Analyzing model in vertex form

WEBVTT
Kind: captions
Language: en

00:00:00.486 --> 00:00:03.373
- [Sal] An object is
launched from a platform.

00:00:03.373 --> 00:00:07.102
Its height in meters, x
seconds after the launch,

00:00:07.102 --> 00:00:10.888
is modeled by: h of x is
equal to negative five

00:00:10.888 --> 00:00:14.138
times x minus four squared plus 180.

00:00:15.520 --> 00:00:18.198
Normally, when they talk
about seconds or time,

00:00:18.198 --> 00:00:19.610
they usually would use the variable t,

00:00:19.610 --> 00:00:22.334
but we can roll with x being that.

00:00:22.334 --> 00:00:25.579
Let's think about what's
going to happen here.

00:00:25.579 --> 00:00:27.323
Lemme just visualize it.

00:00:27.323 --> 00:00:30.323
Lemme draw an h axis for our height.

00:00:32.603 --> 00:00:35.411
Let me draw an x axis.

00:00:35.411 --> 00:00:36.494
An x axis.

00:00:37.929 --> 00:00:40.216
At time x is equal to zero.

00:00:40.216 --> 00:00:41.239
We're on a platform,

00:00:41.239 --> 00:00:43.852
so we're already gonna have some height.

00:00:43.852 --> 00:00:45.933
At time x is equal to zero.

00:00:45.933 --> 00:00:47.689
I'm used to saying time t equals zero,

00:00:47.689 --> 00:00:49.675
but at time x is equal to zero,

00:00:49.675 --> 00:00:50.805
we're already gonna have some height

00:00:50.805 --> 00:00:52.614
'cause we're on some platform.

00:00:52.614 --> 00:00:55.743
And then we're gonna
launch this projectile.

00:00:55.743 --> 00:01:00.006
And, it's gonna go in
the shape of a parabola,

00:01:00.006 --> 00:01:01.750
and it's gonna be a
downward-opening parabola.

00:01:01.750 --> 00:01:03.069
You might say: "Sal, how do you know

00:01:03.069 --> 00:01:05.834
"it's gonna be a
downward-opening parabola?"

00:01:05.834 --> 00:01:07.963
Gonna look something like that.

00:01:07.963 --> 00:01:09.734
I didn't draw it exactly perfectly,

00:01:09.734 --> 00:01:12.133
but you get, hopefully, the point.

00:01:12.133 --> 00:01:13.470
The reason why I knew it was a parabola,

00:01:13.470 --> 00:01:15.505
in particular a downward-opening parabola,

00:01:15.505 --> 00:01:18.267
is when you look at what's going on here.

00:01:18.267 --> 00:01:21.316
This is written in vertex
form but it's a quadratic.

00:01:21.316 --> 00:01:24.851
In vertex form, you have an
expression with x squared,

00:01:24.851 --> 00:01:26.973
and then you're multiplying by

00:01:26.973 --> 00:01:28.630
negative five right over here.

00:01:28.630 --> 00:01:31.879
This tells us that it's
gonna be downward-opening.

00:01:31.879 --> 00:01:33.589
If you were to multiply this out,

00:01:33.589 --> 00:01:35.824
x minus four squared is gonna be x squared

00:01:35.824 --> 00:01:37.124
plus something else plus something else,

00:01:37.124 --> 00:01:37.957
then you're gonna have to multiply

00:01:37.957 --> 00:01:39.738
all those terms by negative five,

00:01:39.738 --> 00:01:43.681
your leading term is gonna
be negative five x squared.

00:01:43.681 --> 00:01:46.266
Once again, it's gonna be
a downward-opening parabola

00:01:46.266 --> 00:01:48.826
that looks something like that.

00:01:48.826 --> 00:01:52.522
So, given this visual
intuition that we have,

00:01:52.522 --> 00:01:55.244
let's see if we can answer
some questions about it.

00:01:55.244 --> 00:01:56.545
The first one I'd like to answer

00:01:56.545 --> 00:01:58.737
is how high is the platform.

00:01:58.737 --> 00:02:00.820
How high is the platform?

00:02:03.500 --> 00:02:05.435
How high is the platform?

00:02:05.435 --> 00:02:07.267
I encourage you to pause the video,

00:02:07.267 --> 00:02:09.441
and try to figure that out.

00:02:09.441 --> 00:02:12.701
What is that value right over there?

00:02:12.701 --> 00:02:14.504
Well, as you can see,

00:02:14.504 --> 00:02:17.844
we are at that value
at time x equals zero.

00:02:17.844 --> 00:02:20.441
So to figure out how high is the platform,

00:02:20.441 --> 00:02:24.380
we essentially just have
to evaluate h of zero.

00:02:24.380 --> 00:02:26.619
That's going to be negative five

00:02:26.619 --> 00:02:29.702
times negative four squared plus 180.

00:02:31.739 --> 00:02:34.261
I just substituted x with zero.

00:02:34.261 --> 00:02:36.886
Negative four squared is 16.

00:02:36.886 --> 00:02:40.053
Negative five times 16 is negative 80.

00:02:41.196 --> 00:02:42.292
Plus 180.

00:02:42.292 --> 00:02:45.620
So this is going to be equal to 100.

00:02:45.620 --> 00:02:48.537
So the platform is 100 meters tall.

00:02:49.764 --> 00:02:53.097
Remember, the height is given in meters.

00:02:53.949 --> 00:02:56.592
Now, the next question I have is,

00:02:56.592 --> 00:03:00.759
how many seconds after launch
do we hit our maximum height?

00:03:01.830 --> 00:03:03.902
So our maximum height,
if we're talking about

00:03:03.902 --> 00:03:07.191
a downward-opening parabola,
it's going to be our vertex,

00:03:07.191 --> 00:03:09.307
is going to be our maximum height.

00:03:09.307 --> 00:03:12.974
And so, the x value of that would tell us

00:03:14.302 --> 00:03:18.469
how long after takeoff,
how long after, or launch,

00:03:21.221 --> 00:03:24.230
do we hit the maximum height.

00:03:24.230 --> 00:03:26.058
Trying to use a color you can see.

00:03:26.058 --> 00:03:28.991
What is this x value right over here?

00:03:28.991 --> 00:03:30.783
Once again, pause the video,

00:03:30.783 --> 00:03:33.475
and see if you can figure it out.

00:03:33.475 --> 00:03:36.879
We're trying to answer how long

00:03:36.879 --> 00:03:37.712
after

00:03:38.811 --> 00:03:39.644
launch

00:03:44.334 --> 00:03:46.167
is the maximum height.

00:03:47.207 --> 00:03:51.668
Well, it's going to be the
x coordinate of our vertex.

00:03:51.668 --> 00:03:53.577
How do we figure that out?

00:03:53.577 --> 00:03:56.304
Well, this quadratic has
actually been written

00:03:56.304 --> 00:04:00.287
already in vertex form, which
makes it sound like it should

00:04:00.287 --> 00:04:04.229
be relatively easy to figure
out the vertex over here.

00:04:04.229 --> 00:04:07.864
To appreciate that, we
have to see the structure

00:04:07.864 --> 00:04:10.345
in the expression, is one
way to think about it.

00:04:10.345 --> 00:04:11.402
Let's think about what's going on.

00:04:11.402 --> 00:04:13.188
You have this 180.

00:04:13.188 --> 00:04:14.896
And then you this other
term right over here.

00:04:14.896 --> 00:04:16.764
Anything squared is gonna be nonnegative.

00:04:16.764 --> 00:04:19.694
So x minus four squared is
always gonna be nonnegative.

00:04:19.694 --> 00:04:21.965
But then you always multiply
that times a negative five,

00:04:21.965 --> 00:04:25.715
so this whole thing is
gonna be non-positive.

00:04:26.996 --> 00:04:29.746
So, it will never add to the 180.

00:04:31.746 --> 00:04:34.622
Your maximum value is when
this term right over here

00:04:34.622 --> 00:04:36.531
is going to be equal to zero.

00:04:36.531 --> 00:04:38.849
And when is this term
going to be equal to zero?

00:04:38.849 --> 00:04:40.596
In order to make this term equal to zero,

00:04:40.596 --> 00:04:43.519
then x minus four needs
to be equal to zero.

00:04:43.519 --> 00:04:46.037
The only way to get x minus
four to be equal to zero

00:04:46.037 --> 00:04:48.632
is if x is equal to four.

00:04:48.632 --> 00:04:49.892
Just by looking at this, you say:

00:04:49.892 --> 00:04:50.855
"Hey, what makes this zero?"

00:04:50.855 --> 00:04:51.688
Four.

00:04:51.688 --> 00:04:53.685
X equals four will make this zero.

00:04:53.685 --> 00:04:55.862
This is right over there.

00:04:55.862 --> 00:05:00.029
If I were to write h of
four, this is going to be,

00:05:00.987 --> 00:05:02.819
this term is gonna go to zero,

00:05:02.819 --> 00:05:06.560
and you're gonna be left with the 180.

00:05:06.560 --> 00:05:08.831
There you go, this right over here.

00:05:08.831 --> 00:05:10.942
The maximum height is 180.

00:05:10.942 --> 00:05:14.760
It happens four seconds after launch.

00:05:14.760 --> 00:05:18.060
Now, the last question I'll ask you is,

00:05:18.060 --> 00:05:22.455
how long after launch do
we get to a height of zero?

00:05:22.455 --> 00:05:25.788
So, for what x makes our height zero?

00:05:26.969 --> 00:05:31.684
To do that, we have to solve
h of x is equal to zero.

00:05:31.684 --> 00:05:34.685
Or, we can write h of x as negative five

00:05:34.685 --> 00:05:38.852
times x minus four squared
plus 180 is equal to zero.

00:05:40.830 --> 00:05:42.618
And, once again, pause the video,

00:05:42.618 --> 00:05:45.149
and see if you can solve this.

00:05:45.149 --> 00:05:46.912
Could subtract 180 from both sides.

00:05:46.912 --> 00:05:50.441
You get negative five
times x minus four squared,

00:05:50.441 --> 00:05:52.469
is equal to negative 180.

00:05:52.469 --> 00:05:54.830
We can divide both sides by negative five.

00:05:54.830 --> 00:05:58.663
We get x minus four
squared is equal to 36.

00:06:02.142 --> 00:06:04.784
Scroll down a little bit.

00:06:04.784 --> 00:06:05.617
Then,

00:06:09.219 --> 00:06:11.079
we could take the plus
and minus square root,

00:06:11.079 --> 00:06:12.153
I guess you could say.

00:06:12.153 --> 00:06:15.036
And so, that will give us x minus four

00:06:15.036 --> 00:06:17.311
could be equal to six.

00:06:17.311 --> 00:06:21.295
Or, x minus four is equal to negative six.

00:06:21.295 --> 00:06:23.693
In this first situation,
add four to both sides,

00:06:23.693 --> 00:06:25.874
you get x is equal to 10.

00:06:25.874 --> 00:06:27.905
Or, you add four to both sides here,

00:06:27.905 --> 00:06:31.277
you get x is equal to negative two.

00:06:31.277 --> 00:06:32.858
Now, we're dealing with time here,

00:06:32.858 --> 00:06:34.481
so negative two would've been in the past

00:06:34.481 --> 00:06:35.985
if it wasn't sitting on the platform

00:06:35.985 --> 00:06:37.869
and if it was just
continuing its trajectory,

00:06:37.869 --> 00:06:39.274
I guess you could say, backwards in time.

00:06:39.274 --> 00:06:42.833
But that's not the x that we
wanna take into consideration.

00:06:42.833 --> 00:06:46.239
We want the positive time value,
and that's right over here.

00:06:46.239 --> 00:06:48.271
That is when x is equal to 10.

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10 seconds after takeoff,

00:06:50.181 --> 00:06:53.062
our height is going to be equal to zero.

00:06:53.062 --> 00:06:54.971
If the ground is at height of zero,

00:06:54.971 --> 00:06:57.287
if it's at sea level, I guess, then,

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that's when our projectile
is going to hit the ground.


Graphing parabola from quadratic in factored form

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:02.523
- [Instructor] We're asked
to graph the equation

00:00:02.523 --> 00:00:06.547
y is equal to one-half times x minus six

00:00:06.547 --> 00:00:08.214
times x plus two.

00:00:09.662 --> 00:00:11.528
So like always, pause this video

00:00:11.528 --> 00:00:12.988
and take out some graph paper

00:00:12.988 --> 00:00:14.575
or even try to do it on
a regular piece of paper

00:00:14.575 --> 00:00:17.339
and see if you can graph this equation.

00:00:17.339 --> 00:00:19.182
All right now, let's work
through this together.

00:00:19.182 --> 00:00:21.000
There's many different ways that you could

00:00:21.000 --> 00:00:22.546
attempt to graph it.

00:00:22.546 --> 00:00:25.549
May be the most basic is
try out a bunch of x values

00:00:25.549 --> 00:00:27.906
and a bunch of y values and
try to connect the curve

00:00:27.906 --> 00:00:29.822
that connects all of those dots.

00:00:29.822 --> 00:00:31.560
But let's try to see if
we can get the essence

00:00:31.560 --> 00:00:35.233
of this graph without
doing that much work.

00:00:35.233 --> 00:00:38.232
The key realization here without
even having to do the math

00:00:38.232 --> 00:00:39.532
is if I multiply this out,

00:00:39.532 --> 00:00:42.257
if I multiplied x minus
six times x plus two,

00:00:42.257 --> 00:00:43.608
I'm going to get a quadratic.

00:00:43.608 --> 00:00:45.774
I'm going to get x-squared,

00:00:45.774 --> 00:00:47.787
plus something, plus something else.

00:00:47.787 --> 00:00:51.281
And so this whole thing
is going to be a parabola.

00:00:51.281 --> 00:00:54.684
We are graphing a quadratic equation.

00:00:54.684 --> 00:00:57.434
Now a parabola you might remember

00:00:58.514 --> 00:01:01.847
can intersect the x-axis multiple times.

00:01:02.742 --> 00:01:04.287
So let's see if we can find out

00:01:04.287 --> 00:01:06.562
where this intersects the x-axis.

00:01:06.562 --> 00:01:08.024
And the form that it's in,

00:01:08.024 --> 00:01:09.485
it's in factored form already,

00:01:09.485 --> 00:01:11.559
it makes it pretty straightforward

00:01:11.559 --> 00:01:14.139
for us to recognize
when does y equal zero?

00:01:14.139 --> 00:01:16.414
Which are going to be the
times that we're intersecting

00:01:16.414 --> 00:01:17.760
the x-axis.

00:01:17.760 --> 00:01:19.766
And then from that, we'll
actually be able to find

00:01:19.766 --> 00:01:21.502
the coordinates of the vertex

00:01:21.502 --> 00:01:23.498
and we're going to be able
to get the general shape

00:01:23.498 --> 00:01:26.755
of this curve which is
going to be a parabola.

00:01:26.755 --> 00:01:27.792
So let's think about it.

00:01:27.792 --> 00:01:30.251
When does y equal zero?

00:01:30.251 --> 00:01:33.155
Well to solve that we just
have to figure out when,

00:01:33.155 --> 00:01:36.046
if we want to know when y equals zero,

00:01:36.046 --> 00:01:37.790
then we have to solve for

00:01:37.790 --> 00:01:39.822
when does this expression equal zero?

00:01:39.822 --> 00:01:41.079
So let's just solve the equation.

00:01:41.079 --> 00:01:44.053
One half times x minus six,

00:01:44.053 --> 00:01:47.136
times x plus two is equal to zero.

00:01:48.051 --> 00:01:50.515
Now in previous videos we've
talked about this idea.

00:01:50.515 --> 00:01:53.022
If I have the product of multiple things

00:01:53.022 --> 00:01:54.793
and it needs to be equal to zero,

00:01:54.793 --> 00:01:56.261
the only way that's going to happen

00:01:56.261 --> 00:01:57.785
is if one or more of these things

00:01:57.785 --> 00:01:59.514
are going to be equal to zero.

00:01:59.514 --> 00:02:01.057
Well one half is one-half,

00:02:01.057 --> 00:02:02.663
it's not going to be equal to zero.

00:02:02.663 --> 00:02:05.229
But x minus six could be equal to zero.

00:02:05.229 --> 00:02:07.317
So if x minus six is equal to zero,

00:02:07.317 --> 00:02:10.271
then that would make this equation true.

00:02:10.271 --> 00:02:13.000
Or if x plus two is equal to zero,

00:02:13.000 --> 00:02:15.323
that would also make this equation true.

00:02:15.323 --> 00:02:17.658
So the x values that
satisfy either of these

00:02:17.658 --> 00:02:20.093
would make y equal zero
and those would be places

00:02:20.093 --> 00:02:23.531
where our curve is
intersecting the x-axis.

00:02:23.531 --> 00:02:26.050
So what x value makes
x minus six equal zero?

00:02:26.050 --> 00:02:28.043
Well you could add six to both sides,

00:02:28.043 --> 00:02:30.038
you're probably able to
do that in your head,

00:02:30.038 --> 00:02:32.608
and you get x is equal to six.

00:02:32.608 --> 00:02:36.588
Or you subtract two from both sides here

00:02:36.588 --> 00:02:39.615
and you get x is equal to,

00:02:39.615 --> 00:02:40.850
these cancel out,

00:02:40.850 --> 00:02:43.529
you get x is equal to negative two.

00:02:43.529 --> 00:02:45.680
These are the two x values

00:02:45.680 --> 00:02:47.140
where y will be equal to zero.

00:02:47.140 --> 00:02:49.330
You can substitute it back
into our original equation.

00:02:49.330 --> 00:02:51.504
If x is equal to six,

00:02:51.504 --> 00:02:54.086
then this right over here is
going to be equal to zero,

00:02:54.086 --> 00:02:56.286
and then y is going to be equal to zero.

00:02:56.286 --> 00:02:58.010
If x is equal to negative two,

00:02:58.010 --> 00:02:59.984
then this right over here is
going to be equal to zero,

00:02:59.984 --> 00:03:02.483
and y would be equal to zero.

00:03:02.483 --> 00:03:05.793
So we know that our parabola
is going to intersect

00:03:05.793 --> 00:03:10.217
the x-axis at x equals
negative two right over there,

00:03:10.217 --> 00:03:12.328
and x is equal to six.

00:03:12.328 --> 00:03:14.578
These are our x-intercepts.

00:03:15.620 --> 00:03:19.361
So given this, how do we
figure out the vertex?

00:03:19.361 --> 00:03:22.118
Well the key idea here is to recognize

00:03:22.118 --> 00:03:24.802
that your axis of
symmetry for your parabola

00:03:24.802 --> 00:03:28.570
is going to sit right between
your two x-intercepts.

00:03:28.570 --> 00:03:30.278
So what is the midpoint between,

00:03:30.278 --> 00:03:33.325
or what is the average
of six and negative two?

00:03:33.325 --> 00:03:34.745
Well, you could do that in your head.

00:03:34.745 --> 00:03:38.912
Six plus negative two is
four divided by two is two.

00:03:40.491 --> 00:03:41.566
Let me do that.

00:03:41.566 --> 00:03:45.246
So I'm just trying to find the
midpoint between the point,

00:03:45.246 --> 00:03:46.786
let's use a new color.

00:03:46.786 --> 00:03:47.979
So I'm trying to find the midpoint

00:03:47.979 --> 00:03:51.494
between the point negative two comma zero

00:03:51.494 --> 00:03:53.077
and six comma zero.

00:03:54.484 --> 00:03:55.541
Well the midpoint, those are just

00:03:55.541 --> 00:03:57.582
the average of the coordinates.

00:03:57.582 --> 00:04:00.023
The average of zero and zero
is just going to be zero,

00:04:00.023 --> 00:04:02.232
it's going to sit on the x-axis.

00:04:02.232 --> 00:04:04.891
But then the midpoint
of negative two and six

00:04:04.891 --> 00:04:09.520
or the average negative
two plus six over two.

00:04:09.520 --> 00:04:10.772
Well let's see, that's four over two,

00:04:10.772 --> 00:04:11.932
that's just going to be two,

00:04:11.932 --> 00:04:13.542
so two comma zero.

00:04:13.542 --> 00:04:14.890
And you see that there.

00:04:14.890 --> 00:04:16.765
You could have done that
without even doing the math.

00:04:16.765 --> 00:04:17.798
You say okay,

00:04:17.798 --> 00:04:19.022
if I want to go right in between the two,

00:04:19.022 --> 00:04:21.993
I want to be two away from each of them.

00:04:21.993 --> 00:04:24.307
And so just like that,
I could draw an axis

00:04:24.307 --> 00:04:26.640
of symmetry for my parabola.

00:04:27.724 --> 00:04:32.436
So my vertex is going to sit
on that axis of symmetry.

00:04:32.436 --> 00:04:35.078
And so how do I know what the y value is?

00:04:35.078 --> 00:04:36.392
Well I can figure out,

00:04:36.392 --> 00:04:38.743
I can substitute back
in my original equation,

00:04:38.743 --> 00:04:41.765
and say well what is y equal
when x is equal to two?

00:04:41.765 --> 00:04:45.492
Because remember the vertex
has a coordinate x equals two.

00:04:45.492 --> 00:04:48.477
It's going to be two comma something.

00:04:48.477 --> 00:04:50.769
So let's go back, let's see what y equals.

00:04:50.769 --> 00:04:54.091
So y will equal to one-half times,

00:04:54.091 --> 00:04:55.521
we're going to see when x equals two,

00:04:55.521 --> 00:04:57.522
so two minus six,

00:04:57.522 --> 00:04:59.105
times two plus two.

00:05:00.331 --> 00:05:03.237
Let's see, this is negative four,

00:05:03.237 --> 00:05:04.792
this is positive four.

00:05:04.792 --> 00:05:06.675
Negative four times four is negative 16.

00:05:06.675 --> 00:05:10.547
So it's equal to one-half
times negative 16,

00:05:10.547 --> 00:05:13.255
which is equal to negative eight.

00:05:13.255 --> 00:05:15.590
So our vertex when x is equal to two,

00:05:15.590 --> 00:05:18.518
y is equal to negative eight.

00:05:18.518 --> 00:05:22.948
And so our vertex is going
to be right over here

00:05:22.948 --> 00:05:24.767
two common negative eight.

00:05:24.767 --> 00:05:27.701
And now we can draw the general shape

00:05:27.701 --> 00:05:29.569
of our actual parabola.

00:05:29.569 --> 00:05:31.786
It's going to look something like,

00:05:31.786 --> 00:05:33.992
once again this is a hand-drawn sketch,

00:05:33.992 --> 00:05:35.565
so take it with a little
bit of a grain of salt,

00:05:35.565 --> 00:05:38.523
but it's going to look
something like this.

00:05:38.523 --> 00:05:41.307
And it's going to be symmetric
around our axis of symmetry.

00:05:41.307 --> 00:05:43.985
That's why it's called
the axis of symmetry.

00:05:43.985 --> 00:05:45.774
This art program I have,
there's a symmetry tool,

00:05:45.774 --> 00:05:48.745
but I'll just use this and there you go.

00:05:48.745 --> 00:05:53.506
That's a pretty good sketch
of what this parabola is

00:05:53.506 --> 00:05:57.269
or what this graph is going
to look like which it is,

00:05:57.269 --> 00:05:59.519
an upward-opening parabola.


Solving equations with zero product property

WEBVTT
Kind: captions
Language: en

00:00:00.642 --> 00:00:02.586
- [Instructor] Let's say
that we've got the equation

00:00:02.586 --> 00:00:03.836
two X minus one

00:00:04.792 --> 00:00:06.209
times X plus four

00:00:07.625 --> 00:00:09.042
is equal to zero.

00:00:09.889 --> 00:00:12.623
Pause this video and see
if you can figure out

00:00:12.623 --> 00:00:15.416
the X values that would
satisfy this equation,

00:00:15.416 --> 00:00:18.787
essentially our solutions
to this equation.

00:00:18.787 --> 00:00:21.730
Alright, now let's work
through this together.

00:00:21.730 --> 00:00:23.009
So at first, you might be tempted

00:00:23.009 --> 00:00:25.331
to multiply these things out,

00:00:25.331 --> 00:00:27.391
or there's multiple ways

00:00:27.391 --> 00:00:29.120
that you might have tried to approach it,

00:00:29.120 --> 00:00:30.742
but the key realization here

00:00:30.742 --> 00:00:33.185
is that you have two
things being multiplied,

00:00:33.185 --> 00:00:35.291
and it's being equal to zero.

00:00:35.291 --> 00:00:37.603
So you have the first
thing being multiplied

00:00:37.603 --> 00:00:39.762
is two X minus one.

00:00:39.762 --> 00:00:42.793
This is expression is being multiplied

00:00:42.793 --> 00:00:44.043
by X plus four,

00:00:45.426 --> 00:00:47.326
and to get it to be equal to zero,

00:00:47.326 --> 00:00:49.538
one or both of these expressions

00:00:49.538 --> 00:00:51.705
needs to be equal to zero.

00:00:52.687 --> 00:00:55.482
Let me really reinforce that idea.

00:00:55.482 --> 00:00:57.568
If I had two variables,

00:00:57.568 --> 00:00:59.068
let's say A and B,

00:01:00.046 --> 00:01:02.046
and I told you A times B

00:01:03.001 --> 00:01:04.418
is equal to zero.

00:01:06.821 --> 00:01:09.654
Well, can you get the
product of two numbers

00:01:09.654 --> 00:01:11.418
to equal zero

00:01:11.418 --> 00:01:12.957
without at least one of them

00:01:12.957 --> 00:01:14.970
being equal to zero?

00:01:14.970 --> 00:01:16.479
And the simple answer is no.

00:01:16.479 --> 00:01:17.566
If A is seven,

00:01:17.566 --> 00:01:18.843
the only way that you would get zero

00:01:18.843 --> 00:01:20.393
is if B is zero,

00:01:20.393 --> 00:01:22.012
or if B was five,

00:01:22.012 --> 00:01:23.348
the only way to get zero

00:01:23.348 --> 00:01:25.087
is if A is zero.

00:01:25.087 --> 00:01:26.796
So you see from this example,

00:01:26.796 --> 00:01:29.077
either, let me write this down,

00:01:29.077 --> 00:01:29.910
either A

00:01:32.374 --> 00:01:33.457
or B or both,

00:01:35.229 --> 00:01:37.173
'cause zero times zero is zero,

00:01:37.173 --> 00:01:38.923
or both must be zero.

00:01:41.546 --> 00:01:44.370
The only way that you get the
product of two quantities,

00:01:44.370 --> 00:01:45.608
and you get zero,

00:01:45.608 --> 00:01:49.329
is if one or both of
them is equal to zero.

00:01:49.329 --> 00:01:51.045
I really wanna reinforce this idea.

00:01:51.045 --> 00:01:54.233
I'm gonna put a red box around it

00:01:54.233 --> 00:01:56.480
so that it really gets
stuck in your brain,

00:01:56.480 --> 00:01:58.138
and I want you to think about why that is.

00:01:58.138 --> 00:01:59.325
Try to come up with two numbers.

00:01:59.325 --> 00:02:01.821
Try to multiply them so that you get zero,

00:02:01.821 --> 00:02:03.748
and you're gonna see
that one of those numbers

00:02:03.748 --> 00:02:06.081
is going to need to be zero.

00:02:09.193 --> 00:02:11.885
So we're gonna use this
idea right over here.

00:02:11.885 --> 00:02:13.608
Now this might look a
little bit different,

00:02:13.608 --> 00:02:17.480
but you could view two
X minus one as our A,

00:02:17.480 --> 00:02:20.963
and you could view X plus four as our B.

00:02:20.963 --> 00:02:25.110
So either two X minus one
needs to be equal to zero,

00:02:25.110 --> 00:02:28.583
or X plus four needs to be equal to zero,

00:02:28.583 --> 00:02:31.275
or both of them needs to be equal to zero.

00:02:31.275 --> 00:02:34.608
So I could write that as two X minus one

00:02:35.881 --> 00:02:37.899
needs to be equal to zero,

00:02:37.899 --> 00:02:39.149
or X plus four,

00:02:42.699 --> 00:02:44.941
or X, let me do that orange.

00:02:44.941 --> 00:02:46.266
Actually, let me do the two X minus one

00:02:46.266 --> 00:02:47.810
in that yellow color.

00:02:47.810 --> 00:02:51.393
So either two X minus
one is equal to zero,

00:02:52.924 --> 00:02:55.591
or X plus four is equal to zero.

00:02:57.342 --> 00:02:59.759
X plus four is equal to zero,

00:03:01.034 --> 00:03:03.640
and so let's solve each of these.

00:03:03.640 --> 00:03:06.066
If two X minus one could be equal to zero,

00:03:06.066 --> 00:03:08.922
well, let's see, you could
add one to both sides,

00:03:08.922 --> 00:03:11.672
and we get two X is equal to one.

00:03:13.561 --> 00:03:15.738
Divide both sides by two,

00:03:15.738 --> 00:03:17.241
and this just straightforward

00:03:17.241 --> 00:03:18.516
solving a linear equation.

00:03:18.516 --> 00:03:20.034
If this looks unfamiliar,

00:03:20.034 --> 00:03:21.311
I encourage you to watch videos

00:03:21.311 --> 00:03:23.836
on solving linear
equations on Khan Academy,

00:03:23.836 --> 00:03:27.307
but you'll get X is equal
to 1/2 as one solution.

00:03:27.307 --> 00:03:28.140
This is interesting

00:03:28.140 --> 00:03:30.110
'cause we're gonna have
two solutions here,

00:03:30.110 --> 00:03:32.083
or over here, if we wanna solve for X,

00:03:32.083 --> 00:03:34.621
we can subtract four from both sides,

00:03:34.621 --> 00:03:38.830
and we would get X is
equal to negative four.

00:03:38.830 --> 00:03:39.758
So it's neat.

00:03:39.758 --> 00:03:41.390
In an equation like this,

00:03:41.390 --> 00:03:43.991
you can actually have two solutions.

00:03:43.991 --> 00:03:46.068
X could be equal to 1/2,

00:03:46.068 --> 00:03:49.039
or X could be equal to negative four.

00:03:49.039 --> 00:03:50.196
I think it's pretty interesting

00:03:50.196 --> 00:03:53.174
to substitute either one of these in.

00:03:53.174 --> 00:03:54.933
If X is equal to 1/2,

00:03:54.933 --> 00:03:56.504
what is going to happen?

00:03:56.504 --> 00:04:00.723
Well, this is going to be
two times 1/2 minus one,

00:04:00.723 --> 00:04:04.033
two times 1/2 minus one.

00:04:04.033 --> 00:04:06.715
That's going to be our first expression,

00:04:06.715 --> 00:04:11.598
and then our second expression
is going to be 1/2 plus four.

00:04:11.598 --> 00:04:13.699
And so what's this going to be equal to?

00:04:13.699 --> 00:04:15.852
Well, two times 1/2 is one.

00:04:15.852 --> 00:04:18.261
One minus one is zero,

00:04:18.261 --> 00:04:19.806
so I don't care what you have over here.

00:04:19.806 --> 00:04:24.512
Zero times anything is
going to be equal to zero.

00:04:24.512 --> 00:04:26.115
So when X equals 1/2,

00:04:26.115 --> 00:04:27.677
the first thing becomes zero,

00:04:27.677 --> 00:04:31.014
making everything, making
the product equal zero.

00:04:31.014 --> 00:04:33.210
And likewise, if X equals negative four,

00:04:33.210 --> 00:04:35.075
it's pretty clear that
this second expression

00:04:35.075 --> 00:04:36.368
is going to be zero,

00:04:36.368 --> 00:04:37.931
and even though this first expression

00:04:37.931 --> 00:04:39.493
isn't going to be zero in that case,

00:04:39.493 --> 00:04:42.826
anything times zero is going to be zero.

00:04:43.870 --> 00:04:46.453
Let's do one more example here.

00:04:48.776 --> 00:04:49.609
So let me

00:04:50.776 --> 00:04:53.711
delete out everything
that I just wrote here,

00:04:53.711 --> 00:04:56.927
and so I'm gonna involve a function.

00:04:56.927 --> 00:04:59.059
So let's say someone told you

00:04:59.059 --> 00:05:00.726
that F of X is equal

00:05:01.860 --> 00:05:05.193
to X minus five, times five X, plus two,

00:05:08.620 --> 00:05:12.642
and someone said, "Find
the zeros of F of X."

00:05:12.642 --> 00:05:14.889
Well, the zeros are, what are the X values

00:05:14.889 --> 00:05:16.962
that make F of X equal to zero?

00:05:16.962 --> 00:05:19.707
When does F of X equal zero?

00:05:19.707 --> 00:05:21.124
For what X values

00:05:25.623 --> 00:05:26.540
does F of X

00:05:28.769 --> 00:05:30.046
equal zero?

00:05:30.046 --> 00:05:31.223
That's what people are really asking

00:05:31.223 --> 00:05:34.723
when they say, "Find the zeros of F of X."

00:05:36.159 --> 00:05:38.177
So to do that, well, when
does F of X equal zero?

00:05:38.177 --> 00:05:39.392
Well, F of X is equal to zero

00:05:39.392 --> 00:05:41.474
when this expression right over here

00:05:41.474 --> 00:05:42.800
is equal to zero,

00:05:42.800 --> 00:05:45.741
and so it sets up just like
the equation we just saw.

00:05:45.741 --> 00:05:46.741
X minus five

00:05:48.285 --> 00:05:50.118
times five X plus two,

00:05:51.223 --> 00:05:53.214
when does that equal zero?

00:05:53.214 --> 00:05:55.025
And like we saw before,

00:05:55.025 --> 00:05:56.999
well, this is just like
what we saw before,

00:05:56.999 --> 00:05:58.257
and I encourage you to pause the video,

00:05:58.257 --> 00:06:01.051
and try to work it out on your own.

00:06:01.051 --> 00:06:03.616
So there's two situations

00:06:03.616 --> 00:06:05.205
where this could happen,

00:06:05.205 --> 00:06:09.294
where either the first
expression equals zero,

00:06:09.294 --> 00:06:10.342
or the second expression,

00:06:10.342 --> 00:06:11.476
or maybe in some cases,

00:06:11.476 --> 00:06:14.692
you'll have a situation where
both expressions equal zero.

00:06:14.692 --> 00:06:18.872
So we could say either X
minus five is equal to zero,

00:06:18.872 --> 00:06:21.872
or five X plus two is equal to zero.

00:06:22.940 --> 00:06:24.860
I'll write an, or, right over here.

00:06:24.860 --> 00:06:25.948
Now if we solve for X,

00:06:25.948 --> 00:06:28.409
you add five to both
sides of this equation.

00:06:28.409 --> 00:06:30.399
You get X is equal to five.

00:06:30.399 --> 00:06:31.422
Here, let's see.

00:06:31.422 --> 00:06:32.255
To solve for X,

00:06:32.255 --> 00:06:33.900
you could subtract two from both sides.

00:06:33.900 --> 00:06:37.123
You get five X is equal to negative two,

00:06:37.123 --> 00:06:40.014
and you could divide both sides by five

00:06:40.014 --> 00:06:41.471
to solve for X,

00:06:41.471 --> 00:06:45.139
and you get X is equal to negative 2/5.

00:06:45.139 --> 00:06:47.612
So here are two zeros.

00:06:47.612 --> 00:06:50.149
You input either one of these into F of X.

00:06:50.149 --> 00:06:51.412
If you input X equals five,

00:06:51.412 --> 00:06:53.321
if you take F of five,

00:06:53.321 --> 00:06:54.692
if you try to evaluate F of five,

00:06:54.692 --> 00:06:56.844
then this first
expression's gonna be zero,

00:06:56.844 --> 00:06:59.476
and so a product of
zero and something else,

00:06:59.476 --> 00:07:01.327
it doesn't matter that
this is gonna be 27.

00:07:01.327 --> 00:07:03.690
Zero times 27 is zero,

00:07:03.690 --> 00:07:06.981
and if you take F of negative 2/5,

00:07:06.981 --> 00:07:09.925
it doesn't matter what
this first expression is.

00:07:09.925 --> 00:07:11.132
The second expression right over here

00:07:11.132 --> 00:07:11.965
is gonna be zero.

00:07:11.965 --> 00:07:14.298
Zero times anything is zero.


Visual introduction to parabolas

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.123
- [Instructor] In this video,
we are going to talk about one

00:00:02.123 --> 00:00:04.765
of the most common types
of curves you will see

00:00:04.765 --> 00:00:08.182
in mathematics, and that is the parabola.

00:00:09.209 --> 00:00:12.251
The word parabola sounds
quite fancy, but we'll see

00:00:12.251 --> 00:00:16.157
it's describing something that
is fairly straightforward.

00:00:16.157 --> 00:00:19.126
Now in terms of why it
is called the parabola,

00:00:19.126 --> 00:00:21.720
I've seen multiple explanations for it.

00:00:21.720 --> 00:00:25.530
It comes from Greek para, that root word,

00:00:25.530 --> 00:00:27.477
similar to parable.

00:00:27.477 --> 00:00:29.239
You could view of something beside,

00:00:29.239 --> 00:00:32.271
alongside, something in parallel.

00:00:32.271 --> 00:00:34.552
Bola, same root as when we're talking

00:00:34.552 --> 00:00:37.656
about ballistics, throwing something.

00:00:37.656 --> 00:00:41.678
So you could interpret
it as beside, alongside,

00:00:41.678 --> 00:00:43.989
something that is being thrown.

00:00:43.989 --> 00:00:46.627
Now how does that relate
to curves like this?

00:00:46.627 --> 00:00:50.296
Well my brain immediately
imagines this is the trajectory,

00:00:50.296 --> 00:00:53.516
this is the path that is a
pretty good approximation

00:00:53.516 --> 00:00:56.042
for the path of things
that are actually thrown.

00:00:56.042 --> 00:00:57.063
When you study physics,

00:00:57.063 --> 00:00:59.541
you will see the path, you'll approximate,

00:00:59.541 --> 00:01:03.460
the paths of objects being
thrown, as parabolas,

00:01:03.460 --> 00:01:05.292
so maybe that's where it comes from,

00:01:05.292 --> 00:01:07.812
but there are other potential explanations

00:01:07.812 --> 00:01:10.919
for why it is actually
called the parabola.

00:01:10.919 --> 00:01:13.012
It has been lost to history.

00:01:13.012 --> 00:01:15.582
But what exactly is a parabola?

00:01:15.582 --> 00:01:17.268
In future videos, we're gonna describe it

00:01:17.268 --> 00:01:19.314
a little bit more algebraically.

00:01:19.314 --> 00:01:21.330
In this one, we just wanna get a sense

00:01:21.330 --> 00:01:24.678
for what parabolas look
like and introduce ourselves

00:01:24.678 --> 00:01:28.599
to some terminology around a parabola.

00:01:28.599 --> 00:01:33.368
These three curves, they
are all hand-drawn versions

00:01:33.368 --> 00:01:36.468
of a parabola, and so
you immediately notice

00:01:36.468 --> 00:01:38.899
some interesting things about them.

00:01:38.899 --> 00:01:41.732
Some of them are opened
upwards like this yellow one

00:01:41.732 --> 00:01:45.457
and this pink one, and some
of them are open downwards.

00:01:45.457 --> 00:01:49.624
You will hear people say
things like open, opened down,

00:01:51.991 --> 00:01:55.658
open downwards or open
down or open upwards,

00:01:56.964 --> 00:01:59.097
so it's good to know what
they are talking about,

00:01:59.097 --> 00:02:01.938
and it's, hopefully,
fairly self-explanatory.

00:02:01.938 --> 00:02:04.771
Open upwards, the parabola is open

00:02:05.771 --> 00:02:07.564
towards the top of our graph paper.

00:02:07.564 --> 00:02:10.054
Here it's open towards the
bottom of our graph paper.

00:02:10.054 --> 00:02:11.952
This looks like a right-side up U.

00:02:11.952 --> 00:02:14.557
This looks like an upside
down U right over here.

00:02:14.557 --> 00:02:17.431
This pink one would be open upwards.

00:02:17.431 --> 00:02:19.601
Now another term that
you'll see associated

00:02:19.601 --> 00:02:21.574
with the parabola, and
once again, in the future,

00:02:21.574 --> 00:02:22.955
we'll learn how to calculate these things

00:02:22.955 --> 00:02:26.105
and find them precisely, is the vertex.

00:02:26.105 --> 00:02:29.933
The vertex you should view as the maximum

00:02:29.933 --> 00:02:32.009
or minimum point on a parabola.

00:02:32.009 --> 00:02:35.190
So if a parabola opens upwards
like these two on the right,

00:02:35.190 --> 00:02:38.531
the vertex is the minimum point.

00:02:38.531 --> 00:02:42.489
The vertex is the minimum
point right over there,

00:02:42.489 --> 00:02:44.223
and so if someone said what is the vertex

00:02:44.223 --> 00:02:46.692
of this yellow parabola?

00:02:46.692 --> 00:02:50.068
Well it looks like the x,
looks like the x coordinate

00:02:50.068 --> 00:02:54.303
is three and a half, so
it is three and a half.

00:02:54.303 --> 00:02:57.098
It looks like the y
coordinate, it looks like it is

00:02:57.098 --> 00:02:59.783
about negative three and a half.

00:02:59.783 --> 00:03:02.210
Once again, once we start
representing these things

00:03:02.210 --> 00:03:04.096
with equations, we'll have techniques

00:03:04.096 --> 00:03:07.186
for calculating them more
precisely, but the vertex

00:03:07.186 --> 00:03:09.116
of this other upward-opening parabola,

00:03:09.116 --> 00:03:10.314
it is the minimum point.

00:03:10.314 --> 00:03:11.644
It is the low point.

00:03:11.644 --> 00:03:14.298
There is no maximum point on
an upward-opening parabola.

00:03:14.298 --> 00:03:16.467
It just keeps increasing as x gets larger

00:03:16.467 --> 00:03:19.646
in the positive or the negative direction.

00:03:19.646 --> 00:03:22.166
Now if your parabola opens downward,

00:03:22.166 --> 00:03:26.333
then your vertex is going
to be your maximum point.

00:03:27.915 --> 00:03:31.839
Now related to the idea
of a vertex is the idea

00:03:31.839 --> 00:03:33.767
of an axis of symmetry.

00:03:33.767 --> 00:03:36.957
In general when we're talking
about, well not just three,

00:03:36.957 --> 00:03:39.384
two dimensions but even three dimensions,

00:03:39.384 --> 00:03:42.028
but especially in two dimensions,
you can imagine a line

00:03:42.028 --> 00:03:44.861
over which you can flip the graph,

00:03:46.628 --> 00:03:50.251
and so it meets, it folds onto itself.

00:03:50.251 --> 00:03:54.005
The axis of symmetry for this
yellow graph right over here,

00:03:54.005 --> 00:03:58.256
for this yellow parabola,
it would be this line.

00:03:58.256 --> 00:04:02.647
I'm gonna have to draw
it a little bit better.

00:04:02.647 --> 00:04:05.568
It would be that line right over there.

00:04:05.568 --> 00:04:08.195
You could fold the
parabola over that line,

00:04:08.195 --> 00:04:10.107
and it would meet itself.

00:04:10.107 --> 00:04:12.552
And that line, I didn't
draw it as neat as I should,

00:04:12.552 --> 00:04:15.954
that should go directly
through the vertex,

00:04:15.954 --> 00:04:17.952
so to describe that line you'd say

00:04:17.952 --> 00:04:20.785
that line is x is equal to 3.5.

00:04:23.217 --> 00:04:28.155
Similarly the axis of symmetry
for this pink parabola,

00:04:28.155 --> 00:04:32.391
it should go through the
line x equals negative one,

00:04:32.391 --> 00:04:33.865
so let me do that.

00:04:33.865 --> 00:04:35.449
That's the axis of symmetry.

00:04:35.449 --> 00:04:38.611
It goes through the
vertex, and if you were

00:04:38.611 --> 00:04:41.622
to fold the parabola over
it, it would meet itself.

00:04:41.622 --> 00:04:45.124
The axis of symmetry for this green one?

00:04:45.124 --> 00:04:47.763
It should, once again,
go through the vertex.

00:04:47.763 --> 00:04:51.650
It looks like it is x is
equal to negative six.

00:04:51.650 --> 00:04:53.091
This is, let me write that down,

00:04:53.091 --> 00:04:55.508
that is the axis of symmetry.

00:04:58.711 --> 00:05:02.085
Now another concept that
isn't unique to parabolas,

00:05:02.085 --> 00:05:05.117
but we'll talk a lot about it
in the context of parabolas,

00:05:05.117 --> 00:05:08.713
are intercepts, so when
people say y-intercept,

00:05:08.713 --> 00:05:11.000
and you saw this when
you first graphed lines,

00:05:11.000 --> 00:05:12.588
they're saying where is the graph,

00:05:12.588 --> 00:05:17.139
where does the curve intercept
or intersect the y-axis?

00:05:17.139 --> 00:05:20.033
So the y-intercept of this
yellow line would be right there.

00:05:20.033 --> 00:05:23.960
It looks like it's the
point zero comma three,

00:05:23.960 --> 00:05:25.296
zero comma three.

00:05:25.296 --> 00:05:28.068
The y-intercept for the pink
one is right over there.

00:05:28.068 --> 00:05:30.524
At least on this graph paper,
we don't see the y-intercept,

00:05:30.524 --> 00:05:33.161
but it eventually will
intersect the y-axis.

00:05:33.161 --> 00:05:37.044
It just will be way off of this screen.

00:05:37.044 --> 00:05:40.020
You might also be familiar
with the term x-intercept,

00:05:40.020 --> 00:05:41.881
and that's especially
interesting with parabolas

00:05:41.881 --> 00:05:42.979
as we'll see in the future.

00:05:42.979 --> 00:05:44.825
X-intercept is where do you intercept

00:05:44.825 --> 00:05:46.825
or intersect the x-axis?

00:05:48.586 --> 00:05:51.940
Here this yellow one you
see it does it two places,

00:05:51.940 --> 00:05:53.496
and this is where it gets interesting.

00:05:53.496 --> 00:05:57.663
Lines will only intersect
the x-axis once at most,

00:05:58.787 --> 00:06:01.860
but here we see that a
parabola can intersect

00:06:01.860 --> 00:06:05.336
the x-axis twice, because
it curves back around

00:06:05.336 --> 00:06:08.484
to intersect it again, and so for here

00:06:08.484 --> 00:06:11.712
the x-intercepts are going to be the point

00:06:11.712 --> 00:06:14.545
one comma zero and six comma zero.

00:06:16.608 --> 00:06:19.258
You might already notice
something interesting.

00:06:19.258 --> 00:06:23.984
The x-intercepts are symmetric
around the axis of symmetry,

00:06:23.984 --> 00:06:27.693
so they should be equal distant
from that axis of symmetry,

00:06:27.693 --> 00:06:29.918
and you can see they indeed are.

00:06:29.918 --> 00:06:33.280
They are both exactly two and a half away

00:06:33.280 --> 00:06:37.456
from that axis of symmetry,
and so if you know

00:06:37.456 --> 00:06:40.487
where the intercepts are,
you just take, you could say,

00:06:40.487 --> 00:06:43.445
the midpoint of the x
coordinates, and then you're going

00:06:43.445 --> 00:06:46.394
to have the axis of
symmetry, the x coordinate

00:06:46.394 --> 00:06:48.922
of the axis of symmetry
and the x coordinate

00:06:48.922 --> 00:06:50.993
of the actual vertex.

00:06:50.993 --> 00:06:55.724
Similarly the x-intercept
here looks like it's negative,

00:06:55.724 --> 00:06:59.256
the points are negative seven comma zero

00:06:59.256 --> 00:07:01.673
and negative five comma zero,

00:07:03.996 --> 00:07:07.616
and the x coordinate of the
vertex, or the line of symmetry,

00:07:07.616 --> 00:07:10.699
is right in between those two points.

00:07:11.835 --> 00:07:14.279
It's worth noting not
every parabola is going

00:07:14.279 --> 00:07:16.279
to intersect the x-axis.

00:07:17.162 --> 00:07:20.593
Notice this pink upward-opening parabola,

00:07:20.593 --> 00:07:23.480
it's low point is above the x-axis,

00:07:23.480 --> 00:07:26.833
so it's never going to
intersect the actual x-axis,

00:07:26.833 --> 00:07:31.000
so this is actually not going
to have any x-intercepts.

00:07:31.919 --> 00:07:33.172
I'll leave you there.

00:07:33.172 --> 00:07:36.713
Those are actually the core
ideas or the core visual themes

00:07:36.713 --> 00:07:38.844
around parabolas, and
we're going to discuss them

00:07:38.844 --> 00:07:42.307
in a lot more detail when we
represent them with equations.

00:07:42.307 --> 00:07:44.888
As you'll see, these equations are going

00:07:44.888 --> 00:07:47.471
to involve second-degree terms.

00:07:48.577 --> 00:07:51.157
So the most simple parabola is going to be

00:07:51.157 --> 00:07:53.124
y is equal to x squared,

00:07:53.124 --> 00:07:54.784
but then you can
complicate it a little bit.

00:07:54.784 --> 00:07:56.898
You could have things like y is equal to

00:07:56.898 --> 00:08:01.258
two x squared minus five x plus seven.

00:08:01.258 --> 00:08:04.190
These types that we'll talk
about in more general terms,

00:08:04.190 --> 00:08:08.155
these types of equations
sometimes called quadratics,

00:08:08.155 --> 00:08:12.072
they are represented,
generally, by parabolas.


Recognizing quadratic factor methods part 2

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.469
- [Sal] In the last video we looked

00:00:01.469 --> 00:00:03.552
at three different
examples, really as a bit

00:00:03.552 --> 00:00:06.294
of a review of some of
our factoring techniques,

00:00:06.294 --> 00:00:09.307
and also to appreciate when
we might wanna apply them.

00:00:09.307 --> 00:00:12.720
We saw in the first
example that it was just

00:00:12.720 --> 00:00:15.138
a process of recognizing a common factor.

00:00:15.138 --> 00:00:17.490
Once we factored that out, we were done.

00:00:17.490 --> 00:00:21.230
In the second example, there
was a common factor, four.

00:00:21.230 --> 00:00:23.268
Then, after that, we used, you could say,

00:00:23.268 --> 00:00:25.422
our most basic factoring technique,

00:00:25.422 --> 00:00:27.979
or one of our more basic
factoring techniques,

00:00:27.979 --> 00:00:31.297
where we say, what two numbers add up

00:00:31.297 --> 00:00:32.934
to the first-degree coefficient,

00:00:32.934 --> 00:00:35.167
and then their product is the constant.

00:00:35.167 --> 00:00:38.663
And we were able to factor the expression.

00:00:38.663 --> 00:00:41.268
Then, in the third
example, we, once again,

00:00:41.268 --> 00:00:45.121
started off by factoring
out a common value,

00:00:45.121 --> 00:00:46.700
which, in this case, was three.

00:00:46.700 --> 00:00:48.856
We could've done it the same
way we did the second one,

00:00:48.856 --> 00:00:50.562
or we could've immediately recognized

00:00:50.562 --> 00:00:52.758
that this is a perfect-square polynomial.

00:00:52.758 --> 00:00:55.926
But either way, we were able
to factor the expression.

00:00:55.926 --> 00:01:00.678
Let's keep going to see if we
can tackle some other types

00:01:00.678 --> 00:01:04.845
of polynomials that might
require some other techniques.

00:01:07.136 --> 00:01:09.803
Let's say we have the expression

00:01:11.257 --> 00:01:13.498
seven x squared minus 63.

00:01:13.498 --> 00:01:14.638
Like always, pause this video,

00:01:14.638 --> 00:01:17.736
and see if you can factor that.

00:01:17.736 --> 00:01:20.191
I've intentionally designed
all of these so that you

00:01:20.191 --> 00:01:22.507
can check whether there's a
common factor across the terms,

00:01:22.507 --> 00:01:24.700
and here they're all divisible by seven.

00:01:24.700 --> 00:01:26.233
If you factor out a seven, you're gonna

00:01:26.233 --> 00:01:29.812
get seven times x-squared minus nine.

00:01:29.812 --> 00:01:31.719
You might immediately recognize this

00:01:31.719 --> 00:01:33.143
as a difference of squares.

00:01:33.143 --> 00:01:36.390
You have x squared minus
this right over here

00:01:36.390 --> 00:01:38.013
is three squared.

00:01:38.013 --> 00:01:40.133
Minus three squared.

00:01:40.133 --> 00:01:43.158
If the term difference of
squares or how to factor them

00:01:43.158 --> 00:01:45.691
is completely foreign to
you, I encourage you to watch

00:01:45.691 --> 00:01:47.540
the videos on factoring
difference of squares

00:01:47.540 --> 00:01:50.260
or do a search on Khan Academy
for difference of squares.

00:01:50.260 --> 00:01:51.470
But you will see, when you have

00:01:51.470 --> 00:01:52.932
a difference of squares like this,

00:01:52.932 --> 00:01:57.280
it can be factored as seven,
this is just a seven out front,

00:01:57.280 --> 00:02:00.163
and then this part right over
here, get a different color,

00:02:00.163 --> 00:02:02.238
this part right over here can be written

00:02:02.238 --> 00:02:05.738
as x plus three times x minus three.

00:02:08.044 --> 00:02:11.294
It is x squared minus three squared.

00:02:16.052 --> 00:02:17.554
Now, one thing to appreciate.

00:02:17.554 --> 00:02:19.300
This really isn't a different technique

00:02:19.300 --> 00:02:21.737
than the one that we saw
in the previous video.

00:02:21.737 --> 00:02:26.006
If we just focused on
x squared minus nine,

00:02:26.006 --> 00:02:30.173
you could view this as x
squared plus zero x minus nine.

00:02:31.049 --> 00:02:33.245
In that case, you'd say:
"OK, what two numbers

00:02:33.245 --> 00:02:37.412
"get me a product of negative
nine and add up to zero?"

00:02:38.804 --> 00:02:41.049
If I need to get a
product of negative nine,

00:02:41.049 --> 00:02:43.558
that means that they
must be different signs,

00:02:43.558 --> 00:02:44.979
a positive and a negative; otherwise,

00:02:44.979 --> 00:02:47.051
if they were the same sign,
you'd get a positive here.

00:02:47.051 --> 00:02:48.542
So they're different signs.

00:02:48.542 --> 00:02:50.388
Nine only has three factors.

00:02:50.388 --> 00:02:51.443
One.

00:02:51.443 --> 00:02:54.295
You could either have one and nine.

00:02:54.295 --> 00:02:55.653
There's only two combinations here.

00:02:55.653 --> 00:02:58.986
You could either have one or
nine, and three and three.

00:02:58.986 --> 00:03:03.456
And, if you make one
negative or nine negative,

00:03:03.456 --> 00:03:05.487
that's not going to add up to zero.

00:03:05.487 --> 00:03:07.032
But if you make one of
these threes negative,

00:03:07.032 --> 00:03:08.250
that does add up to zero.

00:03:08.250 --> 00:03:09.821
Say, OK, my two numbers are gonna

00:03:09.821 --> 00:03:11.578
be negative three and three.

00:03:11.578 --> 00:03:15.425
So it's gonna be x minus
three times x plus three.

00:03:15.425 --> 00:03:17.211
Once again, I'm just focusing on what

00:03:17.211 --> 00:03:19.037
was inside the parentheses
right over here.

00:03:19.037 --> 00:03:20.254
You'd put that seven out front

00:03:20.254 --> 00:03:23.009
if we were doing this
exact same expression.

00:03:23.009 --> 00:03:25.244
But if you recognize it as
a difference of squares,

00:03:25.244 --> 00:03:28.902
it might happen for you
a little bit faster.

00:03:28.902 --> 00:03:31.019
Let's do one more example.

00:03:31.019 --> 00:03:35.186
Let's say that I have two x
squared plus seven x plus three.

00:03:40.636 --> 00:03:42.423
In general, when my coefficient

00:03:42.423 --> 00:03:45.549
on the second-degree
term here is not a one,

00:03:45.549 --> 00:03:48.840
I try to see is there
a common factor here.

00:03:48.840 --> 00:03:52.091
But, seven isn't divisible
by two, and neither is three.

00:03:52.091 --> 00:03:54.530
So I can't use the techniques that I used

00:03:54.530 --> 00:03:56.480
in the last few videos or even over here,

00:03:56.480 --> 00:03:57.895
where I say: "Oh,
there's a common factor",

00:03:57.895 --> 00:04:00.412
and get a leading coefficient of one.

00:04:00.412 --> 00:04:02.320
If you see a situation like that,

00:04:02.320 --> 00:04:06.487
it's a clue that factoring
by grouping might apply here.

00:04:07.843 --> 00:04:09.060
Factoring by grouping.

00:04:09.060 --> 00:04:11.026
On some level, everything
that we've just done now

00:04:11.026 --> 00:04:15.294
you could view as special
cases of factoring by grouping.

00:04:15.294 --> 00:04:17.367
Factoring by grouping, you say, OK,

00:04:17.367 --> 00:04:21.268
can I think of two numbers that
add up to this coefficient.

00:04:21.268 --> 00:04:24.316
a plus b is equal to seven.

00:04:24.316 --> 00:04:28.052
a times b, instead of just
saying it needs to equal

00:04:28.052 --> 00:04:32.557
to three, it actually needs to
be equal to three times this,

00:04:32.557 --> 00:04:34.341
three times the leading coefficient,

00:04:34.341 --> 00:04:36.660
the coefficient on the x-squared term.

00:04:36.660 --> 00:04:39.177
It needs to be equal to three times two.

00:04:39.177 --> 00:04:42.040
If you think about it, we've
always been doing that.

00:04:42.040 --> 00:04:43.633
The other examples we gave,

00:04:43.633 --> 00:04:45.466
the leading coefficient was a one.

00:04:45.466 --> 00:04:48.173
When you took the constant term,
and multiplied it by a one,

00:04:48.173 --> 00:04:49.675
you were just saying a times b needs

00:04:49.675 --> 00:04:51.505
to be equal to that constant term.

00:04:51.505 --> 00:04:53.215
If you wanna talk about it more generally,

00:04:53.215 --> 00:04:56.627
it should be a times b
should be the constant term

00:04:56.627 --> 00:04:58.823
times the leading coefficient.

00:04:58.823 --> 00:05:01.419
In the introduction to
factoring by grouping

00:05:01.419 --> 00:05:03.087
we explained why that works.

00:05:03.087 --> 00:05:05.814
You should never just accept
this as some magic formula.

00:05:05.814 --> 00:05:08.372
It makes sense for a very
good mathematical reason.

00:05:08.372 --> 00:05:09.793
But once you accept that,

00:05:09.793 --> 00:05:13.202
then it's useful to be able
to apply the technique.

00:05:13.202 --> 00:05:15.965
So can we think of two
numbers that add up to seven

00:05:15.965 --> 00:05:19.497
and whose product is equal to six?

00:05:19.497 --> 00:05:21.366
They're going to have to be the same sign

00:05:21.366 --> 00:05:23.435
since this is a positive value.

00:05:23.435 --> 00:05:26.797
And, they're gonna be positive
'cause they're the same sign,

00:05:26.797 --> 00:05:28.301
and if they're adding
up to a positive value,

00:05:28.301 --> 00:05:30.207
they're both gonna be positive.

00:05:30.207 --> 00:05:32.967
Well, let's see, one
and six seems to work.

00:05:32.967 --> 00:05:34.183
One plus six is seven.

00:05:34.183 --> 00:05:35.891
One times six is six.

00:05:35.891 --> 00:05:38.292
In factoring by grouping
we rewrite our expression

00:05:38.292 --> 00:05:41.340
where we break this up
between the a and the b.

00:05:41.340 --> 00:05:44.665
So I can rewrite this as two x squared

00:05:44.665 --> 00:05:48.202
plus six x plus, I could write one x.

00:05:48.202 --> 00:05:49.137
Actually, lemme just do that.

00:05:49.137 --> 00:05:51.220
Plus one x plus three.

00:05:52.912 --> 00:05:56.662
As you can see, the
seven x, different color,

00:05:57.587 --> 00:06:02.102
the seven x has been broken
up into the six x and one x.

00:06:02.102 --> 00:06:04.544
That whole exercise I
just did is to see how

00:06:04.544 --> 00:06:08.891
we can break up this first-degree
term right over here.

00:06:08.891 --> 00:06:12.474
But then, what's useful
about this, is now,

00:06:15.432 --> 00:06:19.167
we can do the reverse of the
distributive property twice.

00:06:19.167 --> 00:06:21.667
So, for these first two terms,

00:06:23.071 --> 00:06:25.778
in a different color than I just use,

00:06:25.778 --> 00:06:29.228
these first two terms,
you see a common factor.

00:06:29.228 --> 00:06:32.502
Two x squared and six x,
they're both divisible by two x.

00:06:32.502 --> 00:06:34.910
So let's factor out a two x
outta those first two terms.

00:06:34.910 --> 00:06:37.760
If you do that, two x
squared divided by two x,

00:06:37.760 --> 00:06:39.505
you're just gonna have an x left over.

00:06:39.505 --> 00:06:43.485
Six x divided by two x, you're
just going to have a three.

00:06:43.485 --> 00:06:45.716
And then you have plus.

00:06:45.716 --> 00:06:48.643
Over here, this is a
special situation where,

00:06:48.643 --> 00:06:52.380
x plus three, there is no common
factor between x and three,

00:06:52.380 --> 00:06:55.997
so we'll just rewrite that, x plus three.

00:06:55.997 --> 00:06:57.457
When I put a parentheses on it,

00:06:57.457 --> 00:06:59.531
which is equivalent to writing
it without a parentheses,

00:06:59.531 --> 00:07:01.397
you might see something else.

00:07:01.397 --> 00:07:05.564
I can undistribute, or I can
factor out, an x plus three.

00:07:06.561 --> 00:07:08.148
What happens if I do that?

00:07:08.148 --> 00:07:10.381
I'm gonna get an x plus three.

00:07:10.381 --> 00:07:13.263
And then I'm gonna have
leftover in this term,

00:07:13.263 --> 00:07:14.683
if I factored out an x plus three,

00:07:14.683 --> 00:07:16.958
I'm just gonna have a two x left over.

00:07:16.958 --> 00:07:18.257
Two x.

00:07:18.257 --> 00:07:21.264
And then, this term, if I
factor out an x plus three,

00:07:21.264 --> 00:07:23.177
I'm just gonna have a one left over.

00:07:23.177 --> 00:07:24.596
Plus one.

00:07:24.596 --> 00:07:26.497
Lemme do it in that same color.

00:07:26.497 --> 00:07:28.081
Having trouble switching colors today.

00:07:28.081 --> 00:07:30.230
Two x plus one.

00:07:30.230 --> 00:07:32.264
And we are done.

00:07:32.264 --> 00:07:34.583
So, as I said, these are
all various techniques.

00:07:34.583 --> 00:07:36.165
On some level, factoring by grouping

00:07:36.165 --> 00:07:38.197
is sometimes viewed as the hardest one.

00:07:38.197 --> 00:07:40.882
But I'll say hard in parentheses because

00:07:40.882 --> 00:07:43.201
everything we did is
just a variation, really,

00:07:43.201 --> 00:07:45.474
a special case, of factoring by grouping.

00:07:45.474 --> 00:07:46.611
As you can see, it's all about,

00:07:46.611 --> 00:07:48.605
two numbers that add up
to that middle coefficient

00:07:48.605 --> 00:07:51.446
on the first-degree term when
it's written in standard form.

00:07:51.446 --> 00:07:53.353
Their product is equal to the product

00:07:53.353 --> 00:07:57.251
of the constant and the
leading coefficient.

00:07:57.251 --> 00:07:59.369
If you do that, you
break it up, it works out

00:07:59.369 --> 00:08:02.094
quite nicely where you
keep factoring out terms.

00:08:02.094 --> 00:08:04.231
This one, on some levels,
was a little bit more subtle,

00:08:04.231 --> 00:08:06.755
'cause you had to recognize
that this x plus three

00:08:06.755 --> 00:08:09.677
has a one coefficient on there implicitly.

00:08:09.677 --> 00:08:12.193
One times x plus three is the
same thing as x plus three.

00:08:12.193 --> 00:08:14.307
And then see that you can
factor out an x plus three

00:08:14.307 --> 00:08:16.746
outta both of these terms,
and then, once you do that,

00:08:16.746 --> 00:08:19.417
you're gonna be left
with the two x plus one.

00:08:19.417 --> 00:08:21.575
But all of these, if you
really feel comfortable

00:08:21.575 --> 00:08:24.199
with this arsenal of techniques,

00:08:24.199 --> 00:08:25.939
you're gonna do pretty well.

00:08:25.939 --> 00:08:28.456
Frankly, if none of these work, well,

00:08:28.456 --> 00:08:30.933
you might already be familiar
with the quadratic formula

00:08:30.933 --> 00:08:32.883
or you might be soon to learn it,

00:08:32.883 --> 00:08:36.631
but that's when the quadratic
formula might be effective

00:08:36.631 --> 00:08:39.381
if none of these techniques work.


Recognizing quadratic factor methods

WEBVTT
Kind: captions
Language: en

00:00:00.561 --> 00:00:01.720
- [Instructor] We have other videos

00:00:01.720 --> 00:00:05.023
on individual techniques
for factoring quadratics,

00:00:05.023 --> 00:00:06.379
but what I would like to do in this video

00:00:06.379 --> 00:00:08.960
is get some practice figuring out

00:00:08.960 --> 00:00:11.793
which technique to use, so I'm gonna write

00:00:11.793 --> 00:00:13.602
a bunch of quadratics, and I encourage you

00:00:13.602 --> 00:00:15.280
to pause the video, try to see if you can

00:00:15.280 --> 00:00:17.066
factor that quadratic yourself before

00:00:17.066 --> 00:00:19.136
I work through it with you.

00:00:19.136 --> 00:00:22.886
So the first quadratic
is 6x squared plus 3x.

00:00:24.245 --> 00:00:27.818
So pause and see if you can factor this.

00:00:27.818 --> 00:00:29.200
So this one might jump out at you

00:00:29.200 --> 00:00:32.626
that both of these terms
here have a common factor.

00:00:32.626 --> 00:00:34.934
Both of them are divisible by three,

00:00:34.934 --> 00:00:37.026
six is divisible by three and so is three,

00:00:37.026 --> 00:00:39.260
and both of them are divisible by x,

00:00:39.260 --> 00:00:41.702
so you can factor out a 3x.

00:00:41.702 --> 00:00:43.914
So if you factor out a 3x, 6x squared

00:00:43.914 --> 00:00:47.594
divided by 3x, you're gonna
have a 2x left over there

00:00:47.594 --> 00:00:51.524
and then 3x divided by 3x,
you're going to have a one.

00:00:51.524 --> 00:00:52.982
And that's about as much as we can

00:00:52.982 --> 00:00:54.589
actually factor, and you can verify

00:00:54.589 --> 00:00:57.314
that these two expressions are the same

00:00:57.314 --> 00:01:00.857
if you distribute the 3x,
3x times 2x is 6x squared,

00:01:00.857 --> 00:01:05.371
3x times one is 3x, and
that's all we would do.

00:01:05.371 --> 00:01:06.373
We would be done.

00:01:06.373 --> 00:01:09.036
That's all you can
really do to factor that.

00:01:09.036 --> 00:01:11.311
And as we'll see, in this example,

00:01:11.311 --> 00:01:13.797
trying to factor out a common factor

00:01:13.797 --> 00:01:16.527
was all we had to do, but as we'll see

00:01:16.527 --> 00:01:19.885
in future examples, that's
usually a good first step.

00:01:19.885 --> 00:01:22.026
Do all of, check whether the terms

00:01:22.026 --> 00:01:23.956
have a common factor, and if they do

00:01:23.956 --> 00:01:27.093
it never hurts to factor that out.

00:01:27.093 --> 00:01:29.429
So let's do another example.

00:01:29.429 --> 00:01:33.596
Let's say I have the quadratic
4x squared minus 4x minus 48.

00:01:37.438 --> 00:01:41.356
Pause the video and try to
factor that as much as you can.

00:01:41.356 --> 00:01:42.969
Alright, so the first thing
you might have noticed

00:01:42.969 --> 00:01:45.554
is that there is a common
factor amongst the terms.

00:01:45.554 --> 00:01:47.667
All of them are divisible by four.

00:01:47.667 --> 00:01:49.250
Four is clearly divisible by four

00:01:49.250 --> 00:01:51.485
and 48 is also divisible by four.

00:01:51.485 --> 00:01:53.064
So let's factor a four out.

00:01:53.064 --> 00:01:54.188
This would be the same thing as four

00:01:54.188 --> 00:01:57.438
times x squared minus x minus 12.

00:02:00.160 --> 00:02:02.959
I just divided each of these
by four and I factored it out.

00:02:02.959 --> 00:02:04.596
You can distribute the four and verify

00:02:04.596 --> 00:02:06.854
that these two expressions are the same.

00:02:06.854 --> 00:02:08.497
Now are we done?

00:02:08.497 --> 00:02:10.398
Well no, we can factor what we have

00:02:10.398 --> 00:02:14.212
inside the parentheses, we
can factor this further.

00:02:14.212 --> 00:02:16.728
Now how would we do that?

00:02:16.728 --> 00:02:18.550
So over here the key realization

00:02:18.550 --> 00:02:20.793
is alright, I have a one as a coefficient

00:02:20.793 --> 00:02:22.778
on my second degree term.

00:02:22.778 --> 00:02:23.998
I've written it in standard form

00:02:23.998 --> 00:02:25.284
where I have the second degree and then

00:02:25.284 --> 00:02:26.516
if there's a first degree term,

00:02:26.516 --> 00:02:27.706
and then I have my constant term

00:02:27.706 --> 00:02:29.719
or my zero degree term, and if I have

00:02:29.719 --> 00:02:32.055
a one coefficient right over here

00:02:32.055 --> 00:02:34.496
then I say, okay, are there two numbers

00:02:34.496 --> 00:02:37.376
whose sum equals the coefficient

00:02:37.376 --> 00:02:39.503
on the first degree term, on the x term,

00:02:39.503 --> 00:02:43.206
so are there two numbers
that add up to negative one?

00:02:43.206 --> 00:02:46.082
You didn't see a one here
before but it's implicit there.

00:02:46.082 --> 00:02:49.067
Negative x is the same
thing as negative 1x.

00:02:49.067 --> 00:02:51.008
So are there two numbers, a plus b,

00:02:51.008 --> 00:02:53.568
that is equal to negative one and whose

00:02:53.568 --> 00:02:56.235
product is equal to negative 12?

00:02:57.904 --> 00:03:00.221
This is a technique that
we do in other videos,

00:03:00.221 --> 00:03:01.582
and here the key is to realize that hey,

00:03:01.582 --> 00:03:03.605
maybe we can use it here.

00:03:03.605 --> 00:03:06.772
So a times be is equal to negative 12.

00:03:08.800 --> 00:03:10.641
And there's a couple of
key realizations here.

00:03:10.641 --> 00:03:11.932
It's like, okay, if I have two numbers

00:03:11.932 --> 00:03:13.951
and their product is
going to be a negative

00:03:13.951 --> 00:03:15.334
that means one of them, that means

00:03:15.334 --> 00:03:16.429
they're gonna have different signs.

00:03:16.429 --> 00:03:18.341
One's going to be positive and
one's going to be negative.

00:03:18.341 --> 00:03:21.494
If they had the same sign
then this would be positive.

00:03:21.494 --> 00:03:22.663
So let's think about the factors of 12,

00:03:22.663 --> 00:03:24.421
and especially think about them

00:03:24.421 --> 00:03:26.817
in terms of different sign combinations.

00:03:26.817 --> 00:03:29.817
So you could think about one and 12,

00:03:30.722 --> 00:03:32.478
and whether you're
thinking about negative one

00:03:32.478 --> 00:03:37.008
and 12, negative one plus
12 would be positive 11.

00:03:37.008 --> 00:03:38.418
If you went the other way around,

00:03:38.418 --> 00:03:41.715
if you went negative 12 and
one it would be negative 11,

00:03:41.715 --> 00:03:43.590
but either way that doesn't work.

00:03:43.590 --> 00:03:45.815
Two and six, negative two and six

00:03:45.815 --> 00:03:48.183
would be four, negative six and two

00:03:48.183 --> 00:03:50.564
would be negative four,
so that doesn't work.

00:03:50.564 --> 00:03:51.814
Three and four.

00:03:52.709 --> 00:03:54.266
Let's see, negative three and four

00:03:54.266 --> 00:03:57.071
would be positive one, but three

00:03:57.071 --> 00:03:58.592
and negative four works out.

00:03:58.592 --> 00:03:59.880
You add these two together, you take

00:03:59.880 --> 00:04:01.943
their product, you clearly get negative 12

00:04:01.943 --> 00:04:05.250
and then you add them
together, you get negative one.

00:04:05.250 --> 00:04:08.324
So we can write inside the parentheses,

00:04:08.324 --> 00:04:09.589
so let me write, so this is gonna be

00:04:09.589 --> 00:04:13.756
four times, so we can factor
that as two binomials.

00:04:15.662 --> 00:04:18.579
The first is going to be x plus,

00:04:20.471 --> 00:04:22.772
the first is going to be x plus three

00:04:22.772 --> 00:04:23.872
and then the next is going to be,

00:04:23.872 --> 00:04:25.467
we could say x plus negative four

00:04:25.467 --> 00:04:28.854
or we could say x minus
four, and we're done.

00:04:28.854 --> 00:04:30.854
And if any of this seems
intimidating to you

00:04:30.854 --> 00:04:32.645
I encourage you to watch the videos

00:04:32.645 --> 00:04:34.986
on introduction to factoring polynomials.

00:04:34.986 --> 00:04:37.748
The key here is to recognize the method.

00:04:37.748 --> 00:04:39.735
So once again, at first try to

00:04:39.735 --> 00:04:41.209
factor out any common factors.

00:04:41.209 --> 00:04:43.335
We did that in both examples.

00:04:43.335 --> 00:04:44.478
And then we saw here, hey if we have

00:04:44.478 --> 00:04:46.101
a leading one coefficient here

00:04:46.101 --> 00:04:47.366
on the second degree term and we have

00:04:47.366 --> 00:04:49.607
it written in standard form, well let's

00:04:49.607 --> 00:04:51.015
think of two numbers that add up

00:04:51.015 --> 00:04:53.067
to this coefficient and whose product

00:04:53.067 --> 00:04:55.586
is equal to the constant
term, and in this case

00:04:55.586 --> 00:04:57.023
it was three and negative four,

00:04:57.023 --> 00:04:58.357
we were able to factor it this way.

00:04:58.357 --> 00:05:00.909
We prove that in other videos.

00:05:00.909 --> 00:05:02.205
Let's do another example.

00:05:02.205 --> 00:05:04.749
We can't get enough
practice, and like always,

00:05:04.749 --> 00:05:07.451
pause the video and see if you
can work through it yourself.

00:05:07.451 --> 00:05:10.534
Three x squared, plus 30, plus 75.

00:05:13.318 --> 00:05:15.562
Alright, I'm assuming you had a go at it.

00:05:15.562 --> 00:05:16.765
So you might immediately see that

00:05:16.765 --> 00:05:18.662
all of the terms are divisible by three,

00:05:18.662 --> 00:05:20.311
so let's factor three
out, so it's gonna be

00:05:20.311 --> 00:05:23.315
three times x squared plus, oh whoops,

00:05:23.315 --> 00:05:25.170
this should be an x here, my apologies.

00:05:25.170 --> 00:05:27.436
Pause the video again and
see if you can do it (laughs)

00:05:27.436 --> 00:05:30.317
now that I wrote the
actual right thing there.

00:05:30.317 --> 00:05:33.250
So as you imagine it's nice to factor out

00:05:33.250 --> 00:05:36.515
a three first, so it's
three times x squared,

00:05:36.515 --> 00:05:39.432
plus 10x, plus 25, and so you might

00:05:40.936 --> 00:05:42.168
immediately say alright, let's use

00:05:42.168 --> 00:05:43.332
the technique we had here.

00:05:43.332 --> 00:05:45.240
We have a leading one coefficient,

00:05:45.240 --> 00:05:46.986
it's written in standard form.

00:05:46.986 --> 00:05:50.270
Can I think of two numbers
that add up to ten,

00:05:50.270 --> 00:05:53.468
so a plus b is equal to
10, and whose product,

00:05:53.468 --> 00:05:55.635
a times b, is equal to 25?

00:05:56.504 --> 00:05:58.259
And this would work.

00:05:58.259 --> 00:06:01.244
And if you look at the factors of 25

00:06:01.244 --> 00:06:02.972
you'd say, alright, well this thing here

00:06:02.972 --> 00:06:04.484
is positive, this is positive,

00:06:04.484 --> 00:06:05.948
so I'm dealing with two positive numbers

00:06:05.948 --> 00:06:07.625
and to get 25 it's either a one and 25

00:06:07.625 --> 00:06:11.588
or five and five, and
five and five match this.

00:06:11.588 --> 00:06:14.284
Five plus five is equal to 10.

00:06:14.284 --> 00:06:17.323
Five times five is equal to 25.

00:06:17.323 --> 00:06:18.767
And so just using the exact technique

00:06:18.767 --> 00:06:20.779
we just did, you'd say
okay this is three times

00:06:20.779 --> 00:06:22.154
and the stuff in parentheses would be

00:06:22.154 --> 00:06:24.904
x plus five times x plus five,

00:06:26.467 --> 00:06:31.254
or you could say three
times x plus five squared.

00:06:31.254 --> 00:06:32.630
So some of you might
have immediately said,

00:06:32.630 --> 00:06:34.972
well I don't have to do
that exact technique.

00:06:34.972 --> 00:06:36.588
I could have immediately recognized this

00:06:36.588 --> 00:06:39.410
as a perfect square
because I have a square

00:06:39.410 --> 00:06:40.784
constant right over here, and that's

00:06:40.784 --> 00:06:43.343
a good sign that hey,
maybe I should explore

00:06:43.343 --> 00:06:46.662
whether this is going to be
a perfect square polynomial.

00:06:46.662 --> 00:06:49.318
So this is a perfect square, and if I were

00:06:49.318 --> 00:06:51.597
to take the square root of
it, and this coefficient

00:06:51.597 --> 00:06:54.423
is twice that square root, well that's

00:06:54.423 --> 00:06:56.174
a good sign that I'm probably dealing,

00:06:56.174 --> 00:06:58.979
or that I am dealing
with a perfect square.

00:06:58.979 --> 00:07:00.502
But either way, whether you recognize it

00:07:00.502 --> 00:07:03.700
as a perfect square or
whether you use the technique

00:07:03.700 --> 00:07:05.327
that we used in the second problem,

00:07:05.327 --> 00:07:09.494
either one of those would get
you to the appropriate answer.


Perfect square factorization intro

WEBVTT
Kind: captions
Language: en

00:00:00.098 --> 00:00:03.849
- [Narrator] We're going to
learn to recognize and factor

00:00:03.849 --> 00:00:06.927
perfect square polynomials in this video.

00:00:06.927 --> 00:00:08.552
So for example, say I have the polynomial

00:00:08.552 --> 00:00:11.385
x squared plus six x plus nine.

00:00:12.340 --> 00:00:13.970
And then someone asks you,
"Hey, can you factor this

00:00:13.970 --> 00:00:15.558
"into two binomials?"

00:00:15.558 --> 00:00:17.364
Well, using techniques we
learned in other videos,

00:00:17.364 --> 00:00:18.885
say, "Okay, I need to find two numbers

00:00:18.885 --> 00:00:22.907
"whose product is nine
and whose sum is six."

00:00:22.907 --> 00:00:25.051
And so I encourage you to
pause this video and say,

00:00:25.051 --> 00:00:27.385
"Well, what two numbers can add up to six,

00:00:27.385 --> 00:00:30.443
"and if I take their product I get nine?"

00:00:30.443 --> 00:00:32.489
Well, nine only has so many factors,

00:00:32.489 --> 00:00:34.460
really one, three, and nine.

00:00:34.460 --> 00:00:38.163
And one plus nine does not equal six.

00:00:38.163 --> 00:00:40.625
And so, and negative
one plus negative nine

00:00:40.625 --> 00:00:42.055
does not equal six.

00:00:42.055 --> 00:00:45.269
But three times three equals nine,

00:00:45.269 --> 00:00:47.876
and three plus three does equal six.

00:00:47.876 --> 00:00:51.310
Three times three, three plus three.

00:00:51.310 --> 00:00:54.126
And so we can factor this as

00:00:54.126 --> 00:00:57.043
x plus three times x plus three,

00:00:58.958 --> 00:01:02.059
which is of course the same thing as

00:01:02.059 --> 00:01:03.809
x plus three squared.

00:01:06.513 --> 00:01:08.942
And so what was it about this expression

00:01:08.942 --> 00:01:10.807
that made us recognize,

00:01:10.807 --> 00:01:12.991
or maybe now we will start to recognize it

00:01:12.991 --> 00:01:14.924
as being a perfect square?

00:01:14.924 --> 00:01:19.045
Well, I have of course some
variable that is being squared,

00:01:19.045 --> 00:01:20.347
which we need.

00:01:20.347 --> 00:01:23.409
I have some perfect square as a constant,

00:01:23.409 --> 00:01:25.545
and that whatever is being squared there,

00:01:25.545 --> 00:01:28.005
I have two times that as the coefficient

00:01:28.005 --> 00:01:30.691
on this first degree term here.

00:01:30.691 --> 00:01:33.038
Let's see if that is generally true.

00:01:33.038 --> 00:01:35.858
And I'll switch up the variables
just to show that we can.

00:01:35.858 --> 00:01:40.025
So let's say that I have a
squared plus 14 a plus 49.

00:01:43.560 --> 00:01:45.512
So a few interesting
things are happening here.

00:01:45.512 --> 00:01:47.677
All right, I have my variable squared.

00:01:47.677 --> 00:01:49.404
I have a perfect square constant term,

00:01:49.404 --> 00:01:52.095
that is seven squared right over here.

00:01:52.095 --> 00:01:55.351
And my coefficient on my
first degree term here

00:01:55.351 --> 00:01:57.723
that is two times the
thing that's being squared.

00:01:57.723 --> 00:01:59.432
That is two times seven, or you can say

00:01:59.432 --> 00:02:01.431
it's seven plus seven.

00:02:01.431 --> 00:02:02.508
So you can immediately say, "Okay,

00:02:02.508 --> 00:02:03.862
"if I want to factor this,

00:02:03.862 --> 00:02:07.759
"this is going to be
a plus seven squared."

00:02:07.759 --> 00:02:09.571
And you can of course verify that

00:02:09.571 --> 00:02:11.087
by multiplying out,

00:02:11.087 --> 00:02:13.721
by figuring out what a
plus seven squared is.

00:02:13.721 --> 00:02:14.952
Sometimes when you're first learning this,

00:02:14.952 --> 00:02:16.312
you're like, "Hey, isn't
that just a squared

00:02:16.312 --> 00:02:17.230
"plus seven squared?"

00:02:17.230 --> 00:02:18.063
No!

00:02:18.063 --> 00:02:20.398
Remember, this is the same thing as

00:02:20.398 --> 00:02:23.065
a plus seven times a plus seven.

00:02:25.149 --> 00:02:27.595
And you can calculate this by using

00:02:27.595 --> 00:02:29.952
the foil, F-O-I-L technique.

00:02:29.952 --> 00:02:30.857
I don't like that so much

00:02:30.857 --> 00:02:32.126
because you're not thinking mathematically

00:02:32.126 --> 00:02:33.343
about what's happening.

00:02:33.343 --> 00:02:35.993
Really you just have to do
distributive property twice here.

00:02:35.993 --> 00:02:39.618
First you can multiply
a plus seven times a.

00:02:39.618 --> 00:02:41.618
So a plus seven times a.

00:02:42.816 --> 00:02:45.005
And then multiply a
plus seven times seven.

00:02:45.005 --> 00:02:47.755
So plus a plus seven times seven,

00:02:48.781 --> 00:02:52.104
and so this is going to
be a squared plus seven a,

00:02:52.104 --> 00:02:55.025
plus, now we distribute the seven.

00:02:55.025 --> 00:02:56.775
Plus seven a plus 49.

00:02:58.885 --> 00:03:03.023
So now you see where that 14 a came from.

00:03:03.023 --> 00:03:05.933
It's from the seven a plus the seven a.

00:03:05.933 --> 00:03:07.782
You see where the a squared came from.

00:03:07.782 --> 00:03:11.241
And you see where the 49 came from.

00:03:11.241 --> 00:03:15.185
And you can speak of this
in more general terms.

00:03:15.185 --> 00:03:18.755
If I wanted to just take the expression

00:03:18.755 --> 00:03:20.672
a plus b and square it,

00:03:22.687 --> 00:03:25.687
that's just a plus b times a plus b,

00:03:27.522 --> 00:03:29.249
and we do exactly what we did just here,

00:03:29.249 --> 00:03:31.257
but here I'm just doing
in very general terms

00:03:31.257 --> 00:03:32.992
with a or b and you can think of a

00:03:32.992 --> 00:03:34.340
as either a constant number

00:03:34.340 --> 00:03:35.780
or even a variable.

00:03:35.780 --> 00:03:39.987
And so this is going to
be, if we distribute this,

00:03:39.987 --> 00:03:43.154
it's going to be a plus b times that a

00:03:44.308 --> 00:03:46.558
plus a plus b times that b.

00:03:48.864 --> 00:03:50.290
And so this is going to be a squared,

00:03:50.290 --> 00:03:52.467
now I'm just doing the
distributive property again.

00:03:52.467 --> 00:03:55.884
A squared plus ab plus ab plus b squared.

00:04:01.115 --> 00:04:04.865
So it's a squared plus
two ab plus b squared.

00:04:08.669 --> 00:04:11.275
So this is going to be the general form.

00:04:11.275 --> 00:04:15.442
So if a is the variable, which
was x, or a in this case,

00:04:18.412 --> 00:04:20.127
then it's just going
to be whatever squared

00:04:20.127 --> 00:04:22.802
and the constant term is
going to be two times that

00:04:22.802 --> 00:04:24.254
times the variable.

00:04:24.254 --> 00:04:26.311
And I want to show that
there's some variation

00:04:26.311 --> 00:04:28.250
that you can entertain here.

00:04:28.250 --> 00:04:32.417
So if you were to see 25
plus 10x plus x squared,

00:04:38.071 --> 00:04:38.904
and someone wanted you,

00:04:38.904 --> 00:04:40.727
said "Hey, why don't you factor that?"

00:04:40.727 --> 00:04:42.885
we could say, "Look, this
right here is a perfect square.

00:04:42.885 --> 00:04:44.169
"It's five squared.

00:04:44.169 --> 00:04:46.092
"I have the variable
squared right over here,

00:04:46.092 --> 00:04:48.192
"and then this coefficient
on our first degree term

00:04:48.192 --> 00:04:50.087
"is two times five."

00:04:50.087 --> 00:04:52.168
And so you might immediately
recognize this as

00:04:52.168 --> 00:04:54.085
five plus x squared.

00:04:55.471 --> 00:04:57.878
Now of course you could
just rewrite this polynomial

00:04:57.878 --> 00:05:00.878
as x squared plus 10 x plus 25

00:05:01.714 --> 00:05:04.763
in which case you might say,
"Okay, variable squared,

00:05:04.763 --> 00:05:06.667
"some number squared, five squared,

00:05:06.667 --> 00:05:08.903
"two times that number
is the coefficient here.

00:05:08.903 --> 00:05:12.027
"So that's going to be
x plus five squared."

00:05:12.027 --> 00:05:14.321
And that's good because these two things

00:05:14.321 --> 00:05:16.488
are absolutely equivalent.


Difference of squares intro

WEBVTT
Kind: captions
Language: en

00:00:00.217 --> 00:00:02.261
- [Instructor] We're now
going to explore factoring

00:00:02.261 --> 00:00:05.131
a type of expression called
a difference of squares

00:00:05.131 --> 00:00:07.135
and the reason why it's
called a difference of squares

00:00:07.135 --> 00:00:11.506
is 'cause it's expressions
like x squared minus nine.

00:00:11.506 --> 00:00:12.399
This is a difference.

00:00:12.399 --> 00:00:14.119
We're subtracting between two quantities

00:00:14.119 --> 00:00:15.687
that are each squares.

00:00:15.687 --> 00:00:18.270
This is literally x squared.

00:00:19.570 --> 00:00:21.419
Let me do that in a different color.

00:00:21.419 --> 00:00:24.836
This is x squared minus three squared.

00:00:29.825 --> 00:00:32.769
It's the difference between two quantities

00:00:32.769 --> 00:00:35.010
that have been squared and it turns out

00:00:35.010 --> 00:00:38.738
that this is pretty
straightforward to factor.

00:00:38.738 --> 00:00:41.968
And to see how it can be factored,

00:00:41.968 --> 00:00:44.347
let me pause there for a second

00:00:44.347 --> 00:00:47.373
and get a little bit of review
of multiplying binomials.

00:00:47.373 --> 00:00:48.990
So put this on the back
burner a little bit.

00:00:48.990 --> 00:00:51.795
Before I give you the answer
of how you factor this,

00:00:51.795 --> 00:00:53.555
let's do a little bit of an exercise.

00:00:53.555 --> 00:00:57.305
Let's multiply x plus a times x minus a

00:00:59.029 --> 00:01:00.582
where a is some number.

00:01:00.582 --> 00:01:01.521
Now, we can use that,

00:01:01.521 --> 00:01:04.054
do that using either the FOIL method

00:01:04.054 --> 00:01:05.159
but I like just thinking of this

00:01:05.159 --> 00:01:07.159
as a distributive property twice.

00:01:07.159 --> 00:01:08.915
We could take x plus a

00:01:08.915 --> 00:01:12.642
and distribute it onto
the x and onto the a.

00:01:12.642 --> 00:01:15.013
So when we multiply it by x,

00:01:15.013 --> 00:01:18.448
we would get x times x is x squared,

00:01:18.448 --> 00:01:20.365
a times x is plus ax

00:01:21.310 --> 00:01:24.703
and then when we multiply
it by the negative a,

00:01:24.703 --> 00:01:27.786
well, it'll become negative a times x

00:01:28.760 --> 00:01:30.093
minus a squared.

00:01:31.131 --> 00:01:34.401
So these middle two terms cancel out

00:01:34.401 --> 00:01:37.117
and you are left with x
squared minus a squared.

00:01:37.117 --> 00:01:39.636
You're left with a difference of squares.

00:01:39.636 --> 00:01:41.803
x squared minus a squared.

00:01:43.617 --> 00:01:46.155
So we have an interesting
result right over here

00:01:46.155 --> 00:01:50.410
that x squared minus
a squared is equal to,

00:01:50.410 --> 00:01:52.410
is equal to x plus a,

00:01:53.961 --> 00:01:56.294
x plus a times x minus a.

00:01:59.127 --> 00:02:01.950
And so we can use, and this is for any a.

00:02:01.950 --> 00:02:05.853
So we could use this
pattern now to factor this.

00:02:05.853 --> 00:02:07.651
Here, what is our a?

00:02:07.651 --> 00:02:08.901
Our a is three.

00:02:10.067 --> 00:02:12.315
This is x squared minus three squared

00:02:12.315 --> 00:02:16.575
or we could say minus our a
squared if we say three is a

00:02:16.575 --> 00:02:21.000
and so to factor it, this
is just going to be equal to

00:02:21.000 --> 00:02:23.250
x plus our a which is three

00:02:24.896 --> 00:02:27.641
times x minus our a which is three.

00:02:27.641 --> 00:02:31.450
So x plus three times x minus three.

00:02:31.450 --> 00:02:34.637
Now, let's do some examples
to really reinforce this idea

00:02:34.637 --> 00:02:37.745
of factoring differences of squares.

00:02:37.745 --> 00:02:40.853
So let's say we want to factor,

00:02:40.853 --> 00:02:43.270
let me say y squared minus 25

00:02:46.224 --> 00:02:47.712
and it has to be a difference of squares.

00:02:47.712 --> 00:02:51.139
It doesn't work with a sum of squares.

00:02:51.139 --> 00:02:52.171
Well, in this case,

00:02:52.171 --> 00:02:54.584
this is going to be y
and you have to confirm,

00:02:54.584 --> 00:02:56.715
okay, yeah, 25 is five squared

00:02:56.715 --> 00:02:59.377
and y squared is well, y squared.

00:02:59.377 --> 00:03:03.421
So this gonna be y plus
something times y minus something

00:03:03.421 --> 00:03:04.646
and what is that something?

00:03:04.646 --> 00:03:07.795
Well, this right here is five
squared so it's y plus five

00:03:07.795 --> 00:03:09.675
times y minus five

00:03:09.675 --> 00:03:11.723
and the variable doesn't
have to come first.

00:03:11.723 --> 00:03:13.806
We could write 121 minus,

00:03:17.091 --> 00:03:21.317
I'll introduce a new
variable, minus b squared.

00:03:21.317 --> 00:03:22.971
Well, this is a difference of squares

00:03:22.971 --> 00:03:25.138
because 121 is 11 squared.

00:03:26.051 --> 00:03:29.120
So this is going to be 11 plus something

00:03:29.120 --> 00:03:31.486
times 11 minus something and in this case,

00:03:31.486 --> 00:03:35.979
that something is going to be
the thing that was squared.

00:03:35.979 --> 00:03:38.479
So 11 plus b times 11 minus b.

00:03:41.183 --> 00:03:45.311
So in general, if you see
a difference of squares,

00:03:45.311 --> 00:03:48.345
one square being subtracted
from another square

00:03:48.345 --> 00:03:50.385
and it could be a numeric perfect square

00:03:50.385 --> 00:03:53.900
or it could be a variable that
has been squared that can be,

00:03:53.900 --> 00:03:55.783
that you could take the square root of.

00:03:55.783 --> 00:03:56.724
Well, then you could say, alright,

00:03:56.724 --> 00:03:58.653
well, that's just gonna be the first thing

00:03:58.653 --> 00:04:02.132
that squared plus the second
thing that has been squared

00:04:02.132 --> 00:04:04.159
times the first thing that was squared

00:04:04.159 --> 00:04:06.811
minus the second thing that was squared.

00:04:06.811 --> 00:04:09.507
Now, some common mistakes
that I've seen people do

00:04:09.507 --> 00:04:11.878
including my son when
they first learned this

00:04:11.878 --> 00:04:14.369
is they say, okay, it's easy to recognize

00:04:14.369 --> 00:04:16.050
the difference of squares

00:04:16.050 --> 00:04:19.158
but then they say, oh,
is this y squared plus 25

00:04:19.158 --> 00:04:21.711
times y squared minus 25?

00:04:21.711 --> 00:04:24.514
No, the important thing to realize is

00:04:24.514 --> 00:04:26.896
is that what is getting squared?

00:04:26.896 --> 00:04:30.313
Over here, y is the thing getting squared

00:04:31.427 --> 00:04:35.311
and over here it is five
that is getting squared.

00:04:35.311 --> 00:04:37.708
Those are the things
that are getting squared

00:04:37.708 --> 00:04:39.362
in this difference of squares

00:04:39.362 --> 00:04:44.278
and so it's gonna be y plus
five times y minus five.

00:04:44.278 --> 00:04:46.360
I encourage you to just try this out.

00:04:46.360 --> 00:04:48.136
We have a whole practice
section on Khan Academy

00:04:48.136 --> 00:04:52.303
where you can do many many more
of these to become familiar.