Non-linear systems of equations 1

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.320
Solve the system of equations
by graphing.

00:00:03.320 --> 00:00:05.800
Check your solution
algebraically.

00:00:05.800 --> 00:00:08.810
Let's graph each of these, and
let's start-- let me find a

00:00:08.810 --> 00:00:10.620
nice dark color to
graph these with.

00:00:10.620 --> 00:00:15.180
Let me graph this top equation
in blue, this parabola.

00:00:15.180 --> 00:00:17.170
The first thing to think about
is this going to be an upward

00:00:17.170 --> 00:00:19.690
opening-- one, how did I
know it's a parabola?

00:00:19.690 --> 00:00:22.180
That's because it's a quadratic
function: we have an

00:00:22.180 --> 00:00:25.510
x squared term, a second
degree term, here.

00:00:25.510 --> 00:00:26.870
Then we have to think about:
it is going to be upward

00:00:26.870 --> 00:00:28.370
opening, or downward
opening parabola?

00:00:28.370 --> 00:00:30.940
You see that it's a negative
coefficient in front of the x

00:00:30.940 --> 00:00:34.520
squared, so it's going to be a
downward opening parabola.

00:00:34.520 --> 00:00:36.500
What is going to be
its maximum point?

00:00:36.500 --> 00:00:38.330
Let's think about that
for a second.

00:00:38.330 --> 00:00:44.410
This whole term right here is
always going to be negative,

00:00:44.410 --> 00:00:46.540
or it's always going
to be non-positive.

00:00:46.540 --> 00:00:49.710
x squared will be non-negative
when you multiply it by a

00:00:49.710 --> 00:00:52.240
negative, so it's going
to be non-positive.

00:00:52.240 --> 00:00:55.560
So, the highest value that this
thing can take on is when

00:00:55.560 --> 00:01:00.100
x is going to be equal to 0--
the vertex of this parabola is

00:01:00.100 --> 00:01:03.560
when x is equal to 0,
and y is equal to 6.

00:01:03.560 --> 00:01:07.640
So, x is equal to 0, and
y is 1, 2, 3, 4, 5, 6.

00:01:07.640 --> 00:01:12.710
So that right there is the
highest point of our parabola.

00:01:12.710 --> 00:01:14.480
Then, if we want, we can a
graph a couple of other

00:01:14.480 --> 00:01:16.810
points, just to see
what happens.

00:01:16.810 --> 00:01:21.950
So let's see what happens when
x is equal to-- let me just

00:01:21.950 --> 00:01:27.680
draw a little table here--
2, what is y?

00:01:27.680 --> 00:01:30.020
It's negative x squared
plus 6.

00:01:30.020 --> 00:01:32.090
So when x is 2, what is y?

00:01:32.090 --> 00:01:35.430
You have 2 squared, which is
4, but you have negative 2

00:01:35.430 --> 00:01:39.200
squared, so it's negative 4
plus 6-- it is equal to 2.

00:01:39.200 --> 00:01:41.120
It's the same thing when
x is negative 2.

00:01:41.120 --> 00:01:43.500
You put negative 2 there, you
square it, then you have

00:01:43.500 --> 00:01:45.430
positive 4, but you have a
negative there, so it's

00:01:45.430 --> 00:01:47.510
negative 4 plus 6 is 2.

00:01:47.510 --> 00:01:51.930
You have both of those points
there, so 2 comma 2, and then

00:01:51.930 --> 00:01:54.450
you have a negative 2 comma 2.

00:01:58.720 --> 00:02:03.590
If I were to graph it, Let's try
it with 3, as well-- if we

00:02:03.590 --> 00:02:06.190
put a 3 there, 3 squared is 9.

00:02:06.190 --> 00:02:09.419
It then becomes a negative 9
plus 3, it becomes negative 3,

00:02:09.419 --> 00:02:12.770
and negative 3 will also
become a negative 3.

00:02:12.770 --> 00:02:15.230
Negative 3 squared is positive
9, you have a negative out

00:02:15.230 --> 00:02:19.250
front, it becomes negative 9
plus 6, which is negative 3.

00:02:19.250 --> 00:02:23.850
You have negative 3, negative
3, and then you have 3,

00:02:23.850 --> 00:02:24.780
negative 3.

00:02:24.780 --> 00:02:26.690
So those are all good points.

00:02:26.690 --> 00:02:30.000
Now we can graph our parabola.

00:02:30.000 --> 00:02:33.820
Our parabola will look
something-- I was doing well

00:02:33.820 --> 00:02:39.480
until that second part --like
that, and let me just do the

00:02:39.480 --> 00:02:40.475
second part.

00:02:40.475 --> 00:02:48.610
That second part is hard to
draw-- let me do it from here.

00:02:48.610 --> 00:02:52.530
It looks something like that.

00:02:52.530 --> 00:02:54.960
We connect to this dot
right here, and then

00:02:54.960 --> 00:02:56.820
let me connect this.

00:02:56.820 --> 00:02:58.310
So that it looks something
like that.

00:02:58.310 --> 00:03:00.150
That's what our parabola looks
like, and obviously it keeps

00:03:00.150 --> 00:03:01.990
going down in that direction.

00:03:01.990 --> 00:03:03.180
So that's that first graph.

00:03:03.180 --> 00:03:06.180
Let's graph this second one
over here: y is equal to

00:03:06.180 --> 00:03:07.660
negative 2x minus 2.

00:03:07.660 --> 00:03:09.830
This is just going
to be a line.

00:03:09.830 --> 00:03:12.510
It's a linear equation, and the
highest degree here is 1.

00:03:12.510 --> 00:03:16.420
Our y-intercept is negative
2, so 0, 1, 2.

00:03:16.420 --> 00:03:18.440
Our y-intercept is negative 2.

00:03:18.440 --> 00:03:20.820
Our slope is negative 2.

00:03:20.820 --> 00:03:23.420
If we move 1 in the x direction,
we're going to go 2

00:03:23.420 --> 00:03:26.480
in the y-direction, and if we
move 2 in the x direction,

00:03:26.480 --> 00:03:31.600
we're going to move down
4 in the y direction.

00:03:31.600 --> 00:03:36.580
If we move back 2, we're going
to move up 2 in the y

00:03:36.580 --> 00:03:38.800
direction, and it looks like we
found one of our points of

00:03:38.800 --> 00:03:40.090
intersection.

00:03:40.090 --> 00:03:43.370
Let's just draw that line, so
that line will look something

00:03:43.370 --> 00:03:47.135
like-- It's hard for my hand to
draw that, but let me try

00:03:47.135 --> 00:03:48.520
as best as I can.

00:03:56.280 --> 00:03:57.780
This is the hardest part.

00:03:57.780 --> 00:04:02.690
It will look something like
that right there.

00:04:02.690 --> 00:04:04.820
The question is, where
do they intersect?

00:04:04.820 --> 00:04:08.100
One point of intersection does
immediately pop out at us,

00:04:08.100 --> 00:04:10.330
because they asked us to
do it graphically.

00:04:10.330 --> 00:04:14.420
That point right there, which
is the point negative 2, 2.

00:04:17.675 --> 00:04:19.690
It seems to pop out at
us, so this is the

00:04:19.690 --> 00:04:22.190
point negative 2, 2.

00:04:22.190 --> 00:04:23.440
Let's see if that makes sense.

00:04:23.440 --> 00:04:27.150
When you have the point negative
2, when you put x is

00:04:27.150 --> 00:04:29.540
equal to negative 2 here,
negative 2 times negative 2 is

00:04:29.540 --> 00:04:31.370
4 minus 2, and y
is equal to 2.

00:04:31.370 --> 00:04:34.860
When you put negative 2 here, y
is also equal to 2, so that

00:04:34.860 --> 00:04:35.860
makes sense.

00:04:35.860 --> 00:04:39.070
There's going to be some other
point way out here where they

00:04:39.070 --> 00:04:40.520
also intersect.

00:04:40.520 --> 00:04:43.090
There's also going to be some
other point way out here if we

00:04:43.090 --> 00:04:45.210
keep making this parabola.

00:04:45.210 --> 00:04:50.600
When y is equal to positive 4,
and you have negative 16 plus

00:04:50.600 --> 00:04:52.680
6, you get negative 10.

00:04:52.680 --> 00:04:55.760
So, positive 1, 2, 3, 4, and
then you go down 10.

00:04:55.760 --> 00:05:02.290
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

00:05:02.290 --> 00:05:04.300
That looks like that might
be our other point of

00:05:04.300 --> 00:05:08.680
intersection, so let me connect
this right there.

00:05:08.680 --> 00:05:10.140
Our other point of intersection

00:05:10.140 --> 00:05:11.210
looks to be right there.

00:05:11.210 --> 00:05:13.670
If we just follow this red
line it looks like we

00:05:13.670 --> 00:05:14.850
intersect there.

00:05:14.850 --> 00:05:17.490
Let's verify that
it works out.

00:05:17.490 --> 00:05:19.130
So 4, negative 10.

00:05:19.130 --> 00:05:20.980
We know that that's on this
blue line, so let's see if

00:05:20.980 --> 00:05:21.940
it's on this other line.

00:05:21.940 --> 00:05:28.370
So negative 2 times 4 minus 2,
that is negative 8 minus 2,

00:05:28.370 --> 00:05:30.180
which is equal to negative 10.

00:05:30.180 --> 00:05:33.000
The point 4, negative 10,
is on both of them.

00:05:33.000 --> 00:05:37.070
When x is equal to 4, y is
negative 10 for both equations

00:05:37.070 --> 00:05:43.670
here, so they both definitely
work out.


Applying Quadratic Functions 3

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:04.080
Suppose marketing experts have
determined the relationship

00:00:04.080 --> 00:00:07.330
between the selling price of an
item and the quantity sold

00:00:07.330 --> 00:00:10.670
of an item can be represented
by the linear equation q is

00:00:10.670 --> 00:00:13.490
equal to negative
30s plus 800.

00:00:13.490 --> 00:00:16.260
Where q is the quantity sold
in a year, and s is the

00:00:16.260 --> 00:00:16.970
selling price.

00:00:16.970 --> 00:00:17.585
That makes sense.

00:00:17.585 --> 00:00:20.200
If our selling price is higher,
then this whole thing

00:00:20.200 --> 00:00:21.580
is going to become more
negative, and we're

00:00:21.580 --> 00:00:22.740
going to sell less.

00:00:22.740 --> 00:00:24.780
If we lower our price, we're
going to sell more.

00:00:24.780 --> 00:00:25.950
Fair enough.

00:00:25.950 --> 00:00:29.350
They tell us that if the cost of
to produce the item is $20,

00:00:29.350 --> 00:00:33.690
what is the selling price that
optimizes the yearly profit?

00:00:33.690 --> 00:00:37.050
Let's think a little bit about
what yearly profit means.

00:00:37.050 --> 00:00:38.990
Let's think a little bit
about yearly profit.

00:00:38.990 --> 00:00:42.360
The profit is going to be equal
to the quantity that you

00:00:42.360 --> 00:00:49.040
sell times what?

00:00:49.040 --> 00:00:52.090
For each unit, what you care
about is the profit per unit,

00:00:52.090 --> 00:00:58.720
and the profit per unit is the
selling price per unit minus

00:00:58.720 --> 00:01:01.210
the cost of producing
that unit.

00:01:01.210 --> 00:01:02.940
The cost of producing
the unit right there

00:01:02.940 --> 00:01:06.170
is $20-- minus $20.

00:01:06.170 --> 00:01:07.260
That makes sense.

00:01:07.260 --> 00:01:12.800
If you're selling something for
$30, and it costs you $20

00:01:12.800 --> 00:01:14.910
to make it, every unit that
you sell, you'll make

00:01:14.910 --> 00:01:17.200
$10-- 30 minus 20.

00:01:17.200 --> 00:01:21.240
If you sell two units, it's
going to be 2 times $10, and

00:01:21.240 --> 00:01:22.750
you're going to make $20.

00:01:22.750 --> 00:01:28.220
Nothing really groundbreaking
right here.

00:01:28.220 --> 00:01:31.570
Right now we have profit as a
function of q, of quantity and

00:01:31.570 --> 00:01:34.250
selling price, but we have
quantity as a function of

00:01:34.250 --> 00:01:38.380
selling price right here, so we
could substitute this whole

00:01:38.380 --> 00:01:41.400
expression right here
in for quantity.

00:01:41.400 --> 00:01:43.900
What happens to our profit
function then?

00:01:43.900 --> 00:01:47.610
Then we can rewrite our profit
function as being equal to--

00:01:47.610 --> 00:01:51.340
instead of writing q, we could
say it as the same thing as

00:01:51.340 --> 00:02:02.090
negative 30s plus 800
times s minus 20.

00:02:02.090 --> 00:02:06.130
Let's multiply this out so we
have a nice, clean trinomial.

00:02:06.130 --> 00:02:10.160
Our profit is going to be equal
to negative 30s times s

00:02:10.160 --> 00:02:13.550
and that's negative
30s squared.

00:02:13.550 --> 00:02:20.000
Negative 30s times negative 20
is positive 600s, and 800

00:02:20.000 --> 00:02:23.030
times s is 800s.

00:02:23.030 --> 00:02:31.530
Then, 800 times negative 20
times negative 2 would be

00:02:31.530 --> 00:02:39.640
negative 1,600, so it's going
to be negative 16,000, if I

00:02:39.640 --> 00:02:40.890
did it right.

00:02:43.205 --> 00:02:45.770
This right here is our profit
function, and we can merge

00:02:45.770 --> 00:02:47.400
these middle two terms
right here.

00:02:47.400 --> 00:02:52.800
We have our profit is equal to
negative 30s squared plus

00:02:52.800 --> 00:02:55.170
1,400s-- just merging
these middle two

00:02:55.170 --> 00:02:58.300
terms-- minus 16,000.

00:02:58.300 --> 00:03:00.450
Now, we have a negative
coefficient on the s squared

00:03:00.450 --> 00:03:02.410
term, which tells us this will
be a downward opening

00:03:02.410 --> 00:03:05.500
parabola, and so let's think
a little bit about it.

00:03:05.500 --> 00:03:14.300
If this is our profit axis,
and this right here is our

00:03:14.300 --> 00:03:16.910
selling price axis, this
parabola is going to look

00:03:16.910 --> 00:03:18.750
something like this-- I don't
know exactly what it will look

00:03:18.750 --> 00:03:20.280
like, but it will look
something like this.

00:03:20.280 --> 00:03:22.150
It will be downward opening.

00:03:22.150 --> 00:03:25.250
We're asked to find the selling
price that optimizes

00:03:25.250 --> 00:03:26.130
the yearly profit.

00:03:26.130 --> 00:03:28.880
It's optimal if it's as high as
possible, so we're going to

00:03:28.880 --> 00:03:30.730
look for a maximum point.

00:03:30.730 --> 00:03:33.660
We're going to look for that
maximum point right there,

00:03:33.660 --> 00:03:37.130
find the selling point that
gives us that maximum point.

00:03:37.130 --> 00:03:37.650
we've seen.

00:03:37.650 --> 00:03:38.360
This before.

00:03:38.360 --> 00:03:42.840
That's the selling point, or
the s coordinate of the

00:03:42.840 --> 00:03:48.490
vertex-- we need to find the
s coordinate of the vertex.

00:03:48.490 --> 00:03:52.020
The easiest way to think of
doing that is to put this

00:03:52.020 --> 00:03:55.580
function into our vertex
form, so let's do that.

00:03:55.580 --> 00:03:57.860
The easiest way to do that I
can think of-- well, we're

00:03:57.860 --> 00:03:59.650
going to come a little bit of
completing the square and all

00:03:59.650 --> 00:04:01.590
that, but we've seen all of this
before, so let's take it

00:04:01.590 --> 00:04:03.230
step by step.

00:04:03.230 --> 00:04:05.120
First thing I like to do is
factor out the negative 30, so

00:04:05.120 --> 00:04:08.180
we have a clean coefficient
on the s squared.

00:04:08.180 --> 00:04:13.600
We could write price is equal to
negative 30 times s squared

00:04:13.600 --> 00:04:19.670
plus-- 1,400 hundred divided
by negative 30 is negative

00:04:19.670 --> 00:04:24.520
140/3, and you can divide
both by 10 s.

00:04:24.520 --> 00:04:31.040
Then, 16,000 divided by negative
30 is positive 1,600

00:04:31.040 --> 00:04:34.260
divided by 3-- you take a 0 off
of each of those, you have

00:04:34.260 --> 00:04:36.810
a negative divided by a
negative, they cancel out.

00:04:36.810 --> 00:04:38.880
That's what we're left with--
actually, let me write this

00:04:38.880 --> 00:04:46.180
positive 1,600/3 a little bit
further out, so plus 1,600/3.

00:04:46.180 --> 00:04:49.700
All I did so far is that I
just factored out this

00:04:49.700 --> 00:04:50.850
negative 30.

00:04:50.850 --> 00:04:54.110
What we can do now is we can add
and subtract a term here

00:04:54.110 --> 00:04:56.020
so that the first three
terms here become a

00:04:56.020 --> 00:04:57.390
perfect square trinomial.

00:04:57.390 --> 00:04:59.890
That way, we can get into
that vertex form.

00:04:59.890 --> 00:05:03.620
So what do we add
and subtract?

00:05:03.620 --> 00:05:10.120
Negative 140/3 can be rewritten
as 2 times negative

00:05:10.120 --> 00:05:14.020
70 over 3, so what you want to
add and subtract is negative

00:05:14.020 --> 00:05:18.030
70 over 3 squared, or half
of negative 140/3.

00:05:18.030 --> 00:05:20.740
You want to add that
squared, so plus

00:05:20.740 --> 00:05:24.950
negative 70 over 3 squared.

00:05:24.950 --> 00:05:27.310
We can't just add it to one side
without doing anything

00:05:27.310 --> 00:05:30.140
else, so let's also subtract it
from that same side-- so,

00:05:30.140 --> 00:05:34.630
minus negative 70
over 3 squared.

00:05:34.630 --> 00:05:36.240
We really haven't changed the
equation at all, but we're

00:05:36.240 --> 00:05:39.220
just adding and subtracting the
same thing to both sides.

00:05:39.220 --> 00:05:43.510
What that allows us to do is
to make this into a perfect

00:05:43.510 --> 00:05:47.520
square trinomial-- this is a
perfect square trinomial.

00:05:47.520 --> 00:05:51.430
We can rewrite our profit
function-- our profit function

00:05:51.430 --> 00:05:55.410
is equal to negative 30 times
this thing right here, which

00:05:55.410 --> 00:06:02.160
is a perfect square-- it is
s minus 70 over 3 squared.

00:06:02.160 --> 00:06:05.770
Then we have all of this
business over here: minus

00:06:05.770 --> 00:06:12.220
minus negative 70 over
3 squared plus 1,600

00:06:12.220 --> 00:06:15.365
hundred over 3.

00:06:20.410 --> 00:06:22.520
The next thing we can do is
redistribute this negative

00:06:22.520 --> 00:06:24.300
30-- let me scroll down,
so we have some

00:06:24.300 --> 00:06:26.310
space to do this with.

00:06:26.310 --> 00:06:30.670
We could say price is equal
to negative 30 times this

00:06:30.670 --> 00:06:35.840
expression right here, which is
5 minus 70 over 3-- sorry,

00:06:35.840 --> 00:06:38.200
not 5 minus, that's an s, and
we have to be very careful.

00:06:38.200 --> 00:06:44.180
That's our selling price s minus
70 over 3 squared, and

00:06:44.180 --> 00:06:47.500
then that's negative 3 times
its first term, and then we

00:06:47.500 --> 00:06:52.230
have to multiply negative 30
times each of these terms.

00:06:52.230 --> 00:06:55.040
It's negative 30 times this,
and a negative times a

00:06:55.040 --> 00:06:58.830
negative is a positive-- 30
divided by 3 is 10, so it's

00:06:58.830 --> 00:07:06.360
going to be plus 10 times
negative 70 squared, which we

00:07:06.360 --> 00:07:07.970
could evaluate if we want.

00:07:07.970 --> 00:07:12.870
Negative 30 times 1,600 over 3
will be negative 10 times 3--

00:07:12.870 --> 00:07:18.550
it'll be negative 16,000.

00:07:18.550 --> 00:07:20.530
We can simplify this.

00:07:20.530 --> 00:07:22.490
Negative 70 squared is what?

00:07:22.490 --> 00:07:29.030
It's positive 4,900 times 10,
and you get 49,000 minus

00:07:29.030 --> 00:07:34.850
16,000, which is 33,000.

00:07:34.850 --> 00:07:37.920
We might as well do that, so
this whole thing simplifies to

00:07:37.920 --> 00:07:41.530
our profit function-- our yearly
profit-- is negative 30

00:07:41.530 --> 00:07:45.970
times s minus 70
over 3 squared.

00:07:45.970 --> 00:07:49.690
Let me make sure I did that
right: 7 times 7 is 49, and

00:07:49.690 --> 00:07:52.550
then you have-- no, it's
4,900, so let me

00:07:52.550 --> 00:07:54.020
be very clear here.

00:07:54.020 --> 00:08:00.880
It's 7 times 7 is 4,900, and
then times 10 is 49,000.

00:08:00.880 --> 00:08:04.860
Let me make sure I got it-- yes,
49,000 minus 16,000 is

00:08:04.860 --> 00:08:10.792
33,000, so plus 33,000.

00:08:10.792 --> 00:08:12.110
It's just like that.

00:08:12.110 --> 00:08:13.543
When is this thing maximized?

00:08:16.780 --> 00:08:21.160
If you look right here, this
term right here-- this s minus

00:08:21.160 --> 00:08:24.260
70 over 3 squared-- is always
going to be positive, or it's

00:08:24.260 --> 00:08:27.040
always going to be non-negative,
it could be 0.

00:08:27.040 --> 00:08:28.840
So when you multiply times a
negative number, it's always

00:08:28.840 --> 00:08:31.740
going to be a negative, so this
thing is always going to

00:08:31.740 --> 00:08:37.080
be non-positive-- it could
be equal to 0.

00:08:37.080 --> 00:08:39.799
So, the profit is going to take
its highest value when

00:08:39.799 --> 00:08:41.580
this whole expression is 0.

00:08:41.580 --> 00:08:44.159
And when is this whole
expression going to be 0?

00:08:44.159 --> 00:08:46.145
The whole expression is going
to be 0 when s is

00:08:46.145 --> 00:08:51.530
equal to 70 over 3.

00:08:51.530 --> 00:08:53.440
That's when it equals 0.

00:08:53.440 --> 00:08:57.300
Just so we have a better sense
of that, what is 70/3?

00:08:57.300 --> 00:09:02.680
That is-- 3 goes into 70, 3 goes
into 7 two times, 2 times

00:09:02.680 --> 00:09:07.840
3 is 6, bring down it down and
get a 10, 3 goes into 10 three

00:09:07.840 --> 00:09:11.670
times, 3 times 3 is
9, and subtract.

00:09:11.670 --> 00:09:15.770
You get another 10, so it's
23.3333333-- you can view it

00:09:15.770 --> 00:09:17.980
that way, it just
keeps repeating.

00:09:17.980 --> 00:09:22.740
Or, you could say the ideal
selling price is $23.33.

00:09:22.740 --> 00:09:27.380
So, the ideal selling price, the
maximum selling price, the

00:09:27.380 --> 00:09:36.190
selling price that optimizes
our profit is $23.33.

00:09:36.190 --> 00:09:41.110
When you do sell for $23.33,
this is going to be 0, that

00:09:41.110 --> 00:09:42.865
whole thing is going to
be 0, and your profit

00:09:42.865 --> 00:09:45.250
is going to be $33,000.

00:09:45.250 --> 00:09:47.510
At that point, your profit
is going to be

00:09:47.510 --> 00:09:51.930
$33,000 for the year.


Example 3: Using the quadratic formula

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.300
Determine the number of
solutions to the quadratic

00:00:03.300 --> 00:00:08.200
equation, x squared plus 14x
plus 49 is equal to 0.

00:00:08.200 --> 00:00:09.760
There's a bunch of ways
we could do it.

00:00:09.760 --> 00:00:12.325
We could factor it and just
figure out the values of x

00:00:12.325 --> 00:00:13.560
that satisfy it and
just count them.

00:00:13.560 --> 00:00:14.310
That will be the number
of solutions.

00:00:14.310 --> 00:00:16.400
We could just apply the
quadratic formula.

00:00:16.400 --> 00:00:18.690
But what I want to do here is
actually explore the quadratic

00:00:18.690 --> 00:00:21.290
formula, and think about how we
can determine the number of

00:00:21.290 --> 00:00:23.290
solutions without even maybe
necessarily finding them

00:00:23.290 --> 00:00:24.280
explicitly.

00:00:24.280 --> 00:00:26.950
So the quadratic formula tells
us that if we have an equation

00:00:26.950 --> 00:00:32.830
of the form ax squared plus bx
plus c is equal to 0, that the

00:00:32.830 --> 00:00:36.840
solutions are going to be-- or
the solution if it exists is

00:00:36.840 --> 00:00:41.860
going to be-- negative b plus or
minus the square root of b

00:00:41.860 --> 00:00:44.570
squared minus 4ac.

00:00:44.570 --> 00:00:46.770
All of that over 2a.

00:00:46.770 --> 00:00:51.220
Now the reason why this can be
2 solutions is that we have a

00:00:51.220 --> 00:00:53.070
plus or minus here.

00:00:53.070 --> 00:00:57.620
If this b squared minus 4ac is
a positive number-- so let's

00:00:57.620 --> 00:00:59.250
think about this a little bit.

00:00:59.250 --> 00:01:05.220
If b squared minus 4ac
is greater than 0,

00:01:05.220 --> 00:01:06.470
what's going to happen?

00:01:06.470 --> 00:01:08.410
Well, then it's a
positive number.

00:01:08.410 --> 00:01:09.950
It's going to have
a square root.

00:01:09.950 --> 00:01:13.920
And then when you add it to
negative b you're going to get

00:01:13.920 --> 00:01:15.910
one value for the numerator, and
when you subtract it from

00:01:15.910 --> 00:01:17.400
negative b you are going
to get another

00:01:17.400 --> 00:01:18.450
value in the numerator.

00:01:18.450 --> 00:01:19.935
So this is going to lead
to two solutions.

00:01:24.250 --> 00:01:31.550
Now what happens if b squared
minus 4ac is equal to 0?

00:01:31.550 --> 00:01:34.390
If this expression under the
radical is equal to 0, you're

00:01:34.390 --> 00:01:35.860
just going to have the
square root of 0.

00:01:35.860 --> 00:01:38.790
So it's going to be negative
b plus or minus 0.

00:01:38.790 --> 00:01:40.673
And it doesn't matter whether
you add or subtract 0, you're

00:01:40.673 --> 00:01:42.270
going to get the same value.

00:01:42.270 --> 00:01:44.870
So in that situation, the
actual solution of the

00:01:44.870 --> 00:01:47.260
equation is going to be
negative b over 2a.

00:01:47.260 --> 00:01:49.060
There's not going to be this
plus or minus, it's not going

00:01:49.060 --> 00:01:49.540
to be relevant.

00:01:49.540 --> 00:01:51.770
You're only going to
have one solution.

00:01:51.770 --> 00:01:54.170
So if b squared minus 4ac
is equal to 0, you

00:01:54.170 --> 00:01:57.920
only have one solution.

00:01:57.920 --> 00:02:03.240
And then what happens if
b squared minus 4ac

00:02:03.240 --> 00:02:05.150
is less than 0?

00:02:05.150 --> 00:02:07.800
Well if b squared minus 4ac is
less than 0, this is going to

00:02:07.800 --> 00:02:09.863
be a negative number right here
and you're going to have

00:02:09.863 --> 00:02:11.880
to take the square root
of a negative number.

00:02:11.880 --> 00:02:14.430
And we know, from dealing with
real numbers, you can't take

00:02:14.430 --> 00:02:15.020
the square root.

00:02:15.020 --> 00:02:18.010
There is no real number
squared that becomes a

00:02:18.010 --> 00:02:19.560
negative number.

00:02:19.560 --> 00:02:24.560
So in this situation there is
no solutions, or no real--

00:02:24.560 --> 00:02:26.370
when I say real I literally
mean a real

00:02:26.370 --> 00:02:28.195
number-- no real solution.

00:02:31.460 --> 00:02:33.530
So let's think about it
in the context of this

00:02:33.530 --> 00:02:34.670
equation right here.

00:02:34.670 --> 00:02:38.840
And just in case you're curious
if whether this

00:02:38.840 --> 00:02:41.170
expression right here,
b squared minus 4ac,

00:02:41.170 --> 00:02:42.740
has a name, it does.

00:02:42.740 --> 00:02:43.990
It's called the discriminant.

00:02:50.190 --> 00:02:51.045
This is the discriminant.

00:02:51.045 --> 00:02:53.032
That's that part of the
quadratic equation.

00:02:53.032 --> 00:02:55.450
It determines the number
of solutions we have.

00:02:55.450 --> 00:02:57.185
So if we want to figure out the
number of solutions for

00:02:57.185 --> 00:02:59.180
this equation, we don't have
to go through the whole

00:02:59.180 --> 00:03:01.360
quadratic equation, although
it's not that much work.

00:03:01.360 --> 00:03:05.400
We just have to evaluate
b squared minus 4ac.

00:03:05.400 --> 00:03:08.530
So what is b squared
minus 4ac?

00:03:08.530 --> 00:03:11.830
So b is right here, it's 14.

00:03:11.830 --> 00:03:19.590
So it's 14 squared minus 4 times
a, which is 1, times c,

00:03:19.590 --> 00:03:21.120
which is 49.

00:03:21.120 --> 00:03:23.480
That c, right there, times 49.

00:03:23.480 --> 00:03:25.972
What's 14 times 14?

00:03:25.972 --> 00:03:27.275
Let me do it over here.

00:03:31.480 --> 00:03:33.870
14 times 14.

00:03:33.870 --> 00:03:36.230
4 times 4 is 16.

00:03:36.230 --> 00:03:38.120
4 times 1 is 4.

00:03:38.120 --> 00:03:39.540
Plus 1 is 56.

00:03:39.540 --> 00:03:40.180
Put a 0.

00:03:40.180 --> 00:03:43.100
1 times 14 is 14.

00:03:43.100 --> 00:03:44.850
It is 6, 9, 1.

00:03:44.850 --> 00:03:45.800
It's 196.

00:03:45.800 --> 00:03:48.750
So this right here is 196.

00:03:48.750 --> 00:03:51.070
And we can ignore the 1.

00:03:51.070 --> 00:03:52.845
What's 4 times 49?

00:03:52.845 --> 00:03:56.080
So 49 times 4.

00:03:56.080 --> 00:04:01.840
4 times 9 is 36.

00:04:01.840 --> 00:04:05.090
4 times 4 is 16 plus
3 is 190-- or is

00:04:05.090 --> 00:04:07.210
19, so you get 196.

00:04:07.210 --> 00:04:10.200
So this right here is 196.

00:04:10.200 --> 00:04:14.540
So b squared minus 4ac
is 196 minus 196.

00:04:14.540 --> 00:04:18.250
So 196 minus 196
is equal to 0.

00:04:18.250 --> 00:04:19.850
So we're dealing with a
situation where the

00:04:19.850 --> 00:04:21.720
discriminant is equal to 0.

00:04:21.720 --> 00:04:24.120
We only have one solution.

00:04:24.120 --> 00:04:26.800
And if you want, you could try
to find that one solution.

00:04:26.800 --> 00:04:29.020
This whole part is going to
be the square root of 0.

00:04:29.020 --> 00:04:30.020
It's just going to be 0.

00:04:30.020 --> 00:04:33.790
So the solution is going to
be negative b over 2a.

00:04:33.790 --> 00:04:35.920
And negative b is-- we
could just solve it.

00:04:35.920 --> 00:04:41.170
Negative b is negative 14 over 2
times a. a is just 1 over 2.

00:04:41.170 --> 00:04:43.420
So it's equal to negative 7.

00:04:43.420 --> 00:04:45.560
That's the only solution
to this equation.

00:04:45.560 --> 00:04:47.480
But if you just wanted to know
how many solutions, you just

00:04:47.480 --> 00:04:50.460
have to find out that b squared
minus 4ac is 0.

00:04:50.460 --> 00:04:52.250
So it's only going to
have one solution.

00:04:52.250 --> 00:04:52.990
And there's other ways.

00:04:52.990 --> 00:04:54.530
You could have actually factored
this pretty easily

00:04:54.530 --> 00:04:58.270
into x plus 7 times x plus 7
and gotten the same result.


Example 1: Using the quadratic formula

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.810
Use the quadratic formula to
solve the equation, 0 is equal

00:00:04.810 --> 00:00:10.500
to negative 7q squared
plus 2q plus 9.

00:00:10.500 --> 00:00:14.230
Now, the quadratic formula, it
applies to any quadratic

00:00:14.230 --> 00:00:16.750
equation of the form--
we could put the 0 on

00:00:16.750 --> 00:00:17.600
the left hand side.

00:00:17.600 --> 00:00:24.160
0 is equal to ax squared
plus bx plus c.

00:00:24.160 --> 00:00:27.170
And we generally deal with x's,
in this problem we're

00:00:27.170 --> 00:00:28.230
dealing with q's.

00:00:28.230 --> 00:00:29.980
But the quadratic formula says,
look, if you have a

00:00:29.980 --> 00:00:34.150
quadratic equation of this form,
that the solutions of

00:00:34.150 --> 00:00:37.610
this equation are going to be
x is going to be equal to

00:00:37.610 --> 00:00:44.330
negative b plus or minus the
square root of b squared minus

00:00:44.330 --> 00:00:49.150
4ac-- all of that over 2a.

00:00:49.150 --> 00:00:51.000
And this is actually two
solutions here, because

00:00:51.000 --> 00:00:53.350
there's one solution where you
take the positive square root

00:00:53.350 --> 00:00:54.910
and there's another solution
where you take

00:00:54.910 --> 00:00:57.030
the negative root.

00:00:57.030 --> 00:00:58.680
So it gives you both
roots of this.

00:00:58.680 --> 00:01:02.030
So if we look at the quadratic
equation that we need to solve

00:01:02.030 --> 00:01:03.250
here, we can just
pattern match.

00:01:03.250 --> 00:01:05.980
We're dealing with q's, not
x's, but this is the same

00:01:05.980 --> 00:01:06.850
general idea.

00:01:06.850 --> 00:01:08.420
It could be x's if you like.

00:01:08.420 --> 00:01:13.550
And if we look at it, negative
7 corresponds to a.

00:01:13.550 --> 00:01:14.600
That is our a.

00:01:14.600 --> 00:01:18.320
It's the coefficient on the
second degree term.

00:01:18.320 --> 00:01:21.290
2 corresponds to b.

00:01:21.290 --> 00:01:24.250
It is the coefficient on
the first degree term.

00:01:24.250 --> 00:01:26.880
And then 9 corresponds to c.

00:01:26.880 --> 00:01:29.250
It's the constant.

00:01:29.250 --> 00:01:31.440
So, let's just apply the
quadratic formula.

00:01:31.440 --> 00:01:34.510
The quadratic formula will tell
us that the solutions--

00:01:34.510 --> 00:01:38.600
the q's that satisfy this
equation-- q will be equal to

00:01:38.600 --> 00:01:40.440
negative b.

00:01:40.440 --> 00:01:42.390
b is 2.

00:01:42.390 --> 00:01:53.840
Plus or minus the square root
of b squared, of 2 squared,

00:01:53.840 --> 00:02:08.220
minus 4 times a times negative
7 times c, which is 9.

00:02:08.220 --> 00:02:10.639
And all of that over 2a.

00:02:10.639 --> 00:02:16.020
All of that over 2 times
a, which is once

00:02:16.020 --> 00:02:17.400
again negative 7.

00:02:20.256 --> 00:02:22.760
And then we just have
to evaluate this.

00:02:22.760 --> 00:02:29.190
So this is going to be equal
to negative 2 plus or minus

00:02:29.190 --> 00:02:35.160
the square root of-- let's see,
2 squared is 4-- and then

00:02:35.160 --> 00:02:37.962
if we just take this part right
here, if we just take

00:02:37.962 --> 00:02:42.470
the negative 4 times negative
7 times 9, this negative and

00:02:42.470 --> 00:02:43.890
that negative is going
to cancel out.

00:02:43.890 --> 00:02:45.960
So it's just going to become
a positive number.

00:02:45.960 --> 00:02:49.220
And 4 times 7 times 9.

00:02:49.220 --> 00:02:52.720
4 times 9 is 36.

00:02:52.720 --> 00:02:54.270
36 times 7.

00:02:54.270 --> 00:02:55.710
Let's do it up here.

00:02:55.710 --> 00:02:58.060
36 times 7.

00:02:58.060 --> 00:03:00.780
7 times 6 is 42.

00:03:00.780 --> 00:03:03.350
7 times 3, or 3 times 7 is 21.

00:03:03.350 --> 00:03:05.500
Plus 4 is 25.

00:03:05.500 --> 00:03:06.420
252.

00:03:06.420 --> 00:03:09.290
So this becomes 4 plus 252.

00:03:09.290 --> 00:03:11.430
Remember, you have a
negative 7 and you

00:03:11.430 --> 00:03:12.830
have a minus out front.

00:03:12.830 --> 00:03:16.020
Those cancel out, that's why
we have a positive 252 for

00:03:16.020 --> 00:03:17.950
that part right there.

00:03:17.950 --> 00:03:22.220
And then our denominator,
2 times negative

00:03:22.220 --> 00:03:24.800
7 is negative 14.

00:03:24.800 --> 00:03:26.050
Now what does this equal?

00:03:28.370 --> 00:03:33.460
Well, we have this is equal to
negative 2 plus or minus the

00:03:33.460 --> 00:03:36.080
square root of-- what's
4 plus 252?

00:03:36.080 --> 00:03:38.410
It's just 256.

00:03:38.410 --> 00:03:40.860
All of that over negative 14.

00:03:40.860 --> 00:03:43.600
And what's 256?

00:03:43.600 --> 00:03:46.580
What's the square root of 256?

00:03:46.580 --> 00:03:47.360
It's 16.

00:03:47.360 --> 00:03:48.420
You can try it out
for yourself.

00:03:48.420 --> 00:03:49.810
This is 16 times 16.

00:03:49.810 --> 00:03:53.160
So the square root
of 256 is 16.

00:03:53.160 --> 00:03:56.400
So we can rewrite this whole
thing as being equal to

00:03:56.400 --> 00:04:04.500
negative 2 plus 16
over negative 14.

00:04:04.500 --> 00:04:07.790
Or negative 2 minus-- right?

00:04:07.790 --> 00:04:10.180
This is plus 16 over
negative 14.

00:04:10.180 --> 00:04:16.490
Or minus 16 over negative 14.

00:04:16.490 --> 00:04:20.740
If you think of it as plus or
minus, that plus is that plus

00:04:20.740 --> 00:04:21.720
right there.

00:04:21.720 --> 00:04:26.510
And if you have that minus,
that minus is

00:04:26.510 --> 00:04:28.250
that minus right there.

00:04:28.250 --> 00:04:29.910
Now we just have to evaluate
these two numbers.

00:04:32.430 --> 00:04:39.010
Negative 2 plus 16 is 14 divided
by negative 14 is

00:04:39.010 --> 00:04:40.170
negative 1.

00:04:40.170 --> 00:04:42.470
So q could be equal
to negative 1.

00:04:42.470 --> 00:04:47.430
Or negative 2 minus 16 is
negative 18 divided by

00:04:47.430 --> 00:04:51.320
negative 14 is equal
to 18 over 14.

00:04:51.320 --> 00:04:54.500
The negatives cancel out, which
is equal to 9 over 7.

00:04:54.500 --> 00:04:57.190
So q could be equal to negative
1, or it could be

00:04:57.190 --> 00:04:59.500
equal to 9 over 7.

00:04:59.500 --> 00:05:02.100
And you could try these out,
substitute these q's back into

00:05:02.100 --> 00:05:04.460
this original equation, and
verify for yourself that they

00:05:04.460 --> 00:05:05.350
satisfy it.

00:05:05.350 --> 00:05:07.900
We could even do it with
the first one.

00:05:07.900 --> 00:05:09.750
So if you take q is equal
to negative 1.

00:05:09.750 --> 00:05:12.795
Negative 7 times negative 1
squared-- negative 1 squared

00:05:12.795 --> 00:05:17.210
is just 1-- so this would be
negative 7 times 1, right?

00:05:17.210 --> 00:05:18.920
That's negative 1 squared.

00:05:18.920 --> 00:05:22.260
Negative 1 times 2 is
minus 2 plus 9.

00:05:22.260 --> 00:05:26.150
So it's negative 7 minus 2,
which is negative 9, plus 9,

00:05:26.150 --> 00:05:27.960
does indeed equal 0.

00:05:27.960 --> 00:05:29.180
So this checks out.

00:05:29.180 --> 00:05:31.780
And I'll leave it up to you
to verify that 9 over

00:05:31.780 --> 00:05:34.150
7 also works out.


Example 2: Using the quadratic formula

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:04.490
Use the quadratic formula to
solve the equation, negative x

00:00:04.490 --> 00:00:08.189
squared plus 8x is equal to 1.

00:00:08.189 --> 00:00:11.300
Now, in order to really use the
quadratic equation, or to

00:00:11.300 --> 00:00:13.920
figure out what our a's, b's and
c's are, we have to have

00:00:13.920 --> 00:00:20.110
our equation in the form, ax
squared plus bx plus c is

00:00:20.110 --> 00:00:21.490
equal to 0.

00:00:21.490 --> 00:00:24.110
And then, if we know our a's,
b's, and c's, we will say that

00:00:24.110 --> 00:00:28.060
the solutions to this equation
are x is equal to negative b

00:00:28.060 --> 00:00:34.480
plus or minus the square root
of b squared minus 4ac-- all

00:00:34.480 --> 00:00:36.730
of that over 2a.

00:00:36.730 --> 00:00:38.910
So the first thing we have to
do for this equation right

00:00:38.910 --> 00:00:40.505
here is to put it
in this form.

00:00:40.505 --> 00:00:42.990
And on one side of this
equation, we have a negative x

00:00:42.990 --> 00:00:46.850
squared plus 8x, so that looks
like the first two terms. But

00:00:46.850 --> 00:00:48.220
our constant is on
the other side.

00:00:48.220 --> 00:00:50.560
So let's get the constant on the
left hand side and get a 0

00:00:50.560 --> 00:00:52.100
here on the right hand side.

00:00:52.100 --> 00:00:57.880
So let's subtract 1 from both
sides of this equation.

00:00:57.880 --> 00:01:00.800
The left hand side of the
equation will become negative

00:01:00.800 --> 00:01:06.010
x squared plus 8x minus 1.

00:01:06.010 --> 00:01:09.460
And then the right hand
side, 1 minus 1 is 0.

00:01:09.460 --> 00:01:10.890
Now we have it in that form.

00:01:10.890 --> 00:01:13.725
We have ax squared
a is negative 1.

00:01:13.725 --> 00:01:17.900
So let me write this down.
a is equal to negative 1.

00:01:17.900 --> 00:01:22.450
a is equal to negative 1.

00:01:22.450 --> 00:01:25.695
It's implicit there, you could
put a 1 here if you like.

00:01:25.695 --> 00:01:26.530
A negative 1.

00:01:26.530 --> 00:01:29.030
Negative x squared is the same
thing as negative 1x squared.

00:01:29.030 --> 00:01:32.210
b is equal to 8.

00:01:32.210 --> 00:01:36.270
So b is equal to 8, that's
the 8 right there.

00:01:36.270 --> 00:01:38.535
And c is equal to negative 1.

00:01:41.380 --> 00:01:43.500
That's the negative
1 right there.

00:01:43.500 --> 00:01:46.390
So now we can just apply
the quadratic formula.

00:01:46.390 --> 00:01:51.290
The solutions to this equation
are x is equal to negative b.

00:01:55.910 --> 00:02:06.190
Plus or minus the square root
of b squared, of 8 squared,

00:02:06.190 --> 00:02:12.110
minus 4ac-- let me do it in that
green color --minus 4,

00:02:12.110 --> 00:02:13.430
the green is the part
of the formula.

00:02:13.430 --> 00:02:14.990
The colored parts are the things
that we're substituting

00:02:14.990 --> 00:02:16.620
into the formula.

00:02:16.620 --> 00:02:22.300
Minus 4 times a, which is
negative 1, times negative 1,

00:02:22.300 --> 00:02:24.640
times c, which is
also negative 1.

00:02:27.430 --> 00:02:30.250
And then all of that-- let me
extend the square root sign a

00:02:30.250 --> 00:02:33.080
little bit further --all
of that is going to be

00:02:33.080 --> 00:02:35.570
over 2 times a.

00:02:35.570 --> 00:02:38.280
In this case a is negative 1.

00:02:38.280 --> 00:02:40.130
So let's simplify this.

00:02:40.130 --> 00:02:43.710
So this becomes negative 8, this
is negative 8, plus or

00:02:43.710 --> 00:02:49.650
minus the square root
of 8 squared is 64.

00:02:49.650 --> 00:02:51.700
And then you have a negative 1
times a negative 1, these just

00:02:51.700 --> 00:02:52.850
cancel out just to be a 1.

00:02:52.850 --> 00:02:55.370
So it's 64 minus is 4.

00:02:55.370 --> 00:02:56.670
That's just that 4 over there.

00:02:56.670 --> 00:02:59.050
All of that over negative 2.

00:02:59.050 --> 00:03:02.790
So this is equal to negative
8 plus or minus the

00:03:02.790 --> 00:03:05.740
square root of 60.

00:03:05.740 --> 00:03:07.850
All of that over negative 2.

00:03:07.850 --> 00:03:11.730
And let's see if we can
simplify the radical

00:03:11.730 --> 00:03:14.530
expression here, the
square root of 60.

00:03:14.530 --> 00:03:19.500
Let's see, 60 is equal
to 2 times 30.

00:03:19.500 --> 00:03:22.620
30 is equal to 2 times 15.

00:03:22.620 --> 00:03:25.310
And then 15 is 3 times 5.

00:03:25.310 --> 00:03:26.500
So we do have a perfect
square here.

00:03:26.500 --> 00:03:28.600
We do have a 2 times
2 in there.

00:03:28.600 --> 00:03:32.330
It is 2 times 2 times
15, or 4 times 15.

00:03:32.330 --> 00:03:35.620
So we could write, the square
root of 60 is equal to the

00:03:35.620 --> 00:03:39.000
square root of 4 times the
square root of 15, right?

00:03:39.000 --> 00:03:42.810
The square root of 4 times
the square root of 15,

00:03:42.810 --> 00:03:43.850
that's what 60 is.

00:03:43.850 --> 00:03:45.420
4 times 15.

00:03:45.420 --> 00:03:49.300
And so this is equal to-- square
root of 4 is 2 times

00:03:49.300 --> 00:03:50.700
the square of 15.

00:03:50.700 --> 00:03:53.220
So we can rewrite this
expression, right here, as

00:03:53.220 --> 00:04:01.000
being equal to negative 8 plus
or minus 2 times the square

00:04:01.000 --> 00:04:09.010
root of 15, all of that
over negative 2.

00:04:09.010 --> 00:04:13.480
Now both of these terms right
here are divisible by either 2

00:04:13.480 --> 00:04:14.180
or negative 2.

00:04:14.180 --> 00:04:15.380
So let's divide it.

00:04:15.380 --> 00:04:18.920
So we have negative 8 divided
by negative 2, which is

00:04:18.920 --> 00:04:19.669
positive 4.

00:04:19.669 --> 00:04:21.029
So let me write it over here.

00:04:21.029 --> 00:04:25.210
Negative 8 divided by negative
2 is positive 4.

00:04:25.210 --> 00:04:26.340
And then you have this
weird thing.

00:04:26.340 --> 00:04:28.680
Plus or minus 2 divided
by negative 2.

00:04:28.680 --> 00:04:31.070
And really what we have
here is 2 expressions.

00:04:31.070 --> 00:04:33.940
But if we're plus 2 and we
divide by negative 2, it will

00:04:33.940 --> 00:04:35.400
be negative 1.

00:04:35.400 --> 00:04:38.530
And if we take negative 2 and
divide by negative 2, we're

00:04:38.530 --> 00:04:39.460
going to have positive 1.

00:04:39.460 --> 00:04:41.380
So instead of plus or minus, you
could imagine it is going

00:04:41.380 --> 00:04:45.400
to be minus or plus.

00:04:45.400 --> 00:04:46.470
But it's really the same thing.

00:04:46.470 --> 00:04:46.720
Right?

00:04:46.720 --> 00:04:48.050
It's really now minus or plus.

00:04:48.050 --> 00:04:49.570
If it was plus, it's now
going to be a minus.

00:04:49.570 --> 00:04:52.595
If it was a minus, it's now
going to be a plus.

00:04:52.595 --> 00:05:00.110
Minus or plus 2 times the
square root of 15.

00:05:00.110 --> 00:05:03.220
Or another way to view it is
that the two solutions here

00:05:03.220 --> 00:05:14.090
are 4 minus two roots of 15,
and 4 plus two roots of 15.

00:05:14.090 --> 00:05:18.420
These are both values of x
that'll satisfy this equation.

00:05:18.420 --> 00:05:20.670
And if this confuses you, what I
did, turning a plus or minus

00:05:20.670 --> 00:05:21.590
into minus plus.

00:05:21.590 --> 00:05:23.550
Let me just take a little
bit of an aside there.

00:05:23.550 --> 00:05:28.110
I could write this expression
up here as two expressions.

00:05:28.110 --> 00:05:29.850
That's what the plus
or minus really is.

00:05:29.850 --> 00:05:36.820
There's a negative 8 plus 2
roots of 15 over negative 2.

00:05:36.820 --> 00:05:42.100
And then there's a negative
8 minus 2 roots of 15 over

00:05:42.100 --> 00:05:43.060
negative 2.

00:05:43.060 --> 00:05:45.030
This one simplifies to--
negative 8 divided by

00:05:45.030 --> 00:05:46.260
negative 2 is 4.

00:05:46.260 --> 00:05:50.680
2 divided by negative
2 is negative 1.

00:05:50.680 --> 00:05:54.150
2 times a 4 minus the
square root of 15.

00:05:54.150 --> 00:05:56.580
And then over here you have
negative 8 divided by negative

00:05:56.580 --> 00:05:58.630
2, which is 4.

00:05:58.630 --> 00:06:00.870
And then negative 2 divided by
negative 2, which is plus the

00:06:00.870 --> 00:06:01.730
square of 15.

00:06:01.730 --> 00:06:04.010
And I just realized I made
a mistake up here.

00:06:04.010 --> 00:06:06.030
When we're dividing a 2 divided
by negative 2, we

00:06:06.030 --> 00:06:08.270
don't have this 2 over here.

00:06:08.270 --> 00:06:12.040
This is just a plus or
minus the root of 15.

00:06:12.040 --> 00:06:13.580
We just saw that when
I did it out here.

00:06:13.580 --> 00:06:16.300
So this is minus the
square root of 15.

00:06:16.300 --> 00:06:18.300
And this is plus the
square root of 15.

00:06:18.300 --> 00:06:20.540
So the two solutions for this
equation-- It's good that I

00:06:20.540 --> 00:06:23.820
took that little hiatus there,
that little aside there.

00:06:23.820 --> 00:06:27.220
The two solutions could be 4
minus the square root of 15,

00:06:27.220 --> 00:06:31.430
or x, or and, x could be 4 plus
the square root of 15.

00:06:31.430 --> 00:06:37.120
Either of those values of x
will satisfy this original

00:06:37.120 --> 00:06:39.210
quadratic equation.


Completing the square for quadratic formula

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.810
Complete the square on the
general quadratic equation.

00:00:03.810 --> 00:00:08.950
We have ax squared plus bx
plus c is equal to 0.

00:00:08.950 --> 00:00:11.160
So whenever I complete the
square, actually whenever I

00:00:11.160 --> 00:00:13.620
deal with any of these types
of quadratic equations, I

00:00:13.620 --> 00:00:18.710
always like to not have an a, or
a non 1 coefficient, on the

00:00:18.710 --> 00:00:21.630
x squared terms. So let's make
it into a 1 coefficient.

00:00:21.630 --> 00:00:23.140
And the easiest way to do
that is just divide

00:00:23.140 --> 00:00:24.330
everything by a.

00:00:24.330 --> 00:00:27.930
So we divide every term on the
left side by a, and of course

00:00:27.930 --> 00:00:31.110
we have to divide the right
side by a as well.

00:00:31.110 --> 00:00:34.300
And so the left side
will become x

00:00:34.300 --> 00:00:36.890
squared plus b over ax.

00:00:39.500 --> 00:00:41.630
And then I'll write c
over a over here.

00:00:41.630 --> 00:00:43.970
So we have some room to add and
subtract things so we can

00:00:43.970 --> 00:00:45.620
really complete the square.

00:00:45.620 --> 00:00:51.770
So plus c over a is equal to 0
divided by a, which is just

00:00:51.770 --> 00:00:54.960
going to be equal to 0.

00:00:54.960 --> 00:00:57.800
Now when we complete the
square, we've seen this

00:00:57.800 --> 00:01:01.710
multiple times before, what
we want to do is take the

00:01:01.710 --> 00:01:04.819
coefficient on the x term right
here, it's b over a,

00:01:04.819 --> 00:01:05.750
take half of it.

00:01:05.750 --> 00:01:10.980
This right here is two
times b over 2a.

00:01:10.980 --> 00:01:11.420
Right?

00:01:11.420 --> 00:01:12.420
The 2's cancel out.

00:01:12.420 --> 00:01:14.910
This is just 2 times
b over 2a.

00:01:14.910 --> 00:01:19.020
So you take half of it, half
of b over a is b over 2a.

00:01:19.020 --> 00:01:22.270
You take half of it, and then
you square it, and you add it

00:01:22.270 --> 00:01:22.830
right here.

00:01:22.830 --> 00:01:26.565
So plus b squared-- Let
me write it this way.

00:01:26.565 --> 00:01:31.780
b over 2a squared.

00:01:31.780 --> 00:01:34.800
And of course I can't just add
something to one side of the

00:01:34.800 --> 00:01:36.510
equation, that would change
the equation.

00:01:36.510 --> 00:01:38.720
I also either have to add it
to the other side, or just

00:01:38.720 --> 00:01:41.140
subtract it from the same side
that I'm adding it to.

00:01:41.140 --> 00:01:46.120
So I'll also subtract
the b over 2a

00:01:46.120 --> 00:01:48.410
squared just like that.

00:01:48.410 --> 00:01:52.770
Now, the whole point of doing
this is so these first three

00:01:52.770 --> 00:01:57.200
terms right here are a perfect
square trinomial.

00:01:57.200 --> 00:01:59.290
That's what completing the
square is all about.

00:01:59.290 --> 00:02:01.150
And we've seen the pattern
multiple times.

00:02:01.150 --> 00:02:08.590
If I have, let's say, m plus n
squared-- and I'm using m and

00:02:08.590 --> 00:02:10.650
n so we don't get confused
with the a's and b's

00:02:10.650 --> 00:02:11.860
and x's over here.

00:02:11.860 --> 00:02:14.430
But if I have m plus n squared,
we've seen multiple

00:02:14.430 --> 00:02:20.250
times, that's going to be equal
to m squared plus 2mn

00:02:20.250 --> 00:02:22.030
plus n squared.

00:02:22.030 --> 00:02:23.760
And here we have that
pattern now.

00:02:23.760 --> 00:02:25.770
That's the whole point behind
completing the square.

00:02:25.770 --> 00:02:29.390
That's the whole point behind
taking half of b over a--

00:02:29.390 --> 00:02:31.780
that's b over 2a-- and then
squaring it and adding it

00:02:31.780 --> 00:02:32.140
right here.

00:02:32.140 --> 00:02:33.480
We now fit that pattern.

00:02:33.480 --> 00:02:35.540
m is x.

00:02:35.540 --> 00:02:37.390
n is b over 2a.

00:02:37.390 --> 00:02:42.140
And 2mn, if I take an x times a
b over 2a, and multiply that

00:02:42.140 --> 00:02:44.420
by 2, I get b over ax.

00:02:44.420 --> 00:02:46.780
So this expression, right here,
this trinomial, the

00:02:46.780 --> 00:02:50.250
first three terms, it is a
perfect square trinomial, and

00:02:50.250 --> 00:02:58.410
we can write it as x plus
b over 2a squared.

00:02:58.410 --> 00:02:59.840
And then of course we
have all this other

00:02:59.840 --> 00:03:01.450
business right here.

00:03:01.450 --> 00:03:04.540
And then of course we
have all of this

00:03:04.540 --> 00:03:05.960
other stuff right here.

00:03:05.960 --> 00:03:11.360
Which is negative b over,
and let me just actually

00:03:11.360 --> 00:03:12.320
square it for you.

00:03:12.320 --> 00:03:17.410
So b over 2a squared
is negative b

00:03:17.410 --> 00:03:20.970
squared over 4a squared.

00:03:20.970 --> 00:03:23.660
And then I have this
plus c over a.

00:03:23.660 --> 00:03:25.980
But let's write it with the
same denominator here.

00:03:25.980 --> 00:03:31.530
So I could have c over a, or I
could multiply the numerator

00:03:31.530 --> 00:03:34.280
and the denominator by 4a.

00:03:34.280 --> 00:03:37.000
So if I multiply the numerator
by 4a, I get 4ac.

00:03:37.000 --> 00:03:43.040
If I multiply the denominator
by 4a, I get 4a squared.

00:03:43.040 --> 00:03:44.290
And the whole reason why I
multiplied the numerator and

00:03:44.290 --> 00:03:48.060
the denominator by 4a was
so that we have the same

00:03:48.060 --> 00:03:49.400
denominator right here.

00:03:49.400 --> 00:03:52.940
And, of course, that is going
to be equal to 0.

00:03:52.940 --> 00:03:55.540
And we could simplify it a
little bit more, or actually,

00:03:55.540 --> 00:03:57.550
well yeah we'll just simplify
it next in the next step.

00:03:57.550 --> 00:03:59.580
We don't want to skip
too many steps here.

00:03:59.580 --> 00:04:05.220
So you have x plus b
over 2a squared.

00:04:05.220 --> 00:04:09.380
And then we could say, plus-- we
could put the 4ac first, so

00:04:09.380 --> 00:04:12.510
we could say-- actually let's
just say, plus negative b

00:04:12.510 --> 00:04:23.230
squared, plus 4ac, all of that
over 4a squared is equal to 0.

00:04:23.230 --> 00:04:25.650
I didn't put the 4ac first,
I just put the negative b

00:04:25.650 --> 00:04:26.280
squared there.

00:04:26.280 --> 00:04:30.880
Now let's isolate this
squared binomial on

00:04:30.880 --> 00:04:31.600
the left hand side.

00:04:31.600 --> 00:04:34.600
And the easiest way we can do
that is to subtract this thing

00:04:34.600 --> 00:04:36.150
from both sides of
the equation.

00:04:36.150 --> 00:04:38.040
So let's do that.

00:04:38.040 --> 00:04:42.540
So you can imagine if we add
b-- let me do this in a

00:04:42.540 --> 00:04:44.060
different color.

00:04:44.060 --> 00:04:51.540
If we were to add positive b
squared minus 4ac over 4a

00:04:51.540 --> 00:04:53.950
squared on the left hand side,
those will cancel out and

00:04:53.950 --> 00:04:55.570
we're also going to add it
on the right hand side.

00:04:55.570 --> 00:05:01.780
Positive b squared minus
4ac over 4a squared.

00:05:01.780 --> 00:05:03.650
Anything I do the left I
have to do the right.

00:05:03.650 --> 00:05:04.900
What do we end up with?

00:05:07.540 --> 00:05:10.970
We end up with, on the left
hand side, these two guys

00:05:10.970 --> 00:05:11.470
cancel out.

00:05:11.470 --> 00:05:13.460
We have the same denomenator,
when you add the numerators,

00:05:13.460 --> 00:05:14.800
that cancels with that.

00:05:14.800 --> 00:05:17.270
The 4ac cancels with the
negative 4ac, these just

00:05:17.270 --> 00:05:18.640
completely cancel out.

00:05:18.640 --> 00:05:22.060
And on the left hand side,
you just have x

00:05:22.060 --> 00:05:25.640
plus b over 2a squared.

00:05:25.640 --> 00:05:30.090
And on the right hand side, you
have that being equal to b

00:05:30.090 --> 00:05:34.660
squared minus-- let me do
that in a blue color.

00:05:34.660 --> 00:05:43.190
b squared minus 4ac, all of
that over 4a squared.

00:05:43.190 --> 00:05:46.330
Now the next thing we probably
want to do, if we want to

00:05:46.330 --> 00:05:51.310
really solve for x, is to take
the square root of both sides

00:05:51.310 --> 00:05:52.510
of this equation.

00:05:52.510 --> 00:05:53.300
So let's do that.

00:05:53.300 --> 00:05:57.050
Let's take the square root of
both sides of this equation.

00:05:57.050 --> 00:06:02.220
And if, when we do that-- We
don't want to only take the

00:06:02.220 --> 00:06:06.120
positive square root because x
plus b be over 2a could be a

00:06:06.120 --> 00:06:08.220
negative number, or it could
be a positive number.

00:06:08.220 --> 00:06:10.070
So we want to take
the positive and

00:06:10.070 --> 00:06:11.740
negative square root.

00:06:11.740 --> 00:06:17.030
So we could say that the square
root, we could put the

00:06:17.030 --> 00:06:19.420
positive or negative here or,
since we're taking the square

00:06:19.420 --> 00:06:21.480
root of both sides,
we could put the

00:06:21.480 --> 00:06:23.120
positive or negative there.

00:06:23.120 --> 00:06:25.800
If you put the positive or
negative on both sides, it's

00:06:25.800 --> 00:06:27.390
really just telling you
the same thing.

00:06:27.390 --> 00:06:30.290
It really is all the different
combinations.

00:06:30.290 --> 00:06:32.720
If the negative square root over
here equals the negative

00:06:32.720 --> 00:06:36.750
square root over here, then it's
just another combination

00:06:36.750 --> 00:06:38.280
of the different positives
and negatives.

00:06:38.280 --> 00:06:42.120
So you could just write it as,
this square root is equal to

00:06:42.120 --> 00:06:44.410
the plus or minus square
root of b squared

00:06:44.410 --> 00:06:47.710
minus 4ac over 4a squared.

00:06:47.710 --> 00:06:49.710
Now what does this
simplify to?

00:06:49.710 --> 00:07:01.140
Well, the left hand side just
becomes x plus b over 2a is

00:07:01.140 --> 00:07:04.100
equal to-- and now it
gets interesting.

00:07:04.100 --> 00:07:06.960
And you might even start
recognizing parts of it.

00:07:06.960 --> 00:07:10.150
So let's take the plus or minus
square root of the top.

00:07:10.150 --> 00:07:12.480
What is that going to be?

00:07:12.480 --> 00:07:14.550
And you could just take the plus
or minus only of the top

00:07:14.550 --> 00:07:16.560
because, once again, the
same principles apply.

00:07:16.560 --> 00:07:18.370
There's no reason why you have
to do a plus or minus over a

00:07:18.370 --> 00:07:22.070
plus and minus and a plus or
minus on the left hand side.

00:07:22.070 --> 00:07:24.940
There's only one combination
here, where there's only one

00:07:24.940 --> 00:07:25.910
plus or minus on
the numerator.

00:07:25.910 --> 00:07:28.630
I apologize if that
confuses you.

00:07:28.630 --> 00:07:34.030
So let's write this as the plus
or minus square root of b

00:07:34.030 --> 00:07:37.770
squared minus 4ac over--
What's the square

00:07:37.770 --> 00:07:40.050
root of 4a a squared?

00:07:40.050 --> 00:07:41.300
Well, it's just going
to be 2a.

00:07:43.865 --> 00:07:44.420
Right?

00:07:44.420 --> 00:07:45.930
The square root of 4 is 2.

00:07:45.930 --> 00:07:47.800
The square root of
a squared is a.

00:07:47.800 --> 00:07:49.100
And we're almost there.

00:07:49.100 --> 00:07:52.620
To solve for x, we just have
to subtract b over 2a from

00:07:52.620 --> 00:07:54.070
both sides.

00:07:54.070 --> 00:07:57.640
We just have to subtract b over
2a from both sides of

00:07:57.640 --> 00:07:58.800
this equation.

00:07:58.800 --> 00:08:01.360
The left hand side, we just
end up with our x.

00:08:01.360 --> 00:08:06.500
And then the right hand side,
we have a negative b over 2a

00:08:06.500 --> 00:08:13.200
plus or minus the square root
of b squared minus 4ac-- all

00:08:13.200 --> 00:08:14.630
of that over 2a.

00:08:14.630 --> 00:08:16.890
Since we have the same
denominator, we can write this

00:08:16.890 --> 00:08:22.710
as negative b plus or minus the
square root of b squared

00:08:22.710 --> 00:08:27.730
minus 4ac-- all of
that over 2a.

00:08:27.730 --> 00:08:28.260
And we're done.

00:08:28.260 --> 00:08:31.300
We've solved for the x's and you
see there's actually two

00:08:31.300 --> 00:08:32.289
solutions here.

00:08:32.289 --> 00:08:34.340
There's one where you take the
positive square root, and

00:08:34.340 --> 00:08:35.620
there's another solution
where you take the

00:08:35.620 --> 00:08:36.380
negative square root.

00:08:36.380 --> 00:08:39.429
If this square root exists,
and if the positive and

00:08:39.429 --> 00:08:41.090
negative-- and if it's
not 0, you're

00:08:41.090 --> 00:08:42.789
going to have two solutions.

00:08:42.789 --> 00:08:46.360
And this, right here, this
result we have is-- Look, you

00:08:46.360 --> 00:08:50.540
give me any quadratic equation,
you give me the a,

00:08:50.540 --> 00:08:54.090
the b, and the c, we could now
substitute it into this

00:08:54.090 --> 00:08:57.340
formula essentially we just
derived right here, and I'll

00:08:57.340 --> 00:09:00.790
give you the roots, I'll give
you the x's for that quadratic

00:09:00.790 --> 00:09:03.290
equation-- the x's that satisfy

00:09:03.290 --> 00:09:04.490
that quadratic equation.

00:09:04.490 --> 00:09:07.960
And this formula, right here,
for solving any quadratic

00:09:07.960 --> 00:09:10.040
equation is called the
quadratic formula.

00:09:13.830 --> 00:09:15.820
And you could see it just
comes straight out of

00:09:15.820 --> 00:09:17.010
completing the square.

00:09:17.010 --> 00:09:19.570
There's no mystery,
magic here.

00:09:19.570 --> 00:09:21.700
But it's easily one of the
most useful formulas in

00:09:21.700 --> 00:09:22.165
mathematics.

00:09:22.165 --> 00:09:23.930
And I'm usually not
a huge proponent

00:09:23.930 --> 00:09:25.070
of memorizing things.

00:09:25.070 --> 00:09:28.570
But it probably will benefit
you in life if you did.

00:09:28.570 --> 00:09:30.390
Hope you enjoyed that.


Completing the square for vertex form

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.520
Use completing the square to
write the quadratic equation y

00:00:04.520 --> 00:00:09.390
is equal to negative 3x squared,
plus 24x, minus 27 in

00:00:09.390 --> 00:00:12.920
vertex form, and then
identify the vertex.

00:00:12.920 --> 00:00:15.340
So we'll see what vertex form
is, but we essentially

00:00:15.340 --> 00:00:18.880
complete the square, and we
generate the function, or we

00:00:18.880 --> 00:00:23.440
algebraically manipulate it so
it's in the form y is equal to

00:00:23.440 --> 00:00:29.970
A times x minus B
squared, plus C.

00:00:29.970 --> 00:00:32.390
We want to get the equation
into this form right here.

00:00:32.390 --> 00:00:37.450
This is vertex form
right there.

00:00:37.450 --> 00:00:40.770
And once you have it in vertex
form, you'll see that you can

00:00:40.770 --> 00:00:44.280
identify the x value of the
vertex as what value will make

00:00:44.280 --> 00:00:45.790
this expression equal to 0.

00:00:45.790 --> 00:00:47.740
So in this case it would be B.

00:00:47.740 --> 00:00:52.540
And the y value of the vertex,
if this is equal to 0, then

00:00:52.540 --> 00:00:54.590
the y value is just
going to be C.

00:00:54.590 --> 00:00:55.470
And we're going to see that.

00:00:55.470 --> 00:00:58.480
We're going to understand why
that is the vertex, why this

00:00:58.480 --> 00:01:00.310
vertex form is useful.

00:01:00.310 --> 00:01:02.310
So let's try to manipulate
this equation to get

00:01:02.310 --> 00:01:03.650
it into that form.

00:01:03.650 --> 00:01:06.060
So if we just rewrite it, the
first thing that immediately

00:01:06.060 --> 00:01:09.490
jumps out at me, at least, is
that all of these numbers are

00:01:09.490 --> 00:01:10.790
divisible by negative 3.

00:01:10.790 --> 00:01:13.170
And I just always find it
easier to manipulate an

00:01:13.170 --> 00:01:15.920
equation if I have a 1
coefficient out in front of

00:01:15.920 --> 00:01:16.680
the x squared.

00:01:16.680 --> 00:01:18.390
So let's just factor
out a negative 3

00:01:18.390 --> 00:01:19.670
right from the get-go.

00:01:19.670 --> 00:01:25.040
So we can rewrite this as y is
equal to negative 3 times x

00:01:25.040 --> 00:01:30.470
squared, minus 8x-- 24 divided
by negative 3 is

00:01:30.470 --> 00:01:32.752
negative 8-- plus 9.

00:01:32.752 --> 00:01:36.790
Negative 27 divided by negative
3 is positive 9.

00:01:36.790 --> 00:01:40.410
Let me actually write the
positive 9 out here.

00:01:40.410 --> 00:01:42.950
You're going to see in a second
why I'm doing that.

00:01:42.950 --> 00:01:46.020
Now, we want to be able to
express part of this

00:01:46.020 --> 00:01:47.290
expression as a perfect
square.

00:01:47.290 --> 00:01:48.950
That's what vertex
form does for us.

00:01:48.950 --> 00:01:52.170
We want to be able to express
part of this expression as a

00:01:52.170 --> 00:01:53.160
perfect square.

00:01:53.160 --> 00:01:54.525
Now how can we do that?

00:01:54.525 --> 00:01:57.650
Well, we have an x
squared minus 8x.

00:01:57.650 --> 00:02:01.250
So if we had a positive 16
here-- because, well, just

00:02:01.250 --> 00:02:03.675
think about it this way, if we
had negative 8, you divide it

00:02:03.675 --> 00:02:05.160
by 2, you get negative 4.

00:02:05.160 --> 00:02:08.110
You square that, it's
positive 16.

00:02:08.110 --> 00:02:09.870
So if you had a positive
16 here, this would

00:02:09.870 --> 00:02:10.639
be a perfect square.

00:02:10.639 --> 00:02:13.450
This would be x minus
4 squared.

00:02:13.450 --> 00:02:17.300
But you can't just willy-nilly
add a 16 there, you would

00:02:17.300 --> 00:02:20.075
either have to add a similar
amount to the other side, and

00:02:20.075 --> 00:02:22.000
you would have to scale it by
the negative 3 and all of

00:02:22.000 --> 00:02:25.580
that, or, you can just subtract
a 16 right here.

00:02:25.580 --> 00:02:26.840
I haven't changed
the expression.

00:02:26.840 --> 00:02:28.630
I'm adding a 16, subtracting
a 16.

00:02:28.630 --> 00:02:29.370
I've added a 0.

00:02:29.370 --> 00:02:30.770
I haven't it changed it.

00:02:30.770 --> 00:02:36.890
But what it allows me to do is
express this part of the

00:02:36.890 --> 00:02:39.160
equation as a perfect square.

00:02:39.160 --> 00:02:43.830
That right there is
x minus 4 squared.

00:02:43.830 --> 00:02:46.240
And if you're confused, how
did I know it was 16?

00:02:46.240 --> 00:02:48.720
Just think, I took negative
8, I divided by 2, I

00:02:48.720 --> 00:02:49.980
got negative 4.

00:02:49.980 --> 00:02:51.300
And I squared negative 4.

00:02:51.300 --> 00:02:53.640
This is negative 4 squared
right there.

00:02:53.640 --> 00:02:56.790
And then I have to subtract that
same amount so I don't

00:02:56.790 --> 00:02:58.130
change the equation.

00:02:58.130 --> 00:03:00.190
So that part is x
minus 4 squared.

00:03:00.190 --> 00:03:06.190
And then we still have this
negative 3 hanging out there.

00:03:06.190 --> 00:03:11.690
And then we have negative 16
plus 9, which is negative 7.

00:03:11.690 --> 00:03:13.230
So we're almost there.

00:03:13.230 --> 00:03:15.860
We have y equal to negative 3
times this whole thing, not

00:03:15.860 --> 00:03:16.350
quite there.

00:03:16.350 --> 00:03:17.880
To get it there, we just
multiply negative 3.

00:03:17.880 --> 00:03:21.480
We distribute the negative 3 on
to both of these terms. So

00:03:21.480 --> 00:03:26.820
we get y is equal to negative
3 times x, minus 4 squared.

00:03:26.820 --> 00:03:32.390
And negative 3 times negative
7 is positive 21.

00:03:32.390 --> 00:03:35.520
So we have it in our vertex
form, we're done with that.

00:03:35.520 --> 00:03:38.860
And if you want to think about
what the vertex is, I told you

00:03:38.860 --> 00:03:39.270
how to do it.

00:03:39.270 --> 00:03:41.670
You say, well, what's the
x value that makes

00:03:41.670 --> 00:03:43.680
this equal to 0?

00:03:43.680 --> 00:03:46.860
Well, in order for this term to
be 0, x minus 4 has to be

00:03:46.860 --> 00:03:48.130
equal to 0.

00:03:48.130 --> 00:03:51.950
x minus 4 has to be equal to 0,
or add 4 to both sides. x

00:03:51.950 --> 00:03:53.530
has to be equal to 4.

00:03:53.530 --> 00:03:56.580
And if x is equal to 4, this is
0, this whole thing becomes

00:03:56.580 --> 00:04:01.000
0, then y is equal to 21.

00:04:01.000 --> 00:04:04.420
So the vertex of this parabola--
I'll just do a

00:04:04.420 --> 00:04:08.850
quick graph right here-- the
vertex of this parabola occurs

00:04:08.850 --> 00:04:11.960
at the point 4, 21.

00:04:11.960 --> 00:04:13.500
So I'll draw it like this.

00:04:13.500 --> 00:04:16.480
Occurs at the point.

00:04:16.480 --> 00:04:21.550
If this is the point 4, if this
right here is the-- so

00:04:21.550 --> 00:04:23.960
this is the y-axis, that's
the x-axis-- so this is

00:04:23.960 --> 00:04:25.920
the point 4, 21.

00:04:25.920 --> 00:04:28.590
Now, that's either going to be
the minimum or the maximum

00:04:28.590 --> 00:04:31.860
point in our parabola, and to
think about whether it's the

00:04:31.860 --> 00:04:34.280
minimum or maximum point, think
about what happens.

00:04:34.280 --> 00:04:36.300
Let's explore this equation
a little bit.

00:04:36.300 --> 00:04:44.040
This thing, this x minus 4
squared is always greater than

00:04:44.040 --> 00:04:45.170
or equal to 0.

00:04:45.170 --> 00:04:45.510
Right?

00:04:45.510 --> 00:04:48.330
At worst it could be 0, but
you're taking a square, so

00:04:48.330 --> 00:04:50.560
it's going to be a non-negative
number.

00:04:50.560 --> 00:04:52.160
But when you take a non-negative
number, and then

00:04:52.160 --> 00:04:55.860
you multiply it by negative 3,
that guarantees that this

00:04:55.860 --> 00:05:00.660
whole thing is going to be
less than or equal to 0.

00:05:04.750 --> 00:05:07.550
So the best, the highest, value
that this function can

00:05:07.550 --> 00:05:10.360
attain, is when this
expression right

00:05:10.360 --> 00:05:11.630
here is equal to 0.

00:05:11.630 --> 00:05:13.680
And this expression is equal
to 0 when x is equal

00:05:13.680 --> 00:05:14.810
to 4 and y is 21.

00:05:14.810 --> 00:05:18.360
So this is the highest value
that the function can attain.

00:05:18.360 --> 00:05:20.660
It can only go down
from there.

00:05:20.660 --> 00:05:24.510
Because if you shift the x
around 4, then this expression

00:05:24.510 --> 00:05:27.850
right here will become, well,
it'll become non-zero.

00:05:27.850 --> 00:05:29.910
When you square it, it'll
become positive.

00:05:29.910 --> 00:05:31.225
When you multiply it
by negative 3,

00:05:31.225 --> 00:05:32.360
it'll become negative.

00:05:32.360 --> 00:05:34.600
So you're going to take a
negative number plus 21, it'll

00:05:34.600 --> 00:05:37.490
be less than 21, so your
parabola is going

00:05:37.490 --> 00:05:38.730
to look like this.

00:05:38.730 --> 00:05:44.160
Your parabola is going
to look like that.

00:05:44.160 --> 00:05:47.330
And that's why vertex
form is useful.

00:05:47.330 --> 00:05:52.970
You break it up into the part
of the equation that changes

00:05:52.970 --> 00:05:54.000
in value, and say,
well, what's its

00:05:54.000 --> 00:05:56.130
maximum value attained?

00:05:56.130 --> 00:05:57.150
That's the vertex.

00:05:57.150 --> 00:05:58.670
That happens when
x is equal to 4.

00:05:58.670 --> 00:06:00.320
And you know its y value.

00:06:00.320 --> 00:06:02.800
And because you have a negative
coefficient out here

00:06:02.800 --> 00:06:05.710
that's a negative 3, you know
that it's going to be a

00:06:05.710 --> 00:06:06.980
downward opening graph.

00:06:06.980 --> 00:06:09.560
If that was a positive 3, then
this thing would be, at

00:06:09.560 --> 00:06:13.160
minimum, 0 and it would be
an upward opening graph.


Parabola vertex and axis of symmetry

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.470
We need to find the vertex
and the axis of

00:00:03.470 --> 00:00:06.390
symmetry of this graph.

00:00:06.390 --> 00:00:07.950
The whole point of doing this
problem is so that you

00:00:07.950 --> 00:00:10.240
understand what the vertex
and axis of symmetry is.

00:00:10.240 --> 00:00:13.690
And just as a bit of a
refresher, if a parabola looks

00:00:13.690 --> 00:00:17.280
like this, the vertex is the
lowest point here, so this

00:00:17.280 --> 00:00:20.260
minimum point here, for an
upward opening parabola.

00:00:20.260 --> 00:00:23.610
If the parabola opens downward
like this, the vertex is the

00:00:23.610 --> 00:00:25.710
topmost point right like that.

00:00:25.710 --> 00:00:26.970
It's the maximum point.

00:00:26.970 --> 00:00:29.110
And the axis of symmetry is
the line that you could

00:00:29.110 --> 00:00:31.900
reflect the parabola around,
and it's symmetric.

00:00:31.900 --> 00:00:33.610
So that's the axis
of symmetry.

00:00:33.610 --> 00:00:37.065
That is a reflection of the
left-hand side along that axis

00:00:37.065 --> 00:00:37.710
of symmetry.

00:00:37.710 --> 00:00:40.330
Same thing if it's a
downward-opening parabola.

00:00:40.330 --> 00:00:42.370
And the general way of telling
the difference between an

00:00:42.370 --> 00:00:44.960
upward-opening and a
downward-opening parabola is

00:00:44.960 --> 00:00:50.420
that this will have a positive
coefficient on the x squared

00:00:50.420 --> 00:00:53.260
term, and this will have
a negative coefficient.

00:00:53.260 --> 00:00:54.910
And we'll see that in a little
bit more detail.

00:00:54.910 --> 00:00:56.460
So let's just work on this.

00:00:56.460 --> 00:00:59.450
Now, in order to figure out the
vertex, there's a quick

00:00:59.450 --> 00:01:02.600
and dirty formula, but I'm not
going to do the formula here

00:01:02.600 --> 00:01:05.000
because the formula really tells
you nothing about how

00:01:05.000 --> 00:01:05.540
you got it.

00:01:05.540 --> 00:01:07.740
But I'll show you how to apply
the formula at the end of this

00:01:07.740 --> 00:01:10.245
video, if you see this on a math
test and just want to do

00:01:10.245 --> 00:01:11.110
it really quickly.

00:01:11.110 --> 00:01:14.660
But we're going to do it the
slow, intuitive way first.

00:01:14.660 --> 00:01:17.940
So let's think about how we can
find either the maximum or

00:01:17.940 --> 00:01:19.710
the minimum point of
this parabola.

00:01:19.710 --> 00:01:21.800
So the best way I can think
of doing it is to

00:01:21.800 --> 00:01:22.900
complete the square.

00:01:22.900 --> 00:01:25.400
And it might seem like a very
foreign concept right now, but

00:01:25.400 --> 00:01:27.490
let's just do it one
step at a time.

00:01:27.490 --> 00:01:31.340
So I can rewrite this as y is
equal to-- well, I can factor

00:01:31.340 --> 00:01:32.330
out a negative 2.

00:01:32.330 --> 00:01:39.320
It's equal to negative 2 times
x squared minus 4x minus 4.

00:01:39.320 --> 00:01:41.065
And I'm going to put the
minus 4 out here.

00:01:41.065 --> 00:01:43.990
And this is where I'm going
to complete the square.

00:01:43.990 --> 00:01:46.810
Now, what I want to do is
express the stuff in the

00:01:46.810 --> 00:01:50.430
parentheses as a sum of a
perfect square and then some

00:01:50.430 --> 00:01:51.610
number over here.

00:01:51.610 --> 00:01:54.400
And I have x squared minus 4x.

00:01:54.400 --> 00:01:57.970
If I wanted this to be a perfect
square, it would be a

00:01:57.970 --> 00:02:00.550
perfect square if I had a
positive 4 over here.

00:02:00.550 --> 00:02:02.870
If I had a positive 4 over
there, then this would be a

00:02:02.870 --> 00:02:03.870
perfect square.

00:02:03.870 --> 00:02:07.070
It would be x minus 2 squared.

00:02:07.070 --> 00:02:09.780
And I got the 4, because I said,
well, I want whatever

00:02:09.780 --> 00:02:12.110
half of this number is,
so half of negative

00:02:12.110 --> 00:02:13.530
4 is negative 2.

00:02:13.530 --> 00:02:14.500
Let me square it.

00:02:14.500 --> 00:02:16.470
That'll give me a positive
4 right there.

00:02:16.470 --> 00:02:18.470
But I can't just add
a 4 willy-nilly to

00:02:18.470 --> 00:02:19.680
one side of an equation.

00:02:19.680 --> 00:02:21.550
I either have to add it to the
other side or I would have to

00:02:21.550 --> 00:02:22.900
then just subtract it.

00:02:22.900 --> 00:02:24.850
So here I haven't changed
equation.

00:02:24.850 --> 00:02:26.890
I added 4 and then
I subtracted 4.

00:02:26.890 --> 00:02:30.050
I just added zero to this little
expression here, so it

00:02:30.050 --> 00:02:30.880
didn't change it.

00:02:30.880 --> 00:02:34.270
But what it does allow me to do
is express this part right

00:02:34.270 --> 00:02:42.170
here as a perfect square. x
squared minus 4x plus 4 is x

00:02:42.170 --> 00:02:45.470
minus 2 squared.

00:02:45.470 --> 00:02:48.380
It is x minus 2 squared.

00:02:48.380 --> 00:02:53.340
And then you have this
negative 2 out front

00:02:53.340 --> 00:02:55.420
multiplying everything, and
then you have a negative 4

00:02:55.420 --> 00:02:59.090
minus negative 4, minus
8, just like that.

00:02:59.090 --> 00:03:02.490
So you have y is equal to
negative 2 times this entire

00:03:02.490 --> 00:03:05.200
thing, and now we can multiply
out the negative 2 again.

00:03:05.200 --> 00:03:06.810
So we can distribute it.

00:03:06.810 --> 00:03:11.050
Y is equal to negative 2 times
x minus 2 squared.

00:03:11.050 --> 00:03:17.420
And then negative 2 times
negative 8 is plus 16.

00:03:17.420 --> 00:03:19.270
Now, all I did is algebraically

00:03:19.270 --> 00:03:21.110
arrange this equation.

00:03:21.110 --> 00:03:23.970
But what this allows us to do
is think about what the

00:03:23.970 --> 00:03:26.370
maximum or minimum point
of this equation is.

00:03:26.370 --> 00:03:29.250
So let's just explore
this a little bit.

00:03:29.250 --> 00:03:32.300
This quantity right here, x
minus 2 squared, if you're

00:03:32.300 --> 00:03:34.360
squaring anything, this
is always going to

00:03:34.360 --> 00:03:36.380
be a positive quantity.

00:03:36.380 --> 00:03:41.220
That right there is
always positive.

00:03:41.220 --> 00:03:43.300
But it's being multiplied
by a negative number.

00:03:43.300 --> 00:03:46.210
So if you look at the larger
context, if you look at the

00:03:46.210 --> 00:03:49.160
always positive multiplied by
the negative 2, that's going

00:03:49.160 --> 00:03:53.450
to be always negative.

00:03:53.450 --> 00:03:59.150
And the more positive that this
number becomes when you

00:03:59.150 --> 00:04:01.740
multiply it by a negative, the
more negative this entire

00:04:01.740 --> 00:04:02.990
expression becomes.

00:04:05.340 --> 00:04:07.560
So if you think about it,
this is going to be a

00:04:07.560 --> 00:04:10.010
downward-opening parabola.

00:04:10.010 --> 00:04:12.040
We have a negative coefficient
out here.

00:04:12.040 --> 00:04:15.920
And the maximum point on this
downward-opening parabola is

00:04:15.920 --> 00:04:21.640
when this expression right here
is as small as possible.

00:04:21.640 --> 00:04:24.290
If this gets any larger, it's
just multiplied by a negative

00:04:24.290 --> 00:04:26.630
number, and then you subtract
it from 16.

00:04:26.630 --> 00:04:29.920
So if this expression right here
is 0, then we have our

00:04:29.920 --> 00:04:32.490
maximum y value, which is 16.

00:04:32.490 --> 00:04:35.250
So how do we get x is
equal to 0 here?

00:04:35.250 --> 00:04:37.860
Well, the way to get x minus
2 equal to 0-- so

00:04:37.860 --> 00:04:39.040
let's just do it.

00:04:39.040 --> 00:04:41.400
x minus 2 is equal to 0,
so that happens when

00:04:41.400 --> 00:04:43.220
x is equal to 2.

00:04:43.220 --> 00:04:46.020
So when x is equal to 2,
this expression is 0.

00:04:46.020 --> 00:04:49.240
0 times a negative number,
it's all 0, and then y is

00:04:49.240 --> 00:04:52.240
equal to 16.

00:04:52.240 --> 00:04:55.320
This is our vertex, this
is our maximum point.

00:04:55.320 --> 00:04:58.170
We just reasoned through it,
just looking at the algebra,

00:04:58.170 --> 00:05:01.300
that the highest value this
can take on is 16.

00:05:01.300 --> 00:05:04.390
As x moves away from 2 in
the positive or negative

00:05:04.390 --> 00:05:07.420
direction, this quantity right
here, it might be negative or

00:05:07.420 --> 00:05:09.630
positive, but when you square
it, it's going to be positive.

00:05:09.630 --> 00:05:11.240
And when you multiply it by
negative 2, it's going to

00:05:11.240 --> 00:05:13.440
become negative and it's going
to subtract from 16.

00:05:13.440 --> 00:05:18.010
So our vertex right here
is x is equal to 2.

00:05:18.010 --> 00:05:20.170
Actually, let's say each
of these units are 2.

00:05:20.170 --> 00:05:28.060
So this is 2, 4, 6,
8, 10, 12, 14, 16.

00:05:28.060 --> 00:05:29.070
So my vertex is here.

00:05:29.070 --> 00:05:33.150
That is the absolute maximum
point for this parabola.

00:05:33.150 --> 00:05:37.020
And its axis of symmetry is
going to be along the line x

00:05:37.020 --> 00:05:40.470
is equal to 2, along the
vertical line x is equal to 2.

00:05:40.470 --> 00:05:46.300
That is going to be its
axis of symmetry.

00:05:46.300 --> 00:05:48.550
And now if we're just curious
for a couple of other points,

00:05:48.550 --> 00:05:51.060
just because we want to plot
this thing, we could say,

00:05:51.060 --> 00:05:54.300
well, what happens when
x is equal to 0?

00:05:54.300 --> 00:05:54.990
That's an easy one.

00:05:54.990 --> 00:05:58.420
When x is equal to 0,
y is equal to 8.

00:05:58.420 --> 00:06:01.930
So when x is equal to 0, we have
1, 2, 3, 4-- oh, well,

00:06:01.930 --> 00:06:02.530
these are 2.

00:06:02.530 --> 00:06:04.050
2, 4, 6, 8.

00:06:04.050 --> 00:06:05.090
It's right there.

00:06:05.090 --> 00:06:06.250
This is an axis of symmetry.

00:06:06.250 --> 00:06:09.760
So when x is equal to 3, y is
also going to be equal to 8.

00:06:09.760 --> 00:06:13.390
So this parabola is a really
steep and narrow one that

00:06:13.390 --> 00:06:16.180
looks something like this, where
this right here is the

00:06:16.180 --> 00:06:17.370
maximum point.

00:06:17.370 --> 00:06:20.260
Now I told you this is the slow
and intuitive way to do

00:06:20.260 --> 00:06:20.950
the problem.

00:06:20.950 --> 00:06:23.350
If you wanted a quick and dirty
way to figure out a

00:06:23.350 --> 00:06:25.300
vertex, there is a formula
that you can derive it

00:06:25.300 --> 00:06:28.330
actually, doing this exact same
process we just did, but

00:06:28.330 --> 00:06:33.130
the formula for the vertex, or
the x-value of the vertex, or

00:06:33.130 --> 00:06:37.500
the axis of symmetry, is x is
equal to negative b over 2a.

00:06:37.500 --> 00:06:40.140
So if we just apply this-- but,
you know, this is just

00:06:40.140 --> 00:06:42.660
kind of mindless application
of a formula.

00:06:42.660 --> 00:06:45.130
I wanted to show you the
intuition why this formula

00:06:45.130 --> 00:06:46.010
even exists.

00:06:46.010 --> 00:06:49.240
But if you just mindlessly
apply this, you'll get--

00:06:49.240 --> 00:06:50.490
what's b here?

00:06:50.490 --> 00:06:55.340
So x is equal to negative--
b here is 8.

00:06:55.340 --> 00:06:59.240
8 over 2 times a.

00:06:59.240 --> 00:07:01.690
a right here is a negative 2.

00:07:01.690 --> 00:07:03.520
2 times negative 2.

00:07:03.520 --> 00:07:05.270
So what is that going
to be equal to?

00:07:05.270 --> 00:07:10.230
It is negative 8 over negative
4, which is equal to 2, which

00:07:10.230 --> 00:07:13.390
is the exact same thing we
got by reasoning it out.

00:07:13.390 --> 00:07:16.400
And when x is equal to
2, y is equal to 16.

00:07:16.400 --> 00:07:18.090
Same exact result there.

00:07:18.090 --> 00:07:21.250
That's the point 2 comma 16.


Example 2: Completing the square

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:04.670
Use completing the square
to find the roots of the

00:00:04.670 --> 00:00:06.870
quadratic equation right here.

00:00:06.870 --> 00:00:11.830
And when anyone talks about
roots, this just means find

00:00:11.830 --> 00:00:17.270
the x's where y is equal to 0.

00:00:17.270 --> 00:00:18.940
That's what a root is.

00:00:18.940 --> 00:00:23.270
A root is an x value that will
make this quadratic function

00:00:23.270 --> 00:00:26.650
equal 0, that will
make y equal 0.

00:00:26.650 --> 00:00:30.205
So to find the x's, let's
just make y equal 0 and

00:00:30.205 --> 00:00:31.150
then solve for x.

00:00:31.150 --> 00:00:39.310
So we get 0 is equal to 4x
squared plus 40x, plus 280.

00:00:39.310 --> 00:00:41.240
Now, the first step that we
might want to do, just because

00:00:41.240 --> 00:00:45.430
it looks like all three of these
terms are divisible by

00:00:45.430 --> 00:00:48.810
4, is just divide both sides
of this equation by 4.

00:00:48.810 --> 00:00:50.970
That'll make our math a
little bit simpler.

00:00:50.970 --> 00:00:54.800
So let's just divide everything
by 4 here.

00:00:54.800 --> 00:01:00.300
If we just divide everything by
4, we get 0 is equal to x

00:01:00.300 --> 00:01:12.160
squared plus 10x, plus-- 280
divided by 4 is 70-- plus 70.

00:01:12.160 --> 00:01:14.320
Now, they say use completing the
square, and actually, let

00:01:14.320 --> 00:01:16.700
me write that 70 a little bit
further out, and you'll see

00:01:16.700 --> 00:01:17.820
why I did that in a second.

00:01:17.820 --> 00:01:21.110
So let me just write a plus 70
over here, just to have kind

00:01:21.110 --> 00:01:22.280
of an awkward space here.

00:01:22.280 --> 00:01:24.690
And you'll see what I'm about
to do with this space, that

00:01:24.690 --> 00:01:27.160
has everything to do with
completing the square.

00:01:27.160 --> 00:01:29.330
So they say use completing the
square, which means, turn

00:01:29.330 --> 00:01:31.410
this, if you can, into
a perfect square.

00:01:31.410 --> 00:01:33.470
Turn at least part of this
expression into a perfect

00:01:33.470 --> 00:01:37.650
square, and then we can use that
to actually solve for x.

00:01:37.650 --> 00:01:39.960
So how do we turn this into
a perfect square?

00:01:39.960 --> 00:01:42.280
Well, we have a 10x here.

00:01:42.280 --> 00:01:44.440
And we know that we can turn
this into a perfect square

00:01:44.440 --> 00:01:48.050
trinomial if we take 1/2 of the
10, which is 5, and then

00:01:48.050 --> 00:01:49.300
we square that.

00:01:49.300 --> 00:01:52.880
So 1/2 of 10 is 5, you square
it, you add a 25.

00:01:52.880 --> 00:01:56.130
Now, you can't just willy-nilly
add a 25 to one

00:01:56.130 --> 00:01:58.540
side of the equation without
doing something to the other,

00:01:58.540 --> 00:02:01.780
or without just subtracting
the 25 right here.

00:02:01.780 --> 00:02:02.160
Right?

00:02:02.160 --> 00:02:04.640
Think about it, I have not
changed the equation.

00:02:04.640 --> 00:02:07.280
I've added 25 and I've
subtracted 25.

00:02:07.280 --> 00:02:09.889
So I've added nothing to
the right-hand side.

00:02:09.889 --> 00:02:12.260
I could add a billion and
subtract a billion and not

00:02:12.260 --> 00:02:13.160
change the equation.

00:02:13.160 --> 00:02:16.610
So I have not changed the
equation at all right here.

00:02:16.610 --> 00:02:19.880
But what I have done is I've
made it possible to express

00:02:19.880 --> 00:02:23.130
these three terms as
a perfect square.

00:02:23.130 --> 00:02:25.460
That right there,
2 times 5 is 10.

00:02:25.460 --> 00:02:28.830
5 squared is 25.

00:02:28.830 --> 00:02:33.580
So that is x plus 5 squared.

00:02:33.580 --> 00:02:35.880
And if you don't believe
me, multiply it out.

00:02:35.880 --> 00:02:39.250
You're going to have an x
squared plus 5x, plus 5x,

00:02:39.250 --> 00:02:43.650
which will give you 10x, plus
5 squared, which is 25.

00:02:43.650 --> 00:02:46.290
So those first three terms
become that, and then the

00:02:46.290 --> 00:02:51.670
second two terms, right there,
you just add them.

00:02:51.670 --> 00:02:53.870
Let's see, negative
25 plus 70.

00:02:53.870 --> 00:02:57.560
Let's see, negative 20 plus 70
would be positive 50, and then

00:02:57.560 --> 00:03:01.680
you have another 5,
so it's plus 45.

00:03:01.680 --> 00:03:05.730
So we've just algebraically
manipulated this equation.

00:03:05.730 --> 00:03:10.460
And we get 0 is equal to x
plus 5 squared, plus 45.

00:03:10.460 --> 00:03:12.810
Now, we could've, from the
beginning if we wanted, we

00:03:12.810 --> 00:03:13.920
could've tried to factor it.

00:03:13.920 --> 00:03:15.970
But what we're going to do here,
this will always work.

00:03:15.970 --> 00:03:18.360
Even if you have crazy decimal
numbers here, you can solve

00:03:18.360 --> 00:03:20.810
for x using the method
we're doing here,

00:03:20.810 --> 00:03:22.420
completing the square.

00:03:22.420 --> 00:03:25.590
So to solve for x, let's just
subtract 45 from both sides of

00:03:25.590 --> 00:03:26.840
this equation.

00:03:29.620 --> 00:03:34.330
And so the left-hand side of
this equation becomes negative

00:03:34.330 --> 00:03:40.730
45, and the right-hand
side will be just

00:03:40.730 --> 00:03:43.260
the x plus 5 squared.

00:03:43.260 --> 00:03:46.570
These guys, right here,
cancel out.

00:03:46.570 --> 00:03:49.250
Now, normally if I look at
something like this I'll say,

00:03:49.250 --> 00:03:51.280
OK, let's just take the
square root of both

00:03:51.280 --> 00:03:52.670
sides of this equation.

00:03:52.670 --> 00:03:56.000
And so you might be tempted to
take the square root of both

00:03:56.000 --> 00:03:58.500
sides of this equation, but
immediately when you do that,

00:03:58.500 --> 00:03:59.950
you'll notice something
strange.

00:03:59.950 --> 00:04:01.800
We're trying to take
the square root

00:04:01.800 --> 00:04:03.760
of a negative number.

00:04:03.760 --> 00:04:06.180
And if we're dealing with real
numbers, which is everything

00:04:06.180 --> 00:04:08.890
we've dealt with so far, you
can't take a square root of a

00:04:08.890 --> 00:04:09.460
negative number.

00:04:09.460 --> 00:04:13.380
There is no real number that if
you square it will give you

00:04:13.380 --> 00:04:15.040
a negative number.

00:04:15.040 --> 00:04:19.740
So it's not possible-- I don't
care what you make x-- it is

00:04:19.740 --> 00:04:22.690
not possible to add x to
5 and square it, and

00:04:22.690 --> 00:04:23.820
get a negative number.

00:04:23.820 --> 00:04:33.680
So there is no x that can
satisfy-- if we assume that is

00:04:33.680 --> 00:04:39.450
x is a real number-- that can
satisfy this equation.

00:04:39.450 --> 00:04:42.200
Because I don't care what x you
put here, what real x you

00:04:42.200 --> 00:04:43.920
put here, you add 5 to it, you
square it, there's no way

00:04:43.920 --> 00:04:45.210
you're going to get
a negative number.

00:04:45.210 --> 00:04:49.000
So there's no x that can satisfy
this equation, so we

00:04:49.000 --> 00:04:52.260
could say there are no-- and
I'm using the word real

00:04:52.260 --> 00:04:56.280
because in Algebra 2 you'll
learn that there are things

00:04:56.280 --> 00:04:58.310
called complex numbers, but
don't worry about that right

00:04:58.310 --> 00:05:08.236
now-- but there no real roots
to the quadratic equation.

00:05:12.450 --> 00:05:13.340
And we're done.

00:05:13.340 --> 00:05:15.070
And actually, if you had tried
to factor it, you would have

00:05:15.070 --> 00:05:16.950
found it very difficult,
because this is not a

00:05:16.950 --> 00:05:21.530
factorable expression right
here, and you know it because

00:05:21.530 --> 00:05:23.680
there's no real roots.


Example 1: Completing the square

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:05.170
Use completing the square to
find the value of c that makes

00:00:05.170 --> 00:00:08.890
x squared minus 44x plus c--
so we can just figure out a

00:00:08.890 --> 00:00:11.690
c-- that makes it a perfect
square trinomial-- and a

00:00:11.690 --> 00:00:14.590
trinomial is just a polynomial
with three terms here.

00:00:14.590 --> 00:00:16.990
Then write the expression as
the square of a binomial.

00:00:16.990 --> 00:00:23.690
So we have x squared
minus 44x plus c.

00:00:23.690 --> 00:00:26.370
So how do we make this into
a perfect square?

00:00:26.370 --> 00:00:28.460
Well, if you just look at the
traditional pattern for a

00:00:28.460 --> 00:00:29.920
perfect square, let's
just think of it in

00:00:29.920 --> 00:00:33.350
terms of x plus a squared.

00:00:33.350 --> 00:00:37.860
That's the same thing as x plus
a times x plus a, and

00:00:37.860 --> 00:00:39.260
we've seen this before.

00:00:39.260 --> 00:00:42.150
And if you were to multiply this
out, that's x times x,

00:00:42.150 --> 00:00:48.170
which is x squared, plus
x times a, which is ax.

00:00:48.170 --> 00:00:50.850
Plus a times x, which is ax.

00:00:50.850 --> 00:00:53.490
Plus a times a, which
is a squared.

00:00:53.490 --> 00:00:59.365
So it's x squared plus 2ax,
these two, you have an ax plus

00:00:59.365 --> 00:01:03.610
an ax gives you 2ax,
plus a squared.

00:01:03.610 --> 00:01:06.820
So if we can get this into this
pattern, where I have

00:01:06.820 --> 00:01:10.120
whatever value is here, if
I take half of it, right?

00:01:10.120 --> 00:01:11.490
This is going to be 2a here.

00:01:11.490 --> 00:01:14.240
If I take half of it and square
it over here, then this

00:01:14.240 --> 00:01:16.150
will be a perfect square.

00:01:16.150 --> 00:01:21.530
So if we look over here, this
thing right here is 2a, if we

00:01:21.530 --> 00:01:23.960
want to pattern match, if we
want to make this look like a

00:01:23.960 --> 00:01:25.070
perfect square.

00:01:25.070 --> 00:01:26.300
That has to be 2a.

00:01:26.300 --> 00:01:29.550
So negative 44 is equal to 2a.

00:01:29.550 --> 00:01:32.790
And this, right here, this c, if
we pattern match, c has to

00:01:32.790 --> 00:01:36.770
be equal to a squared.

00:01:36.770 --> 00:01:38.100
So what's a?

00:01:38.100 --> 00:01:41.000
Well, if we know negative 44
is 2a, we can divide both

00:01:41.000 --> 00:01:45.140
sides of that by 2.

00:01:45.140 --> 00:01:50.310
And we know that negative 22
has got to be equal to a. a

00:01:50.310 --> 00:01:52.770
has got to be equal
to negative 22.

00:01:52.770 --> 00:01:55.190
a is half of the coefficient
right here.

00:01:55.190 --> 00:01:58.100
It's half of negative 44.

00:01:58.100 --> 00:02:00.250
And whenever you complete the
square, it's always going to

00:02:00.250 --> 00:02:02.420
be half of the coefficient
right here.

00:02:02.420 --> 00:02:05.840
Now, if that's a, what
does c need to be?

00:02:05.840 --> 00:02:08.800
Well, c needs to be a squared
in order for this to be a

00:02:08.800 --> 00:02:09.430
perfect square.

00:02:09.430 --> 00:02:14.880
So c needs to equal negative
22 squared.

00:02:14.880 --> 00:02:17.390
And we can figure out
what that is.

00:02:17.390 --> 00:02:20.880
22 times 22, we could put the
negative later-- actually it's

00:02:20.880 --> 00:02:22.390
just going to be the same thing
because the negative

00:02:22.390 --> 00:02:24.040
times a negative
is a positive.

00:02:24.040 --> 00:02:26.900
2 times 22 is 44, put a 0.

00:02:26.900 --> 00:02:29.580
2 times 22 is 44.

00:02:29.580 --> 00:02:32.200
Get a 4, get an 8, get a 4.

00:02:32.200 --> 00:02:33.950
So it's 484.

00:02:33.950 --> 00:02:41.350
So if we were to rewrite this
as x squared minus 44x plus

00:02:41.350 --> 00:02:47.030
484, then this is a perfect
square trinomial.

00:02:47.030 --> 00:02:48.050
Or we could write
it like this.

00:02:48.050 --> 00:02:53.130
This is x squared minus 2
times-- or maybe I should

00:02:53.130 --> 00:03:00.750
write it this way-- plus 2
times negative 22x plus

00:03:00.750 --> 00:03:03.350
negative 22 squared.

00:03:03.350 --> 00:03:06.140
And when you view it that way,
it's pretty clear that this is

00:03:06.140 --> 00:03:08.170
a perfect square, and if you
were to factor it, it's the

00:03:08.170 --> 00:03:14.040
same thing as x minus
22 times x minus 22,

00:03:14.040 --> 00:03:17.980
or x minus 22 squared.

00:03:17.980 --> 00:03:20.530
These are all equivalent
statements.