Example 4: Applying the quadratic formula

WEBVTT
Kind: captions
Language: en

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A ball is shot into the air
from the edge of a building,

00:00:03.360 --> 00:00:05.500
50 feet above the ground.

00:00:05.500 --> 00:00:10.800
Its initial velocity
is 20 feet per second.

00:00:10.800 --> 00:00:12.960
The equation h--
and I'm guessing

00:00:12.960 --> 00:00:17.340
h is for height-- is equal to
negative 16t squared plus 20t

00:00:17.340 --> 00:00:19.890
plus 50 can be used to
model the height of the ball

00:00:19.890 --> 00:00:21.029
after t seconds.

00:00:21.029 --> 00:00:22.570
And I think in this
problem they just

00:00:22.570 --> 00:00:24.361
want us to accept this
formula, although we

00:00:24.361 --> 00:00:26.210
do derive formulas
like this and show

00:00:26.210 --> 00:00:29.260
why it works for this type of
problem in the Khan Academy

00:00:29.260 --> 00:00:30.230
physics playlist.

00:00:30.230 --> 00:00:32.975
But for here, we'll just go
with the flow on this example.

00:00:32.975 --> 00:00:34.350
So they give us
the equation that

00:00:34.350 --> 00:00:37.150
can be used to model the height
of the ball after t seconds,

00:00:37.150 --> 00:00:38.730
and then say about
how long does it

00:00:38.730 --> 00:00:41.380
take for the ball
to hit the ground.

00:00:41.380 --> 00:00:45.010
So if this is the
height, the ground

00:00:45.010 --> 00:00:47.070
is when the height
is equal to 0.

00:00:47.070 --> 00:00:50.070
So hitting the ground
means-- this literally

00:00:50.070 --> 00:00:52.120
means that h is equal to 0.

00:00:52.120 --> 00:00:56.790
So we need to figure out at
which times does h equal 0.

00:00:56.790 --> 00:00:58.810
So we're really
solving the equation

00:00:58.810 --> 00:01:07.230
0 is equal to negative 16t
squared plus 20t plus 50.

00:01:07.230 --> 00:01:10.320
And if you want to simplify
this a little bit-- let's see,

00:01:10.320 --> 00:01:13.480
everything here is
divisible at least by 2.

00:01:13.480 --> 00:01:15.231
And let's divide
everything by negative 2,

00:01:15.231 --> 00:01:17.396
just so that we can get rid
of this negative leading

00:01:17.396 --> 00:01:18.030
coefficient.

00:01:18.030 --> 00:01:20.230
So you divide the left
hand side by negative 2,

00:01:20.230 --> 00:01:21.870
you still get a 0.

00:01:21.870 --> 00:01:25.580
Negative 16 divided
by negative 2 is 8.

00:01:25.580 --> 00:01:27.020
So 8t squared.

00:01:27.020 --> 00:01:29.760
20 divided by negative
2 is negative 10.

00:01:29.760 --> 00:01:30.940
Minus 10t.

00:01:30.940 --> 00:01:34.830
50 divided by negative
2 is minus 25.

00:01:34.830 --> 00:01:38.465
And so we have 8t squared minus
10t minus 25 is equal to 0.

00:01:38.465 --> 00:01:40.840
Or if you're comfortable with
this on the left hand side,

00:01:40.840 --> 00:01:42.214
we can put on the
left hand side.

00:01:42.214 --> 00:01:44.970
We could just say
this is equal to 0.

00:01:44.970 --> 00:01:46.650
And now we solve.

00:01:46.650 --> 00:01:48.310
And we could complete
this square here,

00:01:48.310 --> 00:01:50.470
or we can just apply the
quadratic formula, which

00:01:50.470 --> 00:01:53.090
is derived from
completing the square.

00:01:53.090 --> 00:01:55.150
And we have this
in standard form.

00:01:55.150 --> 00:01:58.340
We know that this is our a.

00:01:58.340 --> 00:02:01.530
This right over here is our b.

00:02:01.530 --> 00:02:03.890
And this over here is our c.

00:02:03.890 --> 00:02:06.420
And the quadratic formula
tells us that the roots--

00:02:06.420 --> 00:02:09.039
and in this case, it's in
terms of the variable t--

00:02:09.039 --> 00:02:11.540
are going to be
equal to negative b

00:02:11.540 --> 00:02:14.950
plus or minus the
square root of b squared

00:02:14.950 --> 00:02:20.900
minus 4ac, all of that over 2a.

00:02:20.900 --> 00:02:25.435
So if we apply it, we get
t is equal to negative b.

00:02:25.435 --> 00:02:26.850
b is negative 10.

00:02:26.850 --> 00:02:30.300
So negative negative 10 is
going to be positive 10.

00:02:30.300 --> 00:02:34.800
Plus or minus the square
root of negative 10 squared.

00:02:34.800 --> 00:02:39.030
Well, that's just positive
100, minus 4 times a,

00:02:39.030 --> 00:02:44.600
which is 8, times c,
which is negative 25.

00:02:44.600 --> 00:02:46.630
And all of that over 2a.

00:02:46.630 --> 00:02:47.730
a Is 8.

00:02:47.730 --> 00:02:50.180
So 2 times 8 is 16.

00:02:50.180 --> 00:02:52.520
And this over here,
we have a-- let's

00:02:52.520 --> 00:02:54.425
see if we can simplify
this a little bit.

00:02:54.425 --> 00:02:56.300
The negative sign,
negative times a negative,

00:02:56.300 --> 00:02:57.890
these are going to be positive.

00:02:57.890 --> 00:03:01.310
4 times 25 is 100,
times 8 is 800.

00:03:01.310 --> 00:03:03.290
So all that simplifies to 800.

00:03:03.290 --> 00:03:06.250
And we have 100 plus 800
under the radical sign.

00:03:06.250 --> 00:03:08.370
So this is equal
to 10 plus or minus

00:03:08.370 --> 00:03:12.780
the square root of 900,
all of that over 16.

00:03:12.780 --> 00:03:16.450
And this is equal to 10 plus
or minus-- square root of 900

00:03:16.450 --> 00:03:20.110
is 30-- over 16.

00:03:20.110 --> 00:03:23.160
And so we get time is equal
to 10 plus 30 over 16,

00:03:23.160 --> 00:03:26.940
is 40 over 16, which is
the same thing if we divide

00:03:26.940 --> 00:03:31.950
the numerator and denominator
by 4 to simplify it as 10 over--

00:03:31.950 --> 00:03:36.040
or actually even better, divide
it by 8-- that's 5 over 2.

00:03:36.040 --> 00:03:38.160
So that's one solution,
if we add the 30.

00:03:38.160 --> 00:03:41.410
If we subtract the 30,
we'd get 10 minus 30.

00:03:41.410 --> 00:03:46.480
Or t is equal to 10 minus 30,
which is negative 20 over 16.

00:03:46.480 --> 00:03:49.470
Divide the numerator and
the denominator by 4,

00:03:49.470 --> 00:03:52.010
you get negative 5 over 4.

00:03:52.010 --> 00:03:54.740
Now, we have to remember,
we're trying to find a time.

00:03:54.740 --> 00:03:57.346
And so a time, at
least in this problem

00:03:57.346 --> 00:03:58.970
that we're dealing
with, we should only

00:03:58.970 --> 00:04:00.094
think about positive times.

00:04:00.094 --> 00:04:02.222
We want to figure out
how much time has taken--

00:04:02.222 --> 00:04:04.430
how long does it take for
the ball to hit the ground?

00:04:04.430 --> 00:04:05.891
We don't want to
go back in time.

00:04:05.891 --> 00:04:07.890
So we don't want our
negative answer right here.

00:04:07.890 --> 00:04:10.470
So we only want to think
about our positive answer.

00:04:10.470 --> 00:04:12.820
And so this tells us that the
only root that should work

00:04:12.820 --> 00:04:14.010
is 5/2.

00:04:14.010 --> 00:04:16.140
And we assume that
this is in seconds.

00:04:16.140 --> 00:04:18.929
So this is in 5/2 seconds.

00:04:18.929 --> 00:04:20.970
I wouldn't worry too much
about the physics here.

00:04:20.970 --> 00:04:23.595
I think they really just want us
to apply the quadratic formula

00:04:23.595 --> 00:04:24.880
to this modeling situation.

00:04:24.880 --> 00:04:26.630
The physics, we go
into a lot more depth

00:04:26.630 --> 00:04:29.730
and give you the conceptual
understanding on our physics

00:04:29.730 --> 00:04:30.530
playlist.

00:04:30.530 --> 00:04:33.590
But let's verify that we
definitely are at a height of 0

00:04:33.590 --> 00:04:36.620
at 5/2 seconds, or
t is equal to 5/2.

00:04:36.620 --> 00:04:40.540
This expression right over here
does give us h is equal to 0.

00:04:40.540 --> 00:04:44.160
So we have-- let's try it out.

00:04:44.160 --> 00:04:49.830
We have negative 16
times 5/2 squared

00:04:49.830 --> 00:04:53.820
plus 20 times 5/2 plus 50.

00:04:58.220 --> 00:04:59.420
This needs to be equal to 0.

00:04:59.420 --> 00:05:05.005
So this is negative
16 times 25/4 plus--

00:05:05.005 --> 00:05:07.920
let's see, if we divide
20 by 2, we get 10.

00:05:07.920 --> 00:05:09.260
If we divide 2 by 2, we get 1.

00:05:09.260 --> 00:05:13.120
So 10 times 5 is going to be 50.

00:05:13.120 --> 00:05:14.460
Plus 50.

00:05:14.460 --> 00:05:16.610
This needs to be equal to 0.

00:05:16.610 --> 00:05:19.700
Negative 16 divided
by 4 is negative 4.

00:05:19.700 --> 00:05:21.350
4 divided by 4 is 1.

00:05:21.350 --> 00:05:25.650
So you have negative 4
times 25, which is 100.

00:05:25.650 --> 00:05:27.680
Plus 50-- oh, sorry.

00:05:27.680 --> 00:05:29.880
Negative 4 times
25 is negative 100.

00:05:29.880 --> 00:05:33.780
Plus 50, plus 50
again is equal to 0.

00:05:33.780 --> 00:05:36.972
And so we have negative
100 plus another 100.

00:05:36.972 --> 00:05:38.930
Well, that's definitely
going to be equal to 0.

00:05:38.930 --> 00:05:40.370
We get 0 equals 0.

00:05:40.370 --> 00:05:41.950
And it all checks out.

00:05:41.950 --> 00:05:45.320
We hit the ground
after 5/2 seconds.

00:05:45.320 --> 00:05:49.090
Or another way to think
about it is 2.5 seconds.

00:05:49.090 --> 00:05:52.930
t is equal to 5/2
seconds, or 2.5 seconds.


Example: Complex roots for a quadratic

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:05.470
We're asked to solve 2x
squared plus 5 is equal to 6x.

00:00:05.470 --> 00:00:07.260
And so we have a
quadratic equation here.

00:00:07.260 --> 00:00:10.716
But just to put it into a form
that we're more familiar with,

00:00:10.716 --> 00:00:12.340
let's try to put it
into standard form.

00:00:12.340 --> 00:00:14.173
And standard form, of
course, is the form ax

00:00:14.173 --> 00:00:19.400
squared plus bx plus
c is equal to 0.

00:00:19.400 --> 00:00:22.000
And to do that, we essentially
have to take the 6x

00:00:22.000 --> 00:00:23.829
and get rid of it from
the right hand side.

00:00:23.829 --> 00:00:25.620
So we just have a 0 on
the right hand side.

00:00:25.620 --> 00:00:28.010
And to do that, let's
just subtract 6x

00:00:28.010 --> 00:00:30.380
from both sides
of this equation.

00:00:30.380 --> 00:00:31.850
And so our left
hand side becomes

00:00:31.850 --> 00:00:38.540
2x squared minus 6x plus
5 is equal to-- and then

00:00:38.540 --> 00:00:41.090
on our right hand side, these
two characters cancel out,

00:00:41.090 --> 00:00:43.150
and we just are left with 0.

00:00:43.150 --> 00:00:44.710
And there's many
ways to solve this.

00:00:44.710 --> 00:00:45.930
We could try to factor it.

00:00:45.930 --> 00:00:47.304
And if I was trying
to factor it,

00:00:47.304 --> 00:00:49.720
I would divide both sides by 2.

00:00:49.720 --> 00:00:53.201
If I divide both sides by 2, I
would get integer coefficients

00:00:53.201 --> 00:00:54.742
on the x squared in
the x term, but I

00:00:54.742 --> 00:00:56.330
would get 5/2 for the constant.

00:00:56.330 --> 00:00:58.540
So it's not one of these
easy things to factor.

00:00:58.540 --> 00:01:00.720
We could complete
the square, or we

00:01:00.720 --> 00:01:02.970
could apply the quadratic
formula, which is really

00:01:02.970 --> 00:01:05.960
just a formula derived
from completing the square.

00:01:05.960 --> 00:01:07.990
So let's do that
in this scenario.

00:01:07.990 --> 00:01:09.480
And the quadratic
formula tells us

00:01:09.480 --> 00:01:12.190
that if we have something
in standard form like this,

00:01:12.190 --> 00:01:15.950
that the roots of it are
going to be negative b

00:01:15.950 --> 00:01:18.030
plus or minus-- so that
gives us two roots right

00:01:18.030 --> 00:01:20.800
over there-- plus or minus
square root of b squared

00:01:20.800 --> 00:01:24.910
minus 4ac over 2a.

00:01:24.910 --> 00:01:27.140
So let's apply that
to this situation.

00:01:27.140 --> 00:01:29.710
Negative b-- this
right here is b.

00:01:29.710 --> 00:01:32.760
So negative b is
negative negative 6.

00:01:32.760 --> 00:01:35.010
So that's going
to be positive 6,

00:01:35.010 --> 00:01:38.900
plus or minus the square
root of b squared.

00:01:38.900 --> 00:01:47.990
Negative 6 squared is 36, minus
4 times a-- which is 2-- times

00:01:47.990 --> 00:01:51.300
2 times c, which is 5.

00:01:51.300 --> 00:01:52.740
Times 5.

00:01:52.740 --> 00:01:56.330
All of that over
2 times a. a is 2.

00:01:56.330 --> 00:01:58.570
So 2 times 2 is 4.

00:01:58.570 --> 00:02:01.260
So this is going to be
equal to 6 plus or minus

00:02:01.260 --> 00:02:06.050
the square root of 36-- so
let me just figure this out.

00:02:06.050 --> 00:02:08.580
36 minus-- so this
is 4 times 2 times 5.

00:02:08.580 --> 00:02:10.270
This is 40 over here.

00:02:10.270 --> 00:02:12.950
So 36 minus 40.

00:02:12.950 --> 00:02:14.390
And you already
might be wondering

00:02:14.390 --> 00:02:15.910
what's going to happen here.

00:02:15.910 --> 00:02:17.390
All of that over 4.

00:02:17.390 --> 00:02:19.750
Or this is equal
to 6 plus or minus

00:02:19.750 --> 00:02:23.930
the square root of negative 4.

00:02:23.930 --> 00:02:27.140
36 minus 40 is
negative 4 over 4.

00:02:27.140 --> 00:02:28.540
And you might say,
hey, wait Sal.

00:02:28.540 --> 00:02:30.140
Negative 4, if I
take a square root,

00:02:30.140 --> 00:02:32.020
I'm going to get an
imaginary number.

00:02:32.020 --> 00:02:33.330
And you would be right.

00:02:33.330 --> 00:02:36.670
The only two roots of this
quadratic equation right

00:02:36.670 --> 00:02:38.830
here are going to turn
out to be complex,

00:02:38.830 --> 00:02:41.456
because when we
evaluate this, we're

00:02:41.456 --> 00:02:42.830
going to get an
imaginary number.

00:02:42.830 --> 00:02:45.246
So we're essentially going to
get two complex numbers when

00:02:45.246 --> 00:02:48.620
we take the positive and
negative version of this root.

00:02:48.620 --> 00:02:49.880
So let's do that.

00:02:49.880 --> 00:02:52.110
So the square root
of negative 4,

00:02:52.110 --> 00:02:55.055
that is the same thing as 2i.

00:02:55.055 --> 00:02:57.220
And we know that's
the same thing as 2i,

00:02:57.220 --> 00:02:59.110
or if you want to
think of it this way.

00:02:59.110 --> 00:03:01.290
Square root of negative
4 is the same thing

00:03:01.290 --> 00:03:05.220
as the square root of negative
1 times the square root of 4,

00:03:05.220 --> 00:03:05.970
which is the same.

00:03:05.970 --> 00:03:08.430
I could even do it
one step-- that's

00:03:08.430 --> 00:03:10.360
the same thing as
negative 1 times 4

00:03:10.360 --> 00:03:12.901
under the radical, which is the
same thing as the square root

00:03:12.901 --> 00:03:15.430
of negative 1 times
the square root of 4.

00:03:15.430 --> 00:03:17.250
And the principal square
root of negative 1

00:03:17.250 --> 00:03:21.470
is i times the principal
square root of 4 is 2.

00:03:21.470 --> 00:03:24.370
So this is 2i, or i times 2.

00:03:24.370 --> 00:03:27.485
So this right over
here is going to be 2i.

00:03:27.485 --> 00:03:35.390
So we are left with x is equal
to 6 plus or minus 2i over 4.

00:03:35.390 --> 00:03:37.300
And if we were to
simplify it, we

00:03:37.300 --> 00:03:40.290
could divide the numerator
and the denominator by 2.

00:03:40.290 --> 00:03:43.310
And so that would be the
same thing as 3 plus or minus

00:03:43.310 --> 00:03:45.310
i over 2.

00:03:45.310 --> 00:03:48.500
Or if you want to write them as
two distinct complex numbers,

00:03:48.500 --> 00:03:56.530
you could write this as 3 plus
i over 2, or 3/2 plus 1/2i.

00:03:56.530 --> 00:03:59.610
That's if I take the positive
version of the i there.

00:03:59.610 --> 00:04:05.480
Or we could view this
as 3/2 minus 1/2i.

00:04:08.100 --> 00:04:11.590
This and these two guys
right here are equivalent.

00:04:11.590 --> 00:04:13.360
Those are the two roots.

00:04:13.360 --> 00:04:16.899
Now what I want to do is
a verify that these work.

00:04:16.899 --> 00:04:18.519
Verify these two roots.

00:04:18.519 --> 00:04:22.640
So this one I can rewrite
as 3 plus i over 2.

00:04:22.640 --> 00:04:24.490
These are equivalent.

00:04:24.490 --> 00:04:26.320
All I did-- you can
see that this is just

00:04:26.320 --> 00:04:28.270
dividing both of these by 2.

00:04:28.270 --> 00:04:32.294
Or if you were to essentially
factor out the 1/2,

00:04:32.294 --> 00:04:34.085
you could go either
way on this expression.

00:04:34.085 --> 00:04:38.982
And this one over here is
going to be 3 minus i over 2.

00:04:38.982 --> 00:04:40.440
Or you could go
directly from this.

00:04:40.440 --> 00:04:42.030
This is 3 plus or
minus i over 2.

00:04:42.030 --> 00:04:43.670
So 3 plus i over 2.

00:04:43.670 --> 00:04:46.260
Or 3 minus i over 2.

00:04:46.260 --> 00:04:48.690
This and this or this
and this, or this.

00:04:48.690 --> 00:04:50.690
These are all equal
representations

00:04:50.690 --> 00:04:51.870
of both of the roots.

00:04:51.870 --> 00:04:53.610
But let's see if they work.

00:04:53.610 --> 00:04:56.745
So I'm first going to try this
character right over here.

00:04:56.745 --> 00:04:58.870
It's going to get a little
bit hairy, because we're

00:04:58.870 --> 00:05:00.370
going to have to square
it and all the rest.

00:05:00.370 --> 00:05:01.730
But let's see if we can do it.

00:05:01.730 --> 00:05:03.396
So what we want to
do is we want to take

00:05:03.396 --> 00:05:06.010
2 times this quantity squared.

00:05:06.010 --> 00:05:13.196
So 2 times 3 plus i
over 2 squared plus 5.

00:05:13.196 --> 00:05:14.570
And we want to
verify that that's

00:05:14.570 --> 00:05:17.480
the same thing as 6
times this quantity, as 6

00:05:17.480 --> 00:05:21.400
times 3 plus i over 2.

00:05:21.400 --> 00:05:24.840
So what is 3 plus i squared?

00:05:24.840 --> 00:05:27.800
So this is 2 times--
let me just square this.

00:05:27.800 --> 00:05:31.080
So 3 plus i, that's going
to be 3 squared, which

00:05:31.080 --> 00:05:34.930
is 9, plus 2 times the
product of three and i.

00:05:34.930 --> 00:05:38.915
So 3 times i is
3i, times 2 is 6i.

00:05:38.915 --> 00:05:40.727
So plus 6i.

00:05:40.727 --> 00:05:42.310
And if that doesn't
make sense to you,

00:05:42.310 --> 00:05:44.351
I encourage you to kind
of multiply it out either

00:05:44.351 --> 00:05:46.910
with the distributive
property or FOIL it out,

00:05:46.910 --> 00:05:49.040
and you'll get the middle term.

00:05:49.040 --> 00:05:50.751
You'll get 3i twice.

00:05:50.751 --> 00:05:52.000
When you add them, you get 6i.

00:05:52.000 --> 00:05:56.520
I And then plus i squared,
and i squared is negative 1.

00:05:56.520 --> 00:05:57.565
Minus 1.

00:05:57.565 --> 00:06:02.492
All of that over 4, plus
5, is equal to-- well,

00:06:02.492 --> 00:06:04.700
if you divide the numerator
and the denominator by 2,

00:06:04.700 --> 00:06:07.010
you get a 3 here and
you get a 1 here.

00:06:07.010 --> 00:06:11.330
And 3 distributed on 3 plus
i is equal to 9 plus 3i.

00:06:14.130 --> 00:06:17.080
And what we have over here,
we can simplify it just

00:06:17.080 --> 00:06:18.550
to save some screen real estate.

00:06:18.550 --> 00:06:20.740
9 minus 1 is 8.

00:06:20.740 --> 00:06:26.490
So if I get rid of this,
this is just 8 plus 6i.

00:06:26.490 --> 00:06:28.670
We can divide the numerator
and the denominator

00:06:28.670 --> 00:06:29.570
right here by 2.

00:06:29.570 --> 00:06:37.400
So the numerator would become 4
plus 3i, if we divided it by 2,

00:06:37.400 --> 00:06:39.800
and the denominator here
is just going to be 2.

00:06:39.800 --> 00:06:42.380
This 2 and this 2 are
going to cancel out.

00:06:42.380 --> 00:06:47.140
So on the left hand side, we're
left with 4 plus 3i plus 5.

00:06:47.140 --> 00:06:51.320
And this needs to be
equal to 9 plus 3i.

00:06:51.320 --> 00:06:54.650
Well, you can see we have a 3i
on both sides of this equation.

00:06:54.650 --> 00:06:57.880
And we have a 4 plus 5,
which is exactly equal to 9.

00:06:57.880 --> 00:07:02.460
So this solution, 3 plus
i, definitely works.

00:07:02.460 --> 00:07:06.572
Now let's try 3 minus i.

00:07:06.572 --> 00:07:08.780
So once again, just looking
at the original equation,

00:07:08.780 --> 00:07:11.310
2x squared plus
5 is equal to 6x.

00:07:11.310 --> 00:07:14.100
Let me write it down over here.

00:07:14.100 --> 00:07:16.020
Let me rewrite the
original equation.

00:07:16.020 --> 00:07:20.380
We have 2x squared
plus 5 is equal to 6x.

00:07:20.380 --> 00:07:24.690
And now we're going to try this
root, verify that it works.

00:07:24.690 --> 00:07:30.050
So we have 2 times
3 minus i over 2

00:07:30.050 --> 00:07:36.320
squared plus 5 needs to be
equal to 6 times this business.

00:07:36.320 --> 00:07:40.306
6 times 3 minus i over 2.

00:07:40.306 --> 00:07:41.430
Once again, a little hairy.

00:07:41.430 --> 00:07:44.055
But as long as we do everything,
we put our head down and focus

00:07:44.055 --> 00:07:46.280
on it, we should be able
to get the right result.

00:07:46.280 --> 00:07:49.940
So 3 minus i squared.

00:07:49.940 --> 00:07:53.270
3 minus i times 3
minus i, which is--

00:07:53.270 --> 00:07:55.130
and you could get
practice taking squares

00:07:55.130 --> 00:07:58.030
of two termed expressions,
or complex numbers

00:07:58.030 --> 00:07:59.480
in this case
actually-- it's going

00:07:59.480 --> 00:08:03.290
to be 9, that's 3 squared,
and then 3 times negative i

00:08:03.290 --> 00:08:04.390
is negative 3i.

00:08:04.390 --> 00:08:06.370
And then you're going
to have two of those.

00:08:06.370 --> 00:08:09.850
So negative 6i.

00:08:09.850 --> 00:08:14.420
So negative i squared
is also negative 1.

00:08:14.420 --> 00:08:17.480
That's negative 1 times
negative 1 times i times i.

00:08:17.480 --> 00:08:19.120
So that's also negative 1.

00:08:19.120 --> 00:08:23.130
Negative i squared is
also equal to negative 1.

00:08:23.130 --> 00:08:25.187
Negative i is also
another square root.

00:08:25.187 --> 00:08:26.770
Not the principal
square root, but one

00:08:26.770 --> 00:08:30.410
of the square roots
of negative 1.

00:08:30.410 --> 00:08:32.505
So now we're going
to have a plus 1,

00:08:32.505 --> 00:08:34.630
because-- oh, sorry, we're
going to have a minus 1.

00:08:34.630 --> 00:08:39.350
Because this is negative i
squared, which is negative 1.

00:08:39.350 --> 00:08:41.570
And all of that over 4.

00:08:41.570 --> 00:08:44.390
All of that over--
that's 2 squared is 4.

00:08:44.390 --> 00:08:49.570
Times 2 over here,
plus 5, needs to be

00:08:49.570 --> 00:08:51.799
equal to-- well, before
I even multiply it out,

00:08:51.799 --> 00:08:54.090
we could divide the numerator
and the denominator by 2.

00:08:54.090 --> 00:08:56.230
So 6 divided by 2 is 3.

00:08:56.230 --> 00:08:58.820
2 divided by 2 is 1.

00:08:58.820 --> 00:09:00.930
So 3 times 3 is 9.

00:09:00.930 --> 00:09:04.120
3 times negative
i is negative 3i.

00:09:04.120 --> 00:09:08.090
And if we simplify it a
little bit more, 9 minus 1

00:09:08.090 --> 00:09:10.590
is going to be--
I'll do this in blue.

00:09:10.590 --> 00:09:13.210
9 minus 1 is going to be 8.

00:09:13.210 --> 00:09:15.780
We have 8 minus 6i.

00:09:15.780 --> 00:09:19.320
And then if we divide
8 minus 6i by 2 and 4

00:09:19.320 --> 00:09:23.950
by 2, in the numerator, we're
going to get 4 minus 3i.

00:09:23.950 --> 00:09:29.770
And in the denominator over
here, we're going to get a 2.

00:09:29.770 --> 00:09:32.790
We divided the numerator
and the denominator by 2.

00:09:32.790 --> 00:09:34.942
Then we have a 2 out here.

00:09:34.942 --> 00:09:36.400
And we have a 2 in
the denominator.

00:09:36.400 --> 00:09:38.247
Those two characters
will cancel out.

00:09:38.247 --> 00:09:39.830
And so this expression
right over here

00:09:39.830 --> 00:09:44.660
cancels or simplifies
to 4 minus 3i.

00:09:44.660 --> 00:09:49.620
Then we have a plus 5 needs
to be equal to 9 minus 3i.

00:09:49.620 --> 00:09:51.870
I We have a negative
3i on the left,

00:09:51.870 --> 00:09:53.510
a negative 3i on the right.

00:09:53.510 --> 00:09:54.416
We have a 4 plus 5.

00:09:54.416 --> 00:09:55.290
We could evaluate it.

00:09:55.290 --> 00:09:59.070
This left hand
side is 9 minus 3i,

00:09:59.070 --> 00:10:01.380
which is the exact same
complex number as we have

00:10:01.380 --> 00:10:03.960
on the right hand
side, 9 minus 3i.

00:10:03.960 --> 00:10:05.690
So it also checks out.

00:10:05.690 --> 00:10:06.900
It is also a root.

00:10:06.900 --> 00:10:11.390
So we verified that both
of these complex roots,

00:10:11.390 --> 00:10:14.131
satisfy this quadratic equation.


Discriminant for types of solutions for a quadratic

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:04.070
Use the discriminant
to state the number

00:00:04.070 --> 00:00:06.520
and type of solutions
for the equation

00:00:06.520 --> 00:00:11.250
negative 3x squared plus
5x minus 4 is equal to 0.

00:00:11.250 --> 00:00:12.640
And so just as a
reminder, you're

00:00:12.640 --> 00:00:14.560
probably wondering what
is the discriminant.

00:00:14.560 --> 00:00:16.360
And we can just
review it by looking

00:00:16.360 --> 00:00:17.720
at the quadratic formula.

00:00:17.720 --> 00:00:20.940
So if I have a quadratic
equation in standard form,

00:00:20.940 --> 00:00:25.690
ax squared plus bx
plus c is equal to 0,

00:00:25.690 --> 00:00:27.910
we know that the quadratic
formula, which is really

00:00:27.910 --> 00:00:30.730
just derived from completing
the square right over here,

00:00:30.730 --> 00:00:33.420
tells us that the roots
of this, or the solutions

00:00:33.420 --> 00:00:34.920
of this quadratic
equation are going

00:00:34.920 --> 00:00:39.530
to be x is equal to negative b
plus or minus the square root

00:00:39.530 --> 00:00:45.210
of b squared minus 4ac,
all of that over 2a.

00:00:45.210 --> 00:00:48.020
Now, you might know
from experience applying

00:00:48.020 --> 00:00:50.580
this little bit,
we're going to get

00:00:50.580 --> 00:00:52.770
different types of
solutions depending

00:00:52.770 --> 00:00:56.340
on what happens under the
radical sign over here.

00:00:56.340 --> 00:00:59.140
As you could imagine, if
what's under the radical sign

00:00:59.140 --> 00:01:01.560
over here is
positive, then we're

00:01:01.560 --> 00:01:04.140
going to get an
actual real number

00:01:04.140 --> 00:01:05.515
as its principal square root.

00:01:05.515 --> 00:01:07.890
And when we take the positive
and negative version of it,

00:01:07.890 --> 00:01:10.510
we're going to get
two real solutions.

00:01:10.510 --> 00:01:13.700
So if b squared minus
4ac-- and this is

00:01:13.700 --> 00:01:16.510
what the discriminant
really is, it's

00:01:16.510 --> 00:01:19.450
just this expression
under the radical sign

00:01:19.450 --> 00:01:20.990
of the quadratic formula.

00:01:20.990 --> 00:01:23.000
If this is greater
than 0, then we're

00:01:23.000 --> 00:01:31.910
going to have two real
roots or two real solutions

00:01:31.910 --> 00:01:33.630
to this equation right here.

00:01:33.630 --> 00:01:39.652
If b squared minus
4ac is equal to 0,

00:01:39.652 --> 00:01:41.860
then this whole thing is
just going to be equal to 0.

00:01:41.860 --> 00:01:44.276
It's going to be the plus or
minus square root of 0, which

00:01:44.276 --> 00:01:44.950
is just 0.

00:01:44.950 --> 00:01:46.020
So it's plus or minus 0.

00:01:46.020 --> 00:01:48.310
Well, when you
add or subtract 0,

00:01:48.310 --> 00:01:49.930
that doesn't change
the solution.

00:01:49.930 --> 00:01:53.800
So the only solution is going
to be negative b over 2a.

00:01:53.800 --> 00:01:57.660
So you're only going to
have one real solution.

00:01:57.660 --> 00:02:02.670
So this is going to be 1-- I'll
just write the number 1-- 1

00:02:02.670 --> 00:02:03.580
real solution.

00:02:03.580 --> 00:02:06.970
Or you could say you have
a repeated root here.

00:02:06.970 --> 00:02:08.630
You could say you're
having it twice.

00:02:08.630 --> 00:02:13.490
Or you could say one real
solution, or one real root.

00:02:13.490 --> 00:02:17.500
Now, if b squared minus
4ac were negative,

00:02:17.500 --> 00:02:20.100
you might already
imagine what will happen.

00:02:20.100 --> 00:02:22.170
If this expression right
over here is negative,

00:02:22.170 --> 00:02:24.270
we're taking the square
root of a negative number.

00:02:24.270 --> 00:02:26.900
So we would then get an
imaginary number right over

00:02:26.900 --> 00:02:27.590
here.

00:02:27.590 --> 00:02:31.230
And so we would add or subtract
the same imaginary number.

00:02:31.230 --> 00:02:33.330
So we'll have two
complex solutions.

00:02:33.330 --> 00:02:35.490
And not only will we have
two complex solutions,

00:02:35.490 --> 00:02:37.750
but they will be the
conjugates of each other.

00:02:37.750 --> 00:02:39.520
So if you have one
complex solution

00:02:39.520 --> 00:02:42.110
for a quadratic equation,
the other solution

00:02:42.110 --> 00:02:43.800
will also be a complex solution.

00:02:43.800 --> 00:02:46.080
And it will be its
complex conjugate.

00:02:46.080 --> 00:02:52.120
So here we would have
two complex solutions.

00:02:52.120 --> 00:02:55.132
So numbers that have a real
part and an imaginary part.

00:02:55.132 --> 00:02:56.590
And not only are
they just complex,

00:02:56.590 --> 00:02:58.410
but they are the
conjugates of each other.

00:02:58.410 --> 00:03:00.900
The imaginary parts
have different signs.

00:03:00.900 --> 00:03:04.080
So let's look at b squared
minus 4ac over here.

00:03:04.080 --> 00:03:07.430
This is our a, this is
our b, and this is our c.

00:03:07.430 --> 00:03:10.869
Let me label them. a, b, c.

00:03:10.869 --> 00:03:13.410
And I can do that, because we've
written it in standard form.

00:03:13.410 --> 00:03:15.700
Everything is on one side.

00:03:15.700 --> 00:03:17.550
Or in particular,
the left hand side.

00:03:17.550 --> 00:03:19.140
We have a 0 on the
right hand side.

00:03:19.140 --> 00:03:21.360
We've written it
in descending, I

00:03:21.360 --> 00:03:24.080
guess, power form, or
the descending degree.

00:03:24.080 --> 00:03:25.890
Or we have our second
degree term first,

00:03:25.890 --> 00:03:28.469
then our first degree term,
then our constant term.

00:03:28.469 --> 00:03:30.135
And so we can evaluate
the discriminant.

00:03:32.720 --> 00:03:33.470
b is 5.

00:03:33.470 --> 00:03:38.270
So b squared is 5 squared, minus
4 times a, which is negative 3,

00:03:38.270 --> 00:03:41.220
times c, which is negative
4-- I have to be careful.

00:03:41.220 --> 00:03:44.650
c is this whole thing.
c is negative 4.

00:03:44.650 --> 00:03:48.990
And so I don't know if I said
4 earlier, but c is negative 4.

00:03:48.990 --> 00:03:52.010
We have to make sure we take
the sign into consideration.

00:03:52.010 --> 00:03:55.230
So times c, which is
negative 4 over here.

00:03:55.230 --> 00:03:56.970
And so this is 25.

00:03:56.970 --> 00:03:58.780
And then negative
3 times negative 4,

00:03:58.780 --> 00:04:00.760
that is positive 12.

00:04:00.760 --> 00:04:02.795
And then 4 times 12 is 48.

00:04:02.795 --> 00:04:04.340
But we have a negative out here.

00:04:04.340 --> 00:04:06.340
So 25 minus 48.

00:04:06.340 --> 00:04:10.295
And 25 minus 48, we don't
even have to do the math.

00:04:10.295 --> 00:04:11.920
We can just say that
this is definitely

00:04:11.920 --> 00:04:13.520
going to be less than 0.

00:04:13.520 --> 00:04:15.540
You can actually figure it out.

00:04:15.540 --> 00:04:20.482
This is equal to negative
13, if I did-- oh no,

00:04:20.482 --> 00:04:26.780
sorry, negative 23, which
is clearly less than 0.

00:04:26.780 --> 00:04:30.010
So our discriminant in this
situation is less than 0.

00:04:30.010 --> 00:04:33.210
So we are going to have
two complex roots here,

00:04:33.210 --> 00:04:35.677
and they're going to be
each other's conjugates.


Example 3: Completing the square

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.230
We're asked to
complete the square

00:00:02.230 --> 00:00:08.950
to solve 4x squared plus
40x minus 300 is equal to 0.

00:00:08.950 --> 00:00:10.240
So let me just rewrite it.

00:00:10.240 --> 00:00:17.830
So 4x squared plus 40x
minus 300 is equal to 0.

00:00:17.830 --> 00:00:19.890
So just as a first
step here, I don't

00:00:19.890 --> 00:00:22.204
like having this 4 out
front as a coefficient

00:00:22.204 --> 00:00:23.120
on the x squared term.

00:00:23.120 --> 00:00:25.100
I'd prefer if that was a 1.

00:00:25.100 --> 00:00:28.180
So let's just divide both
sides of this equation by 4.

00:00:28.180 --> 00:00:30.480
So let's just divide
everything by 4.

00:00:30.480 --> 00:00:32.900
So this divided
by 4, this divided

00:00:32.900 --> 00:00:35.710
by 4, that divided by 4,
and the 0 divided by 4.

00:00:35.710 --> 00:00:37.720
Just dividing both sides by 4.

00:00:37.720 --> 00:00:42.050
So this will simplify
to x squared plus 10x.

00:00:42.050 --> 00:00:44.892
And I can obviously
do that, because

00:00:44.892 --> 00:00:46.850
as long as whatever I do
to the left hand side,

00:00:46.850 --> 00:00:48.440
I also do the right
hand side, that

00:00:48.440 --> 00:00:51.570
will make the equality
continue to be valid.

00:00:51.570 --> 00:00:53.030
So that's why I can do that.

00:00:53.030 --> 00:00:55.280
So 40 divided by 4 is 10x.

00:00:55.280 --> 00:00:58.440
And then 300 divided
by 4 is what?

00:00:58.440 --> 00:01:00.090
That is 75.

00:01:00.090 --> 00:01:03.700
Let me verify that.

00:01:03.700 --> 00:01:05.239
4 goes into 30 seven times.

00:01:05.239 --> 00:01:07.100
7 times 4 is 28.

00:01:07.100 --> 00:01:08.740
You subtract, you
get a remainder of 2.

00:01:08.740 --> 00:01:09.660
Bring down the 0.

00:01:09.660 --> 00:01:11.720
4 goes into 20 five times.

00:01:11.720 --> 00:01:13.430
5 times 4 is 20.

00:01:13.430 --> 00:01:14.580
Subtract zero.

00:01:14.580 --> 00:01:16.340
So it goes 75 times.

00:01:16.340 --> 00:01:21.520
This is minus 75 is equal to 0.

00:01:21.520 --> 00:01:24.340
And right when you look at
this, just the way it's written,

00:01:24.340 --> 00:01:26.860
you might try to factor
this in some way.

00:01:26.860 --> 00:01:29.370
But it's pretty clear this
is not a complete square,

00:01:29.370 --> 00:01:31.347
or this is not a perfect
square trinomial.

00:01:31.347 --> 00:01:33.180
Because if you look at
this term right here,

00:01:33.180 --> 00:01:36.450
this 10, half of this 10 is 5.

00:01:36.450 --> 00:01:40.050
And 5 squared is not 75.

00:01:40.050 --> 00:01:41.490
So this is not a perfect square.

00:01:41.490 --> 00:01:43.957
So what we want to
do is somehow turn

00:01:43.957 --> 00:01:45.540
whatever we have on
the left hand side

00:01:45.540 --> 00:01:47.310
into a perfect square.

00:01:47.310 --> 00:01:50.060
And I'm going to start out by
kind of getting this 75 out

00:01:50.060 --> 00:01:50.560
of the way.

00:01:50.560 --> 00:01:52.059
You'll sometimes
see it where people

00:01:52.059 --> 00:01:53.786
leave the 75 on
the left hand side.

00:01:53.786 --> 00:01:55.410
I'm going to put on
the right hand side

00:01:55.410 --> 00:01:57.630
just so it kind of clears
things up a little bit.

00:01:57.630 --> 00:02:00.810
So let's add 75 to both
sides to get rid of the 75

00:02:00.810 --> 00:02:03.800
from the left hand
side of the equation.

00:02:03.800 --> 00:02:08.610
And so we get x squared plus
10x, and then negative 75

00:02:08.610 --> 00:02:09.229
plus 75.

00:02:09.229 --> 00:02:10.345
Those guys cancel out.

00:02:10.345 --> 00:02:11.970
And I'm going to
leave some space here,

00:02:11.970 --> 00:02:13.678
because we're going
to add something here

00:02:13.678 --> 00:02:16.580
to complete the square
that is equal to 75.

00:02:16.580 --> 00:02:19.970
So all I did is add 75 to
both sides of this equation.

00:02:19.970 --> 00:02:21.880
Now, in this step,
this is really

00:02:21.880 --> 00:02:23.860
the meat of
completing the square.

00:02:23.860 --> 00:02:27.000
I want to add something to
both sides of this equation.

00:02:27.000 --> 00:02:29.550
I can't add to only one
side of the equation.

00:02:29.550 --> 00:02:32.240
So I want to add something to
both sides of this equation

00:02:32.240 --> 00:02:35.370
so that this left hand side
becomes a perfect square.

00:02:35.370 --> 00:02:38.410
And the way we can do that,
and saw this in the last video

00:02:38.410 --> 00:02:41.040
where we constructed a
perfect square trinomial,

00:02:41.040 --> 00:02:43.660
is that this last
term-- or I should

00:02:43.660 --> 00:02:46.110
say, what we see on
the left hand side, not

00:02:46.110 --> 00:02:48.720
the last term, this expression
on the left hand side,

00:02:48.720 --> 00:02:53.210
it will be a perfect square if
we have a constant term that

00:02:53.210 --> 00:02:58.050
is the square of half of the
coefficient on the first degree

00:02:58.050 --> 00:02:58.780
term.

00:02:58.780 --> 00:03:00.900
So the coefficient here is 10.

00:03:00.900 --> 00:03:02.830
Half of 10 is 5.

00:03:02.830 --> 00:03:05.200
5 squared is 25.

00:03:05.200 --> 00:03:07.574
So I'm going to add 25
to the left hand side.

00:03:07.574 --> 00:03:09.990
And of course, in order to
maintain the equality, anything

00:03:09.990 --> 00:03:12.430
I do the left hand
side, I also have

00:03:12.430 --> 00:03:15.435
to do to the right hand side.

00:03:15.435 --> 00:03:17.310
And now we see that this
is a perfect square.

00:03:17.310 --> 00:03:19.940
We say, hey, what two numbers
if I add them I get 10

00:03:19.940 --> 00:03:21.920
and when I multiply
them I get 25?

00:03:21.920 --> 00:03:23.570
Well, that's 5 and 5.

00:03:23.570 --> 00:03:26.380
So when we factor this, what
we see on the left hand side

00:03:26.380 --> 00:03:30.150
simplifies to, this
is x plus 5 squared.

00:03:30.150 --> 00:03:33.250
x plus 5 times x plus 5.

00:03:33.250 --> 00:03:36.110
And you can look at
the videos on factoring

00:03:36.110 --> 00:03:38.312
if you find that confusing.

00:03:38.312 --> 00:03:39.770
Or you could look
at the last video

00:03:39.770 --> 00:03:42.477
on constructing perfect
square trinomials.

00:03:42.477 --> 00:03:44.060
I encourage you to
square this and see

00:03:44.060 --> 00:03:46.060
that you get exactly this.

00:03:46.060 --> 00:03:51.060
And this will be equal to 75
plus 25, which is equal to 100.

00:03:51.060 --> 00:03:53.290
And so now we're saying
that something squared

00:03:53.290 --> 00:03:55.110
is equal to 100.

00:03:55.110 --> 00:03:57.100
So really, this is
something right over here--

00:03:57.100 --> 00:03:59.100
if I say something squared
is equal to 100, that

00:03:59.100 --> 00:04:02.500
means that that something is
one of the square roots of 100.

00:04:02.500 --> 00:04:04.630
And we know that 100
has two square roots.

00:04:04.630 --> 00:04:07.840
It has positive 10 and
it has negative 10.

00:04:07.840 --> 00:04:10.050
So we could say that x
plus 5, the something

00:04:10.050 --> 00:04:12.010
that we were
squaring, that must be

00:04:12.010 --> 00:04:14.220
one of the square roots of 100.

00:04:14.220 --> 00:04:16.540
So that must be equal to the
plus or minus square root

00:04:16.540 --> 00:04:20.293
of 100, or plus or minus 10.

00:04:20.293 --> 00:04:21.459
Or we could separate it out.

00:04:21.459 --> 00:04:25.250
We could say that x
plus 5 is equal to 10,

00:04:25.250 --> 00:04:30.340
or x plus 5 is equal
to negative 10.

00:04:30.340 --> 00:04:32.390
On this side right
here, I can just

00:04:32.390 --> 00:04:35.010
subtract 5 from both
sides of this equation

00:04:35.010 --> 00:04:36.990
and I would get-- I'll
just write it out.

00:04:36.990 --> 00:04:41.790
Subtracting 5 from both
sides, I get x is equal to 5.

00:04:41.790 --> 00:04:43.240
And over here, I
could subtract 5

00:04:43.240 --> 00:04:47.810
from both sides again-- I
subtracted 5 in both cases--

00:04:47.810 --> 00:04:52.330
subtract 5 again and I can
get x is equal to negative 15.

00:04:52.330 --> 00:04:56.530
So those are my two
solutions that I

00:04:56.530 --> 00:04:58.320
got to solve this equation.

00:04:58.320 --> 00:05:02.230
We can verify that they actually
work, and I'll do that in blue.

00:05:02.230 --> 00:05:05.800
So let's try with 5.

00:05:05.800 --> 00:05:06.990
I'll just do one of them.

00:05:06.990 --> 00:05:08.850
I'll leave the
other one for you.

00:05:08.850 --> 00:05:11.450
I'll leave the other one for
you to verify that it works.

00:05:11.450 --> 00:05:15.180
So 4 times x squared.

00:05:15.180 --> 00:05:23.330
So 4 times 25 plus
40 times 5 minus 300

00:05:23.330 --> 00:05:24.860
needs to be equal to 0.

00:05:24.860 --> 00:05:26.762
4 times 25 is 100.

00:05:26.762 --> 00:05:28.920
40 times 5 is 200.

00:05:28.920 --> 00:05:30.570
We're going to
subtract that 300.

00:05:30.570 --> 00:05:34.770
100 plus 200 minus 300,
that definitely equals 0.

00:05:34.770 --> 00:05:36.310
So x equals 5 worked.

00:05:36.310 --> 00:05:39.330
And I think you'll find that
x equals negative 15 will also

00:05:39.330 --> 00:05:44.120
work when you substitute it
into this right over here.


Example 5: Using the quadratic formula

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.700
We're asked to solve
the quadratic equation,

00:00:02.700 --> 00:00:07.620
negative 3x squared plus
10x minus 3 is equal to 0.

00:00:07.620 --> 00:00:09.560
And it's already written
in standard form.

00:00:09.560 --> 00:00:11.060
And there's many
ways to solve this.

00:00:11.060 --> 00:00:13.597
But in particular, all solve
it using the quadratic formula.

00:00:13.597 --> 00:00:14.680
So let me just rewrite it.

00:00:14.680 --> 00:00:19.435
We have negative 3x squared
plus 10x minus 3 is equal to 0.

00:00:19.435 --> 00:00:20.810
And actually, I'll
solve it twice

00:00:20.810 --> 00:00:22.720
using the quadratic
formula to show you

00:00:22.720 --> 00:00:25.350
that as long as we manipulated
this in the valid way,

00:00:25.350 --> 00:00:26.850
the quadratic
formula will give us

00:00:26.850 --> 00:00:29.900
the exact same roots or
the exact same solutions

00:00:29.900 --> 00:00:31.370
to this equation.

00:00:31.370 --> 00:00:34.322
So in this form right over
here, what are our ABCs?

00:00:34.322 --> 00:00:36.780
Let's just remind ourselves
what the quadratic formula even

00:00:36.780 --> 00:00:37.450
is actually.

00:00:37.450 --> 00:00:38.730
That's a good place to start.

00:00:38.730 --> 00:00:40.640
The quadratic formula
tells us that if we

00:00:40.640 --> 00:00:43.170
have a quadratic
equation in the form ax

00:00:43.170 --> 00:00:48.320
squared plus bx plus c is equal
to 0, so in standard form,

00:00:48.320 --> 00:00:52.360
then the roots of this are
x are equal to negative b

00:00:52.360 --> 00:00:55.630
plus or minus the
square root of b

00:00:55.630 --> 00:01:02.430
squared minus 4ac,
all of that over 2a.

00:01:02.430 --> 00:01:05.239
And this is derived from
completing the square

00:01:05.239 --> 00:01:06.410
in a general way.

00:01:06.410 --> 00:01:09.914
So it's no magic here, and I've
derived it in other videos.

00:01:09.914 --> 00:01:11.330
But this is the
quadratic formula.

00:01:11.330 --> 00:01:12.910
This is actually giving
you two solutions,

00:01:12.910 --> 00:01:14.890
because you have the
positive square root here

00:01:14.890 --> 00:01:16.680
and the negative square root.

00:01:16.680 --> 00:01:19.750
So let's apply it here in the
case where-- in this case,

00:01:19.750 --> 00:01:26.890
a is equal to negative
3, b is equal to 10,

00:01:26.890 --> 00:01:31.000
and c is equal to negative 3.

00:01:31.000 --> 00:01:33.240
So applying the quadratic
formula right here,

00:01:33.240 --> 00:01:36.526
we get our solutions to be
x is equal to negative b.

00:01:36.526 --> 00:01:37.530
b is 10.

00:01:37.530 --> 00:01:42.800
So negative b is negative 10
plus or minus the square root

00:01:42.800 --> 00:01:44.690
of b squared.

00:01:44.690 --> 00:01:45.460
b is 10.

00:01:45.460 --> 00:01:50.650
So b squared is 100
minus 4 times a times c.

00:01:50.650 --> 00:01:54.387
So minus 4 times negative
3 times negative 3.

00:01:54.387 --> 00:01:55.470
Let me just write it down.

00:01:55.470 --> 00:01:58.930
Minus 4 times negative
3 times negative 3.

00:01:58.930 --> 00:02:01.100
All of that's under
the radical sign.

00:02:01.100 --> 00:02:02.860
And then all of that is over 2a.

00:02:02.860 --> 00:02:05.540
So 2 times a is negative 6.

00:02:05.540 --> 00:02:07.660
So this is going to be
equal to negative 10

00:02:07.660 --> 00:02:14.710
plus or minus the square root
of 100 minus-- negative 3 times

00:02:14.710 --> 00:02:16.080
negative 3 is positive 9.

00:02:16.080 --> 00:02:18.280
Positive 9 times
4 is positive 36.

00:02:18.280 --> 00:02:19.530
We have a minus sign out here.

00:02:19.530 --> 00:02:21.610
So minus 36.

00:02:21.610 --> 00:02:23.860
All of that over negative 6.

00:02:23.860 --> 00:02:27.310
This is equal to
100 minus 36 is 64.

00:02:27.310 --> 00:02:31.830
So negative 10 plus or
minus the square root of 64.

00:02:31.830 --> 00:02:33.900
All of that over negative 6.

00:02:33.900 --> 00:02:35.939
The principal square
root of 64 is 8.

00:02:35.939 --> 00:02:38.230
But we're taking the positive
and negative square root.

00:02:38.230 --> 00:02:43.940
So this is negative 10 plus
or minus 8 over negative 6.

00:02:43.940 --> 00:02:45.620
So if we take the
positive version,

00:02:45.620 --> 00:02:48.210
we say x could be
equal to-- negative 10

00:02:48.210 --> 00:02:52.560
plus 8 is negative
2 over negative 6.

00:02:52.560 --> 00:02:54.570
So that was taking
the plus version.

00:02:54.570 --> 00:02:56.330
That's this right over here.

00:02:56.330 --> 00:02:58.940
And negative 2 over
negative 6 is equal to 1/3.

00:02:58.940 --> 00:03:00.840
If we take the
negative square root,

00:03:00.840 --> 00:03:04.610
negative 10 minus 8-- So let's
take negative 10 minus 8.

00:03:04.610 --> 00:03:08.030
That would be x is equal
to-- negative 10 minus 8

00:03:08.030 --> 00:03:10.070
is negative 18.

00:03:10.070 --> 00:03:13.490
And that's going to
be over negative 6.

00:03:13.490 --> 00:03:16.930
Negative 18 divided by
negative 6 is positive 3.

00:03:16.930 --> 00:03:19.380
So the two roots for
this quadratic equation

00:03:19.380 --> 00:03:22.492
are positive 1/3 and positive 3.

00:03:22.492 --> 00:03:24.700
And I want to show you the
we'll get the same answer,

00:03:24.700 --> 00:03:25.740
even if we manipulate this.

00:03:25.740 --> 00:03:27.280
Some people might
not like the fact

00:03:27.280 --> 00:03:30.120
that our first coefficient
here is a negative 3.

00:03:30.120 --> 00:03:31.876
Maybe they want a positive 3.

00:03:31.876 --> 00:03:33.250
So to get rid of
that negative 3,

00:03:33.250 --> 00:03:37.010
they can multiply both sides of
this equation times negative 1.

00:03:37.010 --> 00:03:39.270
And then if you did
that, you would get 3x

00:03:39.270 --> 00:03:45.000
squared minus 10x plus 3 is
equal to 0 times negative 1,

00:03:45.000 --> 00:03:46.900
which is still equal to 0.

00:03:46.900 --> 00:03:51.990
So in this case, a is equal to
3, b is equal to negative 10,

00:03:51.990 --> 00:03:54.430
and c is equal to 3 again.

00:03:54.430 --> 00:03:56.400
And we could apply
the quadratic formula.

00:03:56.400 --> 00:04:00.560
We get x is equal to
negative b. b is negative 10.

00:04:00.560 --> 00:04:02.460
So negative negative
10 is positive

00:04:02.460 --> 00:04:05.300
10, plus or minus
the square root

00:04:05.300 --> 00:04:07.950
of b squared, which is
negative 10 squared,

00:04:07.950 --> 00:04:12.110
which is 100, minus
4 times a times c.

00:04:12.110 --> 00:04:16.070
a times c is 9 times 4 is 36.

00:04:16.070 --> 00:04:17.980
So minus 36.

00:04:17.980 --> 00:04:20.250
All of that over 2 times a.

00:04:20.250 --> 00:04:21.990
All of that over 6.

00:04:21.990 --> 00:04:28.030
So this is equal to 10 plus or
minus the square root of 64,

00:04:28.030 --> 00:04:30.660
or really that's
just going to be 8.

00:04:30.660 --> 00:04:32.210
All of that over 6.

00:04:32.210 --> 00:04:36.420
If we add 8 here, we get
10 plus 8 is 18 over 6.

00:04:36.420 --> 00:04:38.310
We get x could be equal to 3.

00:04:38.310 --> 00:04:41.300
Or if we take the negative
square root or the negative 8

00:04:41.300 --> 00:04:43.100
here, 10 minus 8 is 2.

00:04:43.100 --> 00:04:46.280
2 over 6 is 1/3.

00:04:46.280 --> 00:04:50.300
So once again, you get
the exact same solutions.


Example: Quadratics in standard form

WEBVTT
Kind: captions
Language: en

00:00:00.467 --> 00:00:10.800
Rewrite the equation 6x^2 + 3 = 2x - 6 in standard form and identify a, b, and c.

00:00:10.800 --> 00:00:19.585
So standard form for a quadratic equation is ax squared plus bx plus c is equal to zero.

00:00:19.585 --> 00:00:23.333
So essentially you wanna get all of the terms on the left-hand side,

00:00:23.333 --> 00:00:29.585
and then we want to write them so that we have the x terms...where their exponents are in decreasing order.

00:00:29.585 --> 00:00:33.502
So we have the x squared term and then the x term and then we have the constant term.

00:00:33.502 --> 00:00:35.477
So let's try to do this over here.

00:00:35.477 --> 00:00:37.677
So let me rewrite our original equation.

00:00:37.677 --> 00:00:43.190
We have 6x squared plus 3 is equal to 2x minus 6.

00:00:43.190 --> 00:00:45.692
So essentially we wanna get everything on the left-hand side.

00:00:45.708 --> 00:00:51.333
so I could subtract 2x from both sides, so I could subtract 2x from both sides,

00:00:51.333 --> 00:00:53.815
so let me just...I'll take one step at a time.

00:00:53.815 --> 00:00:56.462
So I can subtract 2x from both sides.

00:00:56.462 --> 00:01:01.015
And then I'll get...and I'm gonna write it in descending order for the exponents on x.

00:01:01.031 --> 00:01:04.302
So the highest exponent is x squared. So I'll write that first.

00:01:04.302 --> 00:01:10.671
6x squared, and then we have minus 2x, and then we have plus 3 is equal to...

00:01:10.671 --> 00:01:14.492
the 2 'x's on the right cancel out...equal to negative 6.

00:01:14.492 --> 00:01:20.000
And now, to get rid of this negative 6 on the right-hand side, we can add 6 to both sides.

00:01:20.000 --> 00:01:23.318
So let's add 6 to both sides...

00:01:23.318 --> 00:01:32.431
...and then this simplifies to 6x squared, minus 2x, plus nine is equal to...zero.

00:01:32.431 --> 00:01:35.877
So let's make sure we're already in standard form.

00:01:35.877 --> 00:01:40.775
All of our terms, our non-zero terms are on the left-hand side, we've done that.

00:01:40.775 --> 00:01:43.010
We have a zero on the right-hand side, we've done that.

00:01:43.010 --> 00:01:48.559
And, we have the x squared term first, then the x to the first power term, then the constant term.

00:01:48.559 --> 00:01:51.323
x squared, then x to the first, then the constant term.

00:01:51.323 --> 00:01:53.575
So we are in standard form.

00:01:53.575 --> 00:01:58.252
And so we can say that a is equal to 6, a is equal to 6.

00:01:58.252 --> 00:02:04.031
We could say that b is equal to, and this is key, it's not just the 2, it's the negative 2.

00:02:04.031 --> 00:02:10.087
B is equal to negative 2, 'cause notice this says plus bx, but over here we have minus 2x.

00:02:10.087 --> 00:02:14.600
So the b is a negative 2 here. B is negative 2.

00:02:14.600 --> 00:02:21.229
And then c, c is going to be, c is going to be 9.


Dividing complex numbers

WEBVTT
Kind: captions
Language: en

00:00:00.113 --> 00:00:02.179
We're asked to divide.

00:00:02.179 --> 00:00:06.362
And we're dividing six plus three i by seven minus 5i.

00:00:06.362 --> 00:00:10.200
And in particular, when I divide this, I want to get another complex number.

00:00:10.200 --> 00:00:15.023
So I want to get some real number plus some imaginary number,

00:00:15.023 --> 00:00:16.600
so some multiple of i's.

00:00:16.600 --> 00:00:18.067
So let's think about how we can do this.

00:00:18.067 --> 00:00:20.108
Well, division is the same thing --

00:00:20.108 --> 00:00:26.271
and we rewrite this as six plus three i over seven minus five i.

00:00:26.271 --> 00:00:27.971
These are clearly equivalent;

00:00:27.987 --> 00:00:33.200
dividing by something is the same thing as a rational expression where that something is in the denominator,

00:00:33.200 --> 00:00:34.400
right over here.

00:00:34.400 --> 00:00:36.532
And so how do we simplify this?

00:00:36.532 --> 00:00:42.469
Well, we have a tool in our toolkit that can make sure that we don't have an imaginary or complex number in the denominator.

00:00:42.469 --> 00:00:45.267
And that's the complex conjugate.

00:00:45.267 --> 00:00:48.483
If we multiply both the numerator and the denominator of this expression

00:00:48.483 --> 00:00:51.067
by the complex conjugate of the denominator,

00:00:51.067 --> 00:00:55.933
then we will have a real number in the denominator.

00:00:55.933 --> 00:00:57.067
So let's do that.

00:00:57.067 --> 00:01:00.800
Let's multiply the numerator and the denominator by the conjugate of this.

00:01:00.800 --> 00:01:07.400
So seven PLUS five i. Seven plus five i is the complex conjegate of seven minus five i.

00:01:07.400 --> 00:01:12.067
So we're going to multiply it by seven plus five i over seven plus five i.

00:01:12.067 --> 00:01:16.400
And anything divided by itself is going to be one

00:01:16.400 --> 00:01:19.000
(assuming you're not dealing with zero; zero over zero is undefined).

00:01:19.000 --> 00:01:22.152
But seven plus five i over seven plus five i is one.

00:01:22.152 --> 00:01:24.800
So we're not changing the value of this.

00:01:24.800 --> 00:01:29.067
But what this does is it allows us to get rid of the imaginary part in the denominator.

00:01:29.067 --> 00:01:31.333
So let's multiply this out.

00:01:31.333 --> 00:01:32.578
Our numerator --

00:01:32.578 --> 00:01:38.467
we just have to multiply every part of this complex number times every part of this complex number.

00:01:38.467 --> 00:01:43.000
You can think of it as FOIL if you like; we're really just doing the distributive property twice.

00:01:43.000 --> 00:01:49.667
We have six times seven, which is forty two.

00:01:49.667 --> 00:01:58.867
And then we have six times five i, which is thirty i. So plus thirty i.

00:01:58.867 --> 00:02:09.710
And then we have three i times seven, so that's plus twenty-one i.

00:02:09.741 --> 00:02:12.375
And then finally we have three i times five i.

00:02:12.375 --> 00:02:14.351
Three times five is fifteen.

00:02:14.351 --> 00:02:17.439
But we have i times i, or i squared, which is negative one.

00:02:17.439 --> 00:02:22.369
So it would be fifteen times negative one, or minus 15.

00:02:22.369 --> 00:02:24.159
So that's our numerator.

00:02:24.159 --> 00:02:28.282
And then our denomenator is going to be --

00:02:28.282 --> 00:02:33.067
Well, we have a plus b times a minus b. (You could think of it that way.

00:02:33.067 --> 00:02:36.333
Or we could just do what we did up here. Actually, let's just do what we did up here

00:02:36.333 --> 00:02:39.242
so you don't have to remember that difference of squares pattern and all that.)

00:02:39.242 --> 00:02:42.000
Seven times seven is forty-nine.

00:02:42.000 --> 00:02:45.667
Let's think of it in the FOIL way, if that is helpful for you.

00:02:45.667 --> 00:02:52.153
So first we did the 7X7. And we can do the outer terms. 7 X 5i is +35i.

00:02:52.153 --> 00:02:57.133
Then we can do the inner terms. -5i X 7 is -35i.

00:02:57.133 --> 00:02:59.490
These two are going to cancel out.

00:02:59.490 --> 00:03:04.250
And then -5i X 5i is -25i^2 ("negative twenty five i squared").

00:03:04.250 --> 00:03:09.893
-25i^2 is the same thing as -25 times -1, so that is +25.

00:03:09.893 --> 00:03:11.333
Now let's simplify it.

00:03:11.333 --> 00:03:14.235
These guys down here cancel out.

00:03:14.235 --> 00:03:19.831
Our denominator simplifies to 49 + 25 is 74.

00:03:19.831 --> 00:03:23.000
And our numerator: we can add the real parts,

00:03:23.000 --> 00:03:27.238
so we have a 42 and a -15.

00:03:27.238 --> 00:03:32.933
Let's see: 42 - 5 would be 37, minus another 10 would be 27.

00:03:32.933 --> 00:03:35.829
So that is 27.

00:03:35.829 --> 00:03:42.494
And then we're going to add our 30i, plus the 21i --

00:03:42.494 --> 00:03:50.210
so 30 of something plus another 21 of that same something is going to be 51 of that something,

00:03:50.210 --> 00:03:55.148
in this case that something is the imaginary unit; it is i.

00:03:55.148 --> 00:03:57.200
(We'll do this in magenta; o, I guess that's orange.)

00:03:57.200 --> 00:04:00.652
So this is +51i.

00:04:00.652 --> 00:04:05.467
And I want to write it in the form of "a+bi," the traditional complex number form.

00:04:05.467 --> 00:04:12.006
So this right over here is the same thing as 27/74,

00:04:12.006 --> 00:04:32.347
27/74 + 51/74 times i. (I'm going to write that i in that same orange color.)

00:04:32.347 --> 00:04:34.359
And we are done.

00:04:34.359 --> 00:04:37.084
We have a real part, and we have an imaginary part.

00:04:37.084 --> 00:04:41.000
If this last step confuses you a little bit, just remember, if it's helpful for you

00:04:41.000 --> 00:04:42.862
that this is the same thing.

00:04:42.862 --> 00:04:45.267
We're essentially multiplying both of these terms times 1/74.

00:04:45.267 --> 00:04:47.533
We're dividing both of these terms by 74.

00:04:47.533 --> 00:04:52.267
And we're distributing the 1/74 times both of these, I guess is one way to think about it.

00:04:52.267 --> 00:04:54.404
And that's how we got this thing over here,

00:04:54.404 --> 00:04:57.723
where we have a nice real part and a nice imaginary part.


Example: Solving simple quadratic

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:02.780
We're asked to
solve the equation

00:00:02.780 --> 00:00:07.550
2x squared plus
3 is equal to 75.

00:00:07.550 --> 00:00:09.190
So in this situation,
it looks like we

00:00:09.190 --> 00:00:12.890
might be able to isolate
the x squared pretty simply.

00:00:12.890 --> 00:00:15.150
Because there's only one
term that involves an x here.

00:00:15.150 --> 00:00:16.950
It's only this x squared term.

00:00:16.950 --> 00:00:18.277
So let's try to do that.

00:00:18.277 --> 00:00:19.360
So let me just rewrite it.

00:00:19.360 --> 00:00:24.250
We have 2x squared
plus 3 is equal to 75.

00:00:24.250 --> 00:00:26.610
And we're going to try to
isolate this x squared over

00:00:26.610 --> 00:00:26.820
here.

00:00:26.820 --> 00:00:29.153
And the best way to do that,
or at least the first step,

00:00:29.153 --> 00:00:32.960
would be to subtract 3 from
both sides of this equation.

00:00:32.960 --> 00:00:35.270
So let's subtract
3 from both sides.

00:00:35.270 --> 00:00:38.580
The left hand side, we're
just left with 2x squared.

00:00:38.580 --> 00:00:41.690
That was the whole point of
subtracting 3 from both sides.

00:00:41.690 --> 00:00:46.260
And on the right hand
side, 75 minus 3 is 72.

00:00:46.260 --> 00:00:48.070
Now, I want to isolate
this x squared.

00:00:48.070 --> 00:00:49.780
I have a 2x squared here.

00:00:49.780 --> 00:00:51.610
So I could have just
an x squared here

00:00:51.610 --> 00:00:54.265
if I divide this side or
really both sides by 2.

00:00:54.265 --> 00:00:56.640
Anything I do to one side, I
have to do to the other side

00:00:56.640 --> 00:00:59.370
if I want to maintain
the equality.

00:00:59.370 --> 00:01:01.820
So the left side, just
becomes x squared.

00:01:01.820 --> 00:01:06.540
And the right hand side
is 72 divided by 2 is 36.

00:01:06.540 --> 00:01:09.290
So we're left with x
squared is equal to 36.

00:01:09.290 --> 00:01:13.280
And then to solve for x,
we can take the positive,

00:01:13.280 --> 00:01:16.720
the plus or minus square
root of both sides.

00:01:16.720 --> 00:01:21.190
So we could say the plus or--
let me write it this way--

00:01:21.190 --> 00:01:23.430
If we take the square
root of both sides,

00:01:23.430 --> 00:01:28.230
we would get x is equal to
the plus or minus square root

00:01:28.230 --> 00:01:33.316
of 36, which is equal
to plus or minus 6.

00:01:33.316 --> 00:01:34.940
Let me just write
that on another line.

00:01:34.940 --> 00:01:38.010
So x is equal to
plus or minus 6.

00:01:38.010 --> 00:01:40.877
And remember here, if something
squared is equal to 36,

00:01:40.877 --> 00:01:42.710
that something could
be the negative version

00:01:42.710 --> 00:01:43.900
or the positive version.

00:01:43.900 --> 00:01:45.400
It could be the
principal root or it

00:01:45.400 --> 00:01:46.940
could be the negative root.

00:01:46.940 --> 00:01:50.390
Both negative 6 squared is
36 and positive 6 squared

00:01:50.390 --> 00:01:52.910
is 36, so both of these work.

00:01:52.910 --> 00:01:55.160
And you could put them back
into the original equation

00:01:55.160 --> 00:01:56.370
to verify it.

00:01:56.370 --> 00:01:57.070
Let's do that.

00:01:57.070 --> 00:02:00.210
If you say 2 times
6 squared plus 3,

00:02:00.210 --> 00:02:05.040
that's 2 times 36, which
is 72 plus 3 is 75.

00:02:05.040 --> 00:02:05.647
So that works.

00:02:05.647 --> 00:02:07.230
If you put negative
6 in there, you're

00:02:07.230 --> 00:02:08.688
going to get the
exact same result.

00:02:08.688 --> 00:02:11.720
Because negative 6
squared is also 36.

00:02:11.720 --> 00:02:16.820
2 times 36 is 72 plus 3 is 75.


Multiplying complex numbers

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.520
We're asked to multiply the
complex number 1 minus 3i times

00:00:04.520 --> 00:00:07.250
the complex number 2 plus 5i.

00:00:07.250 --> 00:00:09.170
And the general idea
here is you can multiply

00:00:09.170 --> 00:00:13.230
these complex numbers like
you would have multiplied

00:00:13.230 --> 00:00:14.802
any traditional binomial.

00:00:14.802 --> 00:00:17.010
You just have to remember
that this isn't a variable.

00:00:17.010 --> 00:00:21.880
This is the imaginary
unit i, or it's just i.

00:00:21.880 --> 00:00:23.569
But we could do
that in two ways.

00:00:23.569 --> 00:00:25.610
We could just do the
distributive property twice,

00:00:25.610 --> 00:00:29.100
which I like a little bit more,
just because you're doing it

00:00:29.100 --> 00:00:30.325
from a fundamental principle.

00:00:30.325 --> 00:00:31.340
It's nothing new.

00:00:31.340 --> 00:00:33.320
Or you could use
FOIL, which you also

00:00:33.320 --> 00:00:35.970
used when you first
multiplied binomials.

00:00:35.970 --> 00:00:37.570
And I'll do it both ways.

00:00:37.570 --> 00:00:42.310
So this is just a
number, 1 minus 3i.

00:00:42.310 --> 00:00:45.380
And so we can distribute
it over the two numbers

00:00:45.380 --> 00:00:46.534
inside of this expression.

00:00:46.534 --> 00:00:48.950
So when we're multiplying it
times this entire expression,

00:00:48.950 --> 00:00:53.850
we can multiply 1 minus 3i
times 2 and 1 minus 3i times 5i.

00:00:53.850 --> 00:00:54.890
So let's do that.

00:00:54.890 --> 00:00:59.200
So this can be rewritten
as 1 minus 3i times

00:00:59.200 --> 00:01:09.860
2-- I'll write the 2 out front--
plus 1 minus 3i times 5i.

00:01:12.850 --> 00:01:15.070
All I did is the
distributive property here.

00:01:15.070 --> 00:01:18.430
All I said is, look, if
I have a times b plus c,

00:01:18.430 --> 00:01:21.650
this is the same
thing as ab plus ac.

00:01:21.650 --> 00:01:25.970
I just distributed the
a on the b or the c.

00:01:25.970 --> 00:01:29.716
I distributed the 1 minus
3i on the 2 and the 5i.

00:01:29.716 --> 00:01:30.840
And then I can do it again.

00:01:30.840 --> 00:01:33.290
I have a 2 now times 1 minus 3i.

00:01:33.290 --> 00:01:34.170
I can distribute it.

00:01:34.170 --> 00:01:37.660
2 times 1 is 2.

00:01:37.660 --> 00:01:41.607
2 times negative
3i is negative 6i.

00:01:41.607 --> 00:01:42.940
And over here, I'll do it again.

00:01:42.940 --> 00:01:45.580
5i times 1-- so it's plus.

00:01:45.580 --> 00:01:48.130
5i times 1 is 5i.

00:01:48.130 --> 00:01:50.610
And then 5i times
negative 3i-- so

00:01:50.610 --> 00:01:58.490
let's be careful here-- 5 times
negative 3 is negative 15.

00:01:58.490 --> 00:02:01.329
And then I have an i times an i.

00:02:01.329 --> 00:02:02.370
Let me do this over here.

00:02:02.370 --> 00:02:07.010
5i times negative 3i--
this is the same thing

00:02:07.010 --> 00:02:12.710
as 5 times negative
3 times i times i.

00:02:12.710 --> 00:02:16.450
So the 5 times negative
3 is negative 15.

00:02:16.450 --> 00:02:20.830
And then we have i times
i, which is i squared.

00:02:20.830 --> 00:02:23.300
Now, we know what i squared is.

00:02:23.300 --> 00:02:27.030
By definition, i
squared is negative 1.

00:02:27.030 --> 00:02:29.990
i squared, by definition,
is negative 1.

00:02:29.990 --> 00:02:31.780
So you have negative
15 times negative 1.

00:02:31.780 --> 00:02:35.260
Well that's the same
thing as positive 15.

00:02:35.260 --> 00:02:40.440
So this can be rewritten
as 2 minus 6i plus 5i.

00:02:40.440 --> 00:02:44.280
Negative 15 times
negative 1 is positive 15.

00:02:44.280 --> 00:02:46.700
Now we can add the real parts.

00:02:46.700 --> 00:02:50.530
We have a 2, and we
have a positive 15.

00:02:50.530 --> 00:02:52.170
So 2 plus 15.

00:02:52.170 --> 00:02:55.110
And we can add the
imaginary parts.

00:02:55.110 --> 00:02:56.950
We have a negative 6.

00:02:56.950 --> 00:03:00.460
So we have a negative 6, or
a negative 6i, I should say.

00:03:00.460 --> 00:03:01.780
And then we have plus 5i.

00:03:05.110 --> 00:03:08.970
And 2 plus 15 is 17.

00:03:08.970 --> 00:03:10.830
And if I have negative
six of something

00:03:10.830 --> 00:03:12.664
plus five of that
something, what do I have?

00:03:12.664 --> 00:03:14.121
Or if I have five
of that something

00:03:14.121 --> 00:03:15.900
and I take six of
that something away,

00:03:15.900 --> 00:03:18.520
then I have negative
one of that something.

00:03:18.520 --> 00:03:21.700
Negative 6i plus
5i is negative 1i,

00:03:21.700 --> 00:03:24.470
or I could just say minus i.

00:03:24.470 --> 00:03:26.020
So in this way,
I just multiplied

00:03:26.020 --> 00:03:29.130
these two expressions or these
two complex numbers, really.

00:03:29.130 --> 00:03:32.300
I multiplied them just using
the distributive property twice.

00:03:32.300 --> 00:03:34.450
You could also do it using FOIL.

00:03:34.450 --> 00:03:36.730
And I'll do that
right now really fast.

00:03:36.730 --> 00:03:38.600
It is a little bit faster.

00:03:38.600 --> 00:03:40.340
But it's a little
bit mechanical.

00:03:40.340 --> 00:03:43.669
So you might forget why you're
doing it in the first place.

00:03:43.669 --> 00:03:46.210
But at the end of the day, you
are doing the same thing here.

00:03:46.210 --> 00:03:49.200
You're essentially multiplying
every term of this first number

00:03:49.200 --> 00:03:50.670
or every part of
this first number

00:03:50.670 --> 00:03:52.560
times every part of
the second number.

00:03:52.560 --> 00:03:54.730
And FOIL just makes sure
that we're doing it.

00:03:54.730 --> 00:03:56.710
And let me just
write FOIL out here,

00:03:56.710 --> 00:03:58.960
which I'm not a huge fan
of, but I'll do it just

00:03:58.960 --> 00:04:00.710
in case that's the way
you're learning it.

00:04:00.710 --> 00:04:04.470
So FOIL says, let's
multiply the first numbers.

00:04:04.470 --> 00:04:08.520
So that's going to
be the 1 times the 2.

00:04:08.520 --> 00:04:10.940
That is the F in FOIL.

00:04:10.940 --> 00:04:14.180
Then it says, let's multiply the
outer numbers times each other.

00:04:14.180 --> 00:04:19.640
So that's 1 times 5i,
so plus 1 times 5i.

00:04:19.640 --> 00:04:23.070
This is the O in FOIL,
the outer numbers.

00:04:23.070 --> 00:04:26.440
Then we do the inner
numbers, negative 3i times 2.

00:04:26.440 --> 00:04:30.710
So this is negative 3i times 2.

00:04:30.710 --> 00:04:32.430
Those are the inner numbers.

00:04:32.430 --> 00:04:35.660
And then we do the last
numbers, negative 3i times 5i.

00:04:39.350 --> 00:04:41.280
These are the last numbers.

00:04:41.280 --> 00:04:42.986
So that's all that
FOIL is telling us.

00:04:42.986 --> 00:04:44.610
It's just making sure
we're multiplying

00:04:44.610 --> 00:04:48.280
every part of this number times
every part of that number.

00:04:48.280 --> 00:04:52.160
And then when we simplify
it, 1 times 2 is 2.

00:04:52.160 --> 00:04:55.240
1 times 5i is 5i.

00:04:55.240 --> 00:04:58.420
Negative 3i times
2 is negative 6i.

00:04:58.420 --> 00:05:01.540
And negative 3i times
5i-- well, we already

00:05:01.540 --> 00:05:03.850
figured out what that was.

00:05:03.850 --> 00:05:07.110
Negative 3i times 5i
turns out to be 15.

00:05:07.110 --> 00:05:09.450
Negative 3 times
5 is negative 15.

00:05:09.450 --> 00:05:11.692
But i times i is negative 1.

00:05:11.692 --> 00:05:16.320
Negative 15 times
negative 1 is positive 15.

00:05:16.320 --> 00:05:19.920
Add the real parts, 2 plus 15.

00:05:19.920 --> 00:05:21.430
You get 17.

00:05:21.430 --> 00:05:23.530
Add the imaginary parts.

00:05:23.530 --> 00:05:26.130
You have 5i minus 6i.

00:05:26.130 --> 00:05:27.840
You get negative i.

00:05:27.840 --> 00:05:31.500
And once again, you get
the exact same answer.


Complex conjugates example

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.440
We're asked to find the
conjugate of the complex number

00:00:03.440 --> 00:00:05.440
7 minus 5i.

00:00:05.440 --> 00:00:07.690
And what you're going to
find in this video is finding

00:00:07.690 --> 00:00:11.280
the conjugate of a complex
number is shockingly easy.

00:00:11.280 --> 00:00:13.080
It's really the same
as this number--

00:00:13.080 --> 00:00:15.380
or I should be a little
bit more particular.

00:00:15.380 --> 00:00:16.595
It has the same real part.

00:00:16.595 --> 00:00:17.970
So the conjugate
of this is going

00:00:17.970 --> 00:00:19.880
to have the exact
same real part.

00:00:19.880 --> 00:00:22.580
But its imaginary part is going
to have the opposite sign.

00:00:22.580 --> 00:00:28.250
So instead of having a negative
5i, it will have a positive 5i.

00:00:28.250 --> 00:00:31.280
So that right there is
the complex conjugate

00:00:31.280 --> 00:00:33.670
of 7 minus 5i.

00:00:33.670 --> 00:00:36.370
And sometimes the notation
for doing that is you'll

00:00:36.370 --> 00:00:38.490
take 7 minus 5i.

00:00:38.490 --> 00:00:45.190
If you have 7 minus 5i, and you
put a line over it like that,

00:00:45.190 --> 00:00:48.940
that means I want the
conjugate of 7 minus 5i.

00:00:48.940 --> 00:00:51.410
And that will equal 7 plus 5i.

00:00:51.410 --> 00:00:54.390
Or sometimes someone
will write-- you'll see z

00:00:54.390 --> 00:00:57.420
is the variable that people
often use for complex numbers.

00:00:57.420 --> 00:01:04.650
If z is 7 minus 5i, then they'll
say the complex conjugate

00:01:04.650 --> 00:01:07.130
of z-- you put that
line over the z--

00:01:07.130 --> 00:01:10.100
is going to be 7 plus 5i.

00:01:10.100 --> 00:01:11.990
Now you're probably
saying, OK, fairly

00:01:11.990 --> 00:01:15.940
straightforward to find a
conjugate of a complex number.

00:01:15.940 --> 00:01:18.070
But what is it good for?

00:01:18.070 --> 00:01:20.720
And the simplest reason
or the most basic place

00:01:20.720 --> 00:01:25.820
where this is useful is when
you multiply any complex number

00:01:25.820 --> 00:01:29.280
times its conjugate, you're
going to get a real number.

00:01:29.280 --> 00:01:30.570
And I want to emphasize.

00:01:30.570 --> 00:01:33.060
This right here
is the conjugate.

00:01:33.060 --> 00:01:36.340
7 plus 5i is the
conjugate of 7 minus 5i.

00:01:36.340 --> 00:01:40.260
But 7 minus 5i is also the
conjugate of 7 plus 5i,

00:01:40.260 --> 00:01:41.200
for obvious reasons.

00:01:41.200 --> 00:01:42.825
If you started with
this and you change

00:01:42.825 --> 00:01:46.250
the sign of the imaginary
part, you would get 7 minus 5i.

00:01:46.250 --> 00:01:48.110
They're conjugates
of each other.

00:01:48.110 --> 00:01:51.160
But let me show you that when
I multiply complex conjugates

00:01:51.160 --> 00:01:52.690
that I get a real number.

00:01:52.690 --> 00:01:56.890
So let's multiply 7
minus 5i times 7 plus 5i.

00:01:56.890 --> 00:02:04.190
And I will do that in blue--
7 minus 5i times 7 plus 5i.

00:02:04.190 --> 00:02:06.440
And remember, whenever you
multiply these expressions,

00:02:06.440 --> 00:02:08.981
you really just have to multiply
every term times each other.

00:02:08.981 --> 00:02:10.880
You could do the
distributive property twice.

00:02:10.880 --> 00:02:13.920
You could do something like FOIL
to remind yourself to multiply

00:02:13.920 --> 00:02:17.470
every part of this complex
number times every part

00:02:17.470 --> 00:02:18.780
of this complex number.

00:02:18.780 --> 00:02:20.720
So let's just do
it any which way.

00:02:20.720 --> 00:02:24.370
So you'd have 7
times 7, which is 49.

00:02:24.370 --> 00:02:31.450
7 times 5i, which is 35i.

00:02:31.450 --> 00:02:35.580
Then you have negative 5i
times 7, which is negative 35i.

00:02:35.580 --> 00:02:37.870
You can see the imaginary
part is canceling out.

00:02:37.870 --> 00:02:41.490
Then you have negative
5i times positive 5i.

00:02:41.490 --> 00:02:45.010
Well, that's
negative 25i squared.

00:02:45.010 --> 00:02:47.350
And negative 25i squared--
remember, i squared

00:02:47.350 --> 00:02:48.540
is negative 1.

00:02:48.540 --> 00:02:51.470
So negative 25i squared--
let me write this down.

00:02:51.470 --> 00:02:58.500
Negative 5i times 5i is
negative 25 times i squared.

00:02:58.500 --> 00:03:00.110
i squared is negative 1.

00:03:00.110 --> 00:03:04.110
So negative 25 times
negative 1 is positive 25.

00:03:04.110 --> 00:03:06.390
And these two guys
over here cancel out.

00:03:06.390 --> 00:03:11.160
And we're just left with 49
plus 25-- let's see, 50 plus 25

00:03:11.160 --> 00:03:11.740
is 75.

00:03:11.740 --> 00:03:13.770
So this is just 74.

00:03:13.770 --> 00:03:16.259
So we are just left
with the real number 74.

00:03:16.259 --> 00:03:17.800
Another way to do
it-- you don't even

00:03:17.800 --> 00:03:19.430
have to do all this
distributive property.

00:03:19.430 --> 00:03:20.846
You might just
recognize that this

00:03:20.846 --> 00:03:25.950
looks like this is
something minus something

00:03:25.950 --> 00:03:28.190
times that same
something plus something.

00:03:28.190 --> 00:03:30.460
And we know this pattern
from our early algebra,

00:03:30.460 --> 00:03:37.560
that a plus b times a minus b
is equal to a squared minus b

00:03:37.560 --> 00:03:40.330
squared, is equal to a
difference of squares.

00:03:40.330 --> 00:03:42.570
And so in this case, a is 7.

00:03:42.570 --> 00:03:44.270
a squared is 49.

00:03:44.270 --> 00:03:47.950
And b, in this case, is 5i.

00:03:47.950 --> 00:03:50.900
b squared is 5i
squared, which is

00:03:50.900 --> 00:03:54.210
25i squared, which
is negative 25.

00:03:54.210 --> 00:03:55.400
And we're subtracting that.

00:03:55.400 --> 00:03:57.330
So it's going to be
positive plus 25.

00:03:57.330 --> 00:03:58.510
You add them together.

00:03:58.510 --> 00:04:00.266
You get 74.


Imaginary roots of negative numbers

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.120
We're asked to simplify
the principal square root

00:00:04.120 --> 00:00:05.490
of negative 52.

00:00:05.490 --> 00:00:08.230
And we're going to assume,
because we have a negative 52

00:00:08.230 --> 00:00:10.360
here inside of the
radical, that this

00:00:10.360 --> 00:00:15.160
is the principal branch of the
complex square root function.

00:00:15.160 --> 00:00:18.290
That we can actually put,
input, negative numbers

00:00:18.290 --> 00:00:19.790
in the domain of this function.

00:00:19.790 --> 00:00:24.180
That we can actually get
imaginary, or complex, results.

00:00:24.180 --> 00:00:28.410
So we can rewrite negative
52 as negative 1 times 52.

00:00:28.410 --> 00:00:31.910
So this can be rewritten as
the principal square root

00:00:31.910 --> 00:00:38.080
of negative 1 times 52.

00:00:38.080 --> 00:00:41.120
And then, if we assume that
this is the principal branch

00:00:41.120 --> 00:00:44.780
of the complex square root
function, we can rewrite this.

00:00:44.780 --> 00:00:48.400
This is going to be equal to
the square root of negative 1

00:00:48.400 --> 00:00:51.280
times-- or I should say,
the principal square root

00:00:51.280 --> 00:00:58.330
of negative 1 times the
principal square root of 52.

00:00:58.330 --> 00:01:00.960
Now, I want to be
very, very clear here.

00:01:00.960 --> 00:01:02.750
You can do what we just did.

00:01:02.750 --> 00:01:05.830
If we have the principal square
root of the product of two

00:01:05.830 --> 00:01:08.410
things, we can rewrite that
as the principal square root

00:01:08.410 --> 00:01:10.790
of each, and then
we take the product.

00:01:10.790 --> 00:01:13.320
But you can only do
this, or I should say,

00:01:13.320 --> 00:01:16.240
you can only do this if
either both of these numbers

00:01:16.240 --> 00:01:19.300
are positive, or only
one of them is negative.

00:01:19.300 --> 00:01:22.970
You cannot do this if both
of these were negative.

00:01:22.970 --> 00:01:25.600
For example, you
could not do this.

00:01:25.600 --> 00:01:31.550
You could not say the
principal square root of 52

00:01:31.550 --> 00:01:37.030
is equal to negative
1 times negative 52.

00:01:37.030 --> 00:01:37.897
You could do this.

00:01:37.897 --> 00:01:39.480
So far, I haven't
said anything wrong.

00:01:39.480 --> 00:01:43.260
52 is definitely negative
1 times negative 52.

00:01:43.260 --> 00:01:44.960
But then, since these
are both negative,

00:01:44.960 --> 00:01:50.650
you cannot then say that this
is equal to the square root

00:01:50.650 --> 00:01:54.780
of negative 1 times the
square root of negative 52.

00:01:54.780 --> 00:01:56.190
In fact, I invite
you to continue

00:01:56.190 --> 00:01:57.330
on this train of reasoning.

00:01:57.330 --> 00:01:59.370
You're going to get
a nonsensical answer.

00:01:59.370 --> 00:02:02.790
This is not OK.

00:02:02.790 --> 00:02:04.860
You cannot do this,
right over here.

00:02:04.860 --> 00:02:08.009
And the reason why you cannot do
this is that this property does

00:02:08.009 --> 00:02:12.130
not work when both of
these numbers are negative.

00:02:12.130 --> 00:02:14.570
Now with that said, we
can do it if only one

00:02:14.570 --> 00:02:18.110
of them are negative or both of
them are positive, obviously.

00:02:18.110 --> 00:02:20.119
Now, the principal square
root of negative 1,

00:02:20.119 --> 00:02:21.910
if we're talking about
the principal branch

00:02:21.910 --> 00:02:25.340
of the complex square
root function, is i.

00:02:25.340 --> 00:02:29.456
So this right over here
does simplify to i.

00:02:29.456 --> 00:02:31.080
And then let's think
if we can simplify

00:02:31.080 --> 00:02:33.600
the square root of 52 any.

00:02:33.600 --> 00:02:36.100
And to do that, we can think
about its prime factorization,

00:02:36.100 --> 00:02:39.920
see if we have any perfect
squares sitting in there.

00:02:39.920 --> 00:02:48.140
So 52 is 2 times 26,
and 26 is 2 times 13.

00:02:48.140 --> 00:02:50.800
So we have 2 times
2 there, or 4 there,

00:02:50.800 --> 00:02:52.520
which is a perfect square.

00:02:52.520 --> 00:02:58.050
So we can rewrite this as equal
to-- Well, we have our i, now.

00:02:58.050 --> 00:03:00.720
The principal square
root of negative 1 is i.

00:03:00.720 --> 00:03:03.230
The other square root of
negative 1 is negative i.

00:03:03.230 --> 00:03:05.589
But the principal square
root of negative 1 is i.

00:03:05.589 --> 00:03:07.130
And then we're going
to multiply that

00:03:07.130 --> 00:03:12.840
times the square
root of 4 times 13.

00:03:16.590 --> 00:03:23.020
And this is going to be equal
to i times the square root of 4.

00:03:23.020 --> 00:03:25.780
i times the square root of 4,
or the principal square root

00:03:25.780 --> 00:03:29.030
of 4 times the principal
square root of 13.

00:03:29.030 --> 00:03:31.880
The principal square
root of 4 is 2.

00:03:31.880 --> 00:03:34.650
So this all simplifies, and
we can switch the order,

00:03:34.650 --> 00:03:35.200
over here.

00:03:35.200 --> 00:03:39.330
This is equal to 2 times
the square root of 13.

00:03:39.330 --> 00:03:42.610
2 times the principal square
root of 13, I should say,

00:03:42.610 --> 00:03:45.360
times i.

00:03:45.360 --> 00:03:47.436
And I just switched
around the order.

00:03:47.436 --> 00:03:49.060
It makes it a little
bit easier to read

00:03:49.060 --> 00:03:51.620
if I put the i after
the numbers over here.

00:03:51.620 --> 00:03:54.627
But I'm just multiplying i times
2 times the square root of 13.

00:03:54.627 --> 00:03:56.460
That's the same thing
as multiplying 2 times

00:03:56.460 --> 00:03:59.530
the principal square
root of 13 times i.

00:03:59.530 --> 00:04:03.960
And I think this is about as
simplified as we can get here.


Subtracting complex numbers

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:01.850
We're asked to subtract.

00:00:01.850 --> 00:00:04.710
And we have the complex
number 2 minus 3i.

00:00:04.710 --> 00:00:09.420
And from that, we are
subtracting 6 minus 18i.

00:00:09.420 --> 00:00:11.070
So the first thing
I'd like to do here

00:00:11.070 --> 00:00:13.620
is to just get rid
of these parentheses.

00:00:13.620 --> 00:00:16.379
So we just have a bunch of
real parts and imaginary parts

00:00:16.379 --> 00:00:18.150
that we can then
add up together.

00:00:18.150 --> 00:00:21.140
So we have 2 minus 3i.

00:00:21.140 --> 00:00:23.590
And then we're subtracting
this entire quantity.

00:00:23.590 --> 00:00:25.020
And to get rid of
the parentheses,

00:00:25.020 --> 00:00:26.960
we can just distribute
the negative sign.

00:00:26.960 --> 00:00:28.334
Or another way to
think about it,

00:00:28.334 --> 00:00:31.097
we can say that this is
negative 1 times all of this.

00:00:31.097 --> 00:00:32.930
So we can just distribute
the negative sign.

00:00:32.930 --> 00:00:37.470
And negative 1 times
6 is negative 6.

00:00:37.470 --> 00:00:39.470
Let me do these in magenta.

00:00:39.470 --> 00:00:41.340
So this is negative 6.

00:00:41.340 --> 00:00:44.290
And then negative 1 times
negative 18i-- well, that's

00:00:44.290 --> 00:00:47.260
just going to be positive 18i.

00:00:47.260 --> 00:00:50.010
Negative times a
negative is a positive.

00:00:50.010 --> 00:00:51.740
And now we want to
add the real parts,

00:00:51.740 --> 00:00:54.300
and we want to add
the imaginary parts.

00:00:54.300 --> 00:00:56.630
So here's a real part here, 2.

00:00:56.630 --> 00:00:58.360
And then we have a minus 6.

00:00:58.360 --> 00:01:00.260
So we have 2 minus 6.

00:01:00.260 --> 00:01:02.460
And we want to add
the imaginary parts.

00:01:02.460 --> 00:01:04.019
We have a negative.

00:01:04.019 --> 00:01:05.570
Let me do that in
a different color.

00:01:05.570 --> 00:01:09.370
We have a negative
3i right over here.

00:01:09.370 --> 00:01:12.550
So negative 3i, or minus
3i right over there.

00:01:12.550 --> 00:01:15.120
And then we have a plus
18i or positive 18i.

00:01:18.570 --> 00:01:23.950
If you add the real parts,
2 minus 6 is negative 4.

00:01:23.950 --> 00:01:27.320
And you add the imaginary parts.

00:01:27.320 --> 00:01:29.160
If I have negative
3 of something

00:01:29.160 --> 00:01:31.367
and to that I add
18 of something,

00:01:31.367 --> 00:01:33.950
well, that's just going to leave
me with 15 of that something.

00:01:33.950 --> 00:01:34.860
Or another way you
could think about it,

00:01:34.860 --> 00:01:38.060
if I have 18 of something and
I subtract 3 of that something,

00:01:38.060 --> 00:01:39.850
I'll have 15 of that something.

00:01:39.850 --> 00:01:44.620
And in this case, the something
is i, is the imaginary unit.

00:01:44.620 --> 00:01:47.700
So this is going to be plus 15i.

00:01:50.560 --> 00:01:53.140
And we are done.


Adding complex numbers

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:04.760
We're asked to add the
complex number 5 plus 2i

00:00:04.760 --> 00:00:08.740
to the other complex
number 3 minus 7i.

00:00:08.740 --> 00:00:11.370
And as we'll see, when we're
adding complex numbers,

00:00:11.370 --> 00:00:13.670
you can only add the
real parts to each other

00:00:13.670 --> 00:00:15.970
and you can only add the
imaginary parts to each other.

00:00:15.970 --> 00:00:18.000
So let's add the real parts.

00:00:18.000 --> 00:00:19.730
So we have a 5 plus a 3.

00:00:22.390 --> 00:00:25.790
And then the imaginary
parts-- we have a 2i.

00:00:25.790 --> 00:00:27.440
So plus 2i.

00:00:27.440 --> 00:00:32.580
And then we have a negative
7i, or we're subtracting 7i.

00:00:32.580 --> 00:00:35.730
So minus 7i right over here.

00:00:35.730 --> 00:00:38.200
And 5 plus 3-- that's
pretty straightforward.

00:00:38.200 --> 00:00:40.430
That's just going to be 8.

00:00:40.430 --> 00:00:42.230
And then if I have
two of something

00:00:42.230 --> 00:00:45.124
and from that I subtract
seven of that something--

00:00:45.124 --> 00:00:46.540
and in this case,
the something is

00:00:46.540 --> 00:00:49.240
the imaginary
unit, the number i.

00:00:49.240 --> 00:00:53.650
If I have two i's and I take
away seven i's, then I have

00:00:53.650 --> 00:00:55.150
negative five i's.

00:00:55.150 --> 00:00:57.560
2 minus 7 is negative 5.

00:00:57.560 --> 00:01:00.470
So then I have negative 5i.

00:01:00.470 --> 00:01:02.720
So when you add these
two complex numbers,

00:01:02.720 --> 00:01:06.600
you get 8 minus 5i.

00:01:06.600 --> 00:01:08.080
You get another complex number.

00:01:08.080 --> 00:01:11.370
It has a real part
and an imaginary part.