 plus i is.

00:00:53.660 --> 00:00:56.830
A lot of times, people don't
like either when you have a

00:00:56.830 --> 00:00:59.640
square root in the denominator
or they don't like it when you

00:00:59.640 --> 00:01:01.950
have an a complex number
in the denominator.

00:01:01.950 --> 00:01:04.300
A complex number is just
something that has part real

00:01:04.300 --> 00:01:06.230
and part imaginary.

00:01:06.230 --> 00:01:09.350
So the way you can do it with
this case is you multiply

00:01:09.350 --> 00:01:11.270
times the conjugate
of this number.

00:01:11.270 --> 00:01:14.450
I know that sounds like a fancy
word, but all it means

00:01:14.450 --> 00:01:17.700
is you take the opposite
of the imaginary part.

00:01:17.700 --> 00:01:18.950
So what's the conjugate?

00:01:21.220 --> 00:01:23.610
Remember, the only way you
cannot change a fraction is if

00:01:23.610 --> 00:01:26.220
you multiply the numerator and
the denominator by the same

00:01:26.220 --> 00:01:28.540
thing, because then you're
multiplying by 1.

00:01:28.540 --> 00:01:30.690
So what's the conjugate
of this?

00:01:30.690 --> 00:01:35.980
The conjugate of 3 plus
i is 3 minus i.

00:01:35.980 --> 00:01:39.480
You have to do 3 minus
i over 3 minus i.

00:01:39.480 --> 00:01:41.510
I'm doing that because when you
multiply a complex number

00:01:41.510 --> 00:01:46.150
by its conjugate, you end up
actually with a real number.

00:01:46.150 --> 00:01:49.010
Then you're going to end up with
a complex number up here.

00:01:49.010 --> 00:01:51.050
A conjugate, all it means is
you keep the real part the

00:01:51.050 --> 00:01:54.330
same, and you change the sign
on the imaginary part.

00:01:54.330 --> 00:01:56.160
That's all that word means.

00:01:56.160 --> 00:01:58.540
All right, now let's figure
out what this is equal to.

00:01:58.540 --> 00:02:00.570
2 times 3, just same
distributive

00:02:00.570 --> 00:02:02.440
properties as always.

00:02:02.440 --> 00:02:03.740
Let me do it in green.

00:02:03.740 --> 00:02:05.540
2 times 3 is 6.

00:02:05.540 --> 00:02:11.240
2 times minus i is minus
2i, all of that over

00:02:11.240 --> 00:02:13.790
3 times 3 is 9.

00:02:13.790 --> 00:02:18.960
3 times minus i is minus 3i.

00:02:18.960 --> 00:02:23.680
i times 3 is plus 3i.

00:02:23.680 --> 00:02:29.910
Then i times minus i, you can
view that as minus i squared.

00:02:33.190 --> 00:02:34.930
Let's think about what
this is equal to.

00:02:34.930 --> 00:02:39.390
Well, this plus 3i and the
minus 3i cancel out.

00:02:39.390 --> 00:02:40.230
What's i squared?

00:02:40.230 --> 00:02:41.360
Well, this is equal
to negative 1.

00:02:41.360 --> 00:02:44.980
So you end up with a minus
of a negative 1.

00:02:44.980 --> 00:02:47.360
Minus a negative
1 is a plus 1.

00:02:47.360 --> 00:02:54.690
So you're left with 6 minus
2i over 9 plus 1 over 10.

00:02:54.690 --> 00:02:56.250
Actually, we can reduce
this some.

00:02:56.250 --> 00:02:58.880
We can divide the top
and the bottom by 2.

00:02:58.880 --> 00:03:03.300
So that becomes 3
minus i over 5.

00:03:03.300 --> 00:03:05.690
We just divided both the
numerator and the

00:03:05.690 --> 00:03:06.340
denominator by 2.

00:03:06.340 --> 00:03:07.730
That is choice B.

00:03:10.260 --> 00:03:16.110
Problem 29, these go
fast. No systems of

00:03:16.110 --> 00:03:17.060
equations and all that.

00:03:17.060 --> 00:03:20.530
What is the product of the
complex numbers 3 plus

00:03:20.530 --> 00:03:24.040
i and 3 minus i?

00:03:24.040 --> 00:03:25.020
They just asked us that.

00:03:25.020 --> 00:03:26.035
That was in the last problem.

00:03:26.035 --> 00:03:28.740
When we multiplied 3 plus
i times 3 minus

00:03:28.740 --> 00:03:29.740
i, what did we get?

00:03:29.740 --> 00:03:30.440
We got 10.

00:03:30.440 --> 00:03:31.110
We did this whole thing.

00:03:31.110 --> 00:03:32.590
3 times 3 is 9.

00:03:32.590 --> 00:03:33.800
We did the whole thing.

00:03:33.800 --> 00:03:35.570
They're just repeating
the same question.

00:03:35.570 --> 00:03:36.430
That equals 10.

00:03:36.430 --> 00:03:38.170
That's choice B.

00:03:38.170 --> 00:03:40.300
Rewind the video and see
what I did here.

00:03:40.300 --> 00:03:48.010
3 times 3 is 9 times minus 3i
plus 3i minus i squared.

00:03:48.010 --> 00:03:51.400
These terms cancel out and you
get a minus i squared, which

00:03:51.400 --> 00:03:53.000
is minus negative 1.

00:03:53.000 --> 00:03:58.090
So you have 9 plus 1,
so that equals 10.

00:03:58.090 --> 00:03:59.420
So that is choice B.

00:03:59.420 --> 00:04:00.520
That's crazy.

00:04:00.520 --> 00:04:03.220
They just gave us the same
problem twice in a row.

00:04:13.700 --> 00:04:14.120
All right.

00:04:14.120 --> 00:04:16.360
They're giving us more and more
imaginary problems. But

00:04:16.360 --> 00:04:17.920
these are fun, especially if
they're going to repeat the

00:04:17.920 --> 00:04:19.019
same question.

00:04:19.019 --> 00:04:21.980
Problem 30, if i is equal to
the square root of negative

00:04:21.980 --> 00:04:23.710
1-- well, I'll just keep writing
it because they keep

00:04:23.710 --> 00:04:24.120
writing it.

00:04:24.120 --> 00:04:26.190
It's good as a reminder. i is
equal to the square root of

00:04:26.190 --> 00:04:29.860
negative 1 and a and b are
non-zero real numbers.

00:04:29.860 --> 00:04:35.140
What is 1 over a plus bi?

00:04:35.140 --> 00:04:37.490
Well, once again, if they kind
of want us to rewrite this,

00:04:37.490 --> 00:04:39.980
they probably just want us to
get rid of the i's in the

00:04:39.980 --> 00:04:40.410
denominator.

00:04:40.410 --> 00:04:41.850
So we multiply times
the conjugate of

00:04:41.850 --> 00:04:43.270
this complex number.

00:04:43.270 --> 00:04:47.780
So we multiply times a plus
bi, the conjugate over the

00:04:47.780 --> 00:04:49.580
conjugate, a plus bi.

00:04:49.580 --> 00:04:51.330
The conjugate isn't
this whole thing.

00:04:51.330 --> 00:04:53.570
The conjugate is
just a plus bi.

00:04:53.570 --> 00:04:57.410
Oh sorry, we should be
multiplying times a minus bi.

00:04:57.410 --> 00:05:01.860
a minus bi over a minus bi.

00:05:01.860 --> 00:05:03.720
The conjugate of a plus bi,
just so you get the

00:05:03.720 --> 00:05:06.940
terminology right, is
just the a minus bi.

00:05:06.940 --> 00:05:09.490
But I'm doing a minus
bi over a minus bi.

00:05:09.490 --> 00:05:11.150
So I'm not changing this
fraction because this

00:05:11.150 --> 00:05:13.020
obviously is 1.

00:05:13.020 --> 00:05:14.690
x over x is the same
thing as 1.

00:05:14.690 --> 00:05:16.450
So let's see what this
is equal to.

00:05:16.450 --> 00:05:20.350
Well, the numerator just becomes
a minus bi times 1,

00:05:20.350 --> 00:05:21.840
which is a minus bi.

00:05:21.840 --> 00:05:25.720
Then you have a times a,
which is a squared.

00:05:25.720 --> 00:05:30.810
a times minus bi, so
it's minus abi.

00:05:30.810 --> 00:05:34.830
Then you have bi times
a, which is plus abi.

00:05:34.830 --> 00:05:36.800
I'll write it down here.

00:05:36.800 --> 00:05:38.600
They cancel out.

00:05:38.600 --> 00:05:42.450
Then you have bi
times minus bi.

00:05:42.450 --> 00:05:48.830
So that's minus b squared
i squared.

00:05:48.830 --> 00:05:50.420
These cancel out.

00:05:50.420 --> 00:05:56.810
So you're left with a minus
bi over a squared

00:05:56.810 --> 00:06:00.970
minus b squared i squared.

00:06:00.970 --> 00:06:02.360
We know what i squared is.

00:06:02.360 --> 00:06:03.970
i squared is equal
to negative 1.

00:06:03.970 --> 00:06:05.970
You square both sides of that
and you get i squared is equal

00:06:05.970 --> 00:06:09.470
to negative 1.

00:06:09.470 --> 00:06:13.100
So if we multiply, this
becomes a negative 1.

00:06:13.100 --> 00:06:13.990
But then this negative and this

00:06:13.990 --> 00:06:15.100
negative, they cancel out.

00:06:15.100 --> 00:06:16.270
You get a plus.

00:06:16.270 --> 00:06:23.830
So then we get a minus bi over
a squared plus b squared.

00:06:23.830 --> 00:06:26.980
That is choice B.

00:06:29.750 --> 00:06:31.000
Next problem.

00:06:35.100 --> 00:06:38.510
All right, what are the
solutions to the equation x

00:06:38.510 --> 00:06:46.140
squared plus 2x plus 2?

00:06:46.140 --> 00:06:49.610
All right, now, if you see an
equation like this and you're

00:06:49.610 --> 00:06:52.130
like, OK, let's see,
can I factor it?

00:06:52.130 --> 00:06:54.130
Oh, that is equal to 0.

00:06:54.130 --> 00:06:57.370
You say, OK, what number when
I multiple them equal 2 and

00:06:57.370 --> 00:07:00.540
when I add them equal 2?

00:07:00.540 --> 00:07:02.430
Well, that's a hard one.

00:07:02.430 --> 00:07:04.720
Whenever you get stuck, the
easiest thing to do is just

00:07:04.720 --> 00:07:07.540
use the quadratic equation,
just as a reminder.

00:07:10.260 --> 00:07:15.400
If the equation is Ax squared
plus Bx plus C-- maybe I

00:07:15.400 --> 00:07:17.750
should have done them all in in
capital letters-- is equal

00:07:17.750 --> 00:07:20.560
to 0, the solution to this and
the proof of this we've done

00:07:20.560 --> 00:07:22.805
in the Khan Academy, it comes
from completing the square,

00:07:22.805 --> 00:07:26.580
but it's useful to memorize,
is negative B.

00:07:26.580 --> 00:07:30.200
So the solutions are x are equal
to negative B plus or

00:07:30.200 --> 00:07:37.710
minus the square root of B
squared minus 4AC, all

00:07:37.710 --> 00:07:40.820
of that over 2A.

00:07:40.820 --> 00:07:42.340
So in this case, what is A?

00:07:42.340 --> 00:07:43.720
A is the coefficient
on x squared.

00:07:43.720 --> 00:07:45.650
So A is equal to 1.

00:07:45.650 --> 00:07:46.260
What is B?

00:07:46.260 --> 00:07:47.590
B is the coefficient on x.

00:07:47.590 --> 00:07:49.540
So B is equal to 2.

00:07:49.540 --> 00:07:50.210
What is C?

00:07:50.210 --> 00:07:51.410
C is just a constant term.

00:07:51.410 --> 00:07:52.930
So C is equal to 2.

00:07:52.930 --> 00:07:55.740
We could just solve like that,
just substitute these values.

00:07:55.740 --> 00:08:02.780
So you get x is equal to minus
B, which is minus 2, plus or

00:08:02.780 --> 00:08:07.170
minus the square root of
B squared, so that's 4,

00:08:07.170 --> 00:08:10.230
minus 4 minus A.

00:08:10.230 --> 00:08:20.080
So 4 minus 4 times A, 4 times
1, times 2, all of that--

00:08:20.080 --> 00:08:22.100
that's an ugly looking radical--
all of that over 2A,

00:08:22.100 --> 00:08:23.780
2 times 1 is 2.

00:08:23.780 --> 00:08:29.760
So that equals, let's see, we
get minus 2 plus or minus the

00:08:29.760 --> 00:08:33.929
square root of 4 minus--
this is 8, right?

00:08:33.929 --> 00:08:35.240
4 times 1 times 2.

00:08:35.240 --> 00:08:45.410
4 minus 8, which is minus 4, all
of that over 2, which is

00:08:45.410 --> 00:08:48.990
equal to minus 2
plus or minus.

00:08:48.990 --> 00:08:51.380
So the square root of minus 4,
we could rewrite that as the

00:08:51.380 --> 00:08:56.210
square root of 4
times minus 1.

00:08:56.210 --> 00:09:01.290
So that equals 2 plus or minus
the square root of 4 times the

00:09:01.290 --> 00:09:06.960
square root of minus 1, all of
that over 2, which is equal to

00:09:06.960 --> 00:09:10.750
minus 2 plus or minus-- the
square root of 4 is 2 and the

00:09:10.750 --> 00:09:15.950
plus or minus is dealing with
positive or negative 2-- times

00:09:15.950 --> 00:09:17.560
the square root of negative 1.

00:09:17.560 --> 00:09:19.620
Well, we learned that's
i multiple times.

00:09:19.620 --> 00:09:24.510
The square root of negative 1
is i, all of that over 2.

00:09:24.510 --> 00:09:26.560
We can simplify this
a little bit.

00:09:26.560 --> 00:09:28.750
I'll do it in a different color,
just because I've run

00:09:28.750 --> 00:09:30.600
out of space at the
bottom here.

00:09:30.600 --> 00:09:34.390
So this is equal to divide
everything by 2, you get

00:09:34.390 --> 00:09:39.140
negative 1 plus or minus i.

00:09:39.140 --> 00:09:42.770
Let's see what the
choices are.

00:09:42.770 --> 00:09:45.520
That's choice C, because
negative 1 plus or minus i

00:09:45.520 --> 00:09:49.660
could also be written as
negative 1 plus i as one value

00:09:49.660 --> 00:09:54.670
for x, and then the other one
could be negative 1 minus i.

00:09:54.670 --> 00:09:55.460
They separate it out.

00:09:55.460 --> 00:09:59.010
They don't write plus or minus
in the actual problem.

00:09:59.010 --> 00:09:59.760
I'm out of time.

00:09:59.760 --> 00:10:01.860
I'll see you in the next video.
you in the next video