Quadratic inequalities

WEBVTT
Kind: captions
Language: en

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Welcome to the presentation
on quadratic inequalities.

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I know that sounds very
complicated, but hopefully

00:00:06.450 --> 00:00:09.140
you'll see it's actually
not that difficult.

00:00:09.140 --> 00:00:10.830
Or at least, maybe the problems
we're going to work on

00:00:10.830 --> 00:00:12.170
aren't that difficult.

00:00:12.170 --> 00:00:15.890
Well, let's get started with
some problems and hopefully

00:00:15.890 --> 00:00:17.730
you'll see where this is kind
of slightly different

00:00:17.730 --> 00:00:20.050
than solving regular
quadratic equations.

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So let's say I had the
inequality x squared plus

00:00:27.730 --> 00:00:33.480
3x is greater than 10.

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And remember, whenever you
solve a quadratic or I guess

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you would call it a second
degree equation-- I guess

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this is an inequality.

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I shouldn't use the
word equation.

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It's tempting to sometimes do
it the same way you'd do a

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linear equation, kind of
getting all the x terms

00:00:45.470 --> 00:00:47.070
on one side and all the
constants on the other.

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But it never works because you
actually have an x term and

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then you have an
x squared term.

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So you actually want to get it
in kind of what I would call

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the-- I don't know if it's
actually called this-- the

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standard form where you
actually have all of the terms

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on one side and then a
0 on the other side.

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And then you can either
factor it or use the

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quadratic equation.

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So let's do that.

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Well this is pretty easy.

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We just have to subtract 10
from both sides and we get x

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squared plus 3x minus
10 is greater than 0.

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Now let's see if
we can factor it.

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Are there two numbers that when
you multiply it become negative

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10 and when you add it
become positive 3?

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Well, yeah.

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Positive 5 and negative 2.

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And once again, at this point I
think you already know how to

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do factoring, so this should
be hopefully, obvious to you.

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So it's x plus 5 times x
minus 2 is greater than 0.

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Now this is the part where it's
going to become a little bit

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more difficult than just your
traditional factoring problem.

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We have two numbers, I
guess you could view it.

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We have x plus 5.

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I view that as one number.

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Or I guess we have
two expressions.

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We have x plus 5 and
we have x minus 2.

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And when we're multiplying
them we're getting

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something greater than 0.

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Now let's think about
what happens when you

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multiply numbers.

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If they're both positive and
you multiply them, then

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you get a positive number.

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And if they're both negative
and you multiply them, then you

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also get a positive number.

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So we know that either both of
these expressions are the same

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sign, that they're both greater
than 0, they're both positive.

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Or we know that they're
both negative.

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And I know this might be a
little confusing, but just

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think of it as-- if I told you
that-- I'll do something

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slightly separate out here.

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If I told you that a times b is
greater than 0 we know that

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either a is greater than 0
and b is greater than 0.

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Which just means that
they're both positive.

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Or a is less than 0
and b is less than 0.

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Which means that
they're both negative.

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All we know is that they both
have to be the same sign in

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order for their product
to be greater than 0.

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Now we just do the
same thing here.

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So we know that either both of
these are positive, so x plus

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5 is greater than 0 and x
minus 2 is greater than 0.

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Or-- now this is a little
confusing, but if you work

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through these problems it
actually makes a lot of sense.

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Or they're both negative.

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Or x plus 5 is less than 0 and
x minus 2 is less than 0.

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I know that's confusing, but
just think of it in terms of we

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have two expressions: they're
either both positive or

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they're either both negative.

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Because when you multiple
them you get something

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larger than 0.

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Well, let's solve this side.

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So this says that x is
greater than negative 5

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and x is greater than 2.

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We just 2 both sides
of this equation.

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Or, and if we solve this side--
x is less than negative

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5 and x is less than 2.

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I just solved both of these
inequalities right here.

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Now we can actually simplify
this because here we say that

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x is greater than negative
5 and x is greater than 2.

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So in order for x ti be greater
than negative 5 and for x to be

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greater than 2, this just
simplifies as saying, well,

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x is just greater than 2.

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Because if x is greater
than 2, it's definitely

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greater than negative 5.

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So it just simplifies to this.

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And we'd say or-- and here we
said x is less than negative

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5 or x is less than 2.

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Well, we know if x is less
than negative 5, then x is

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definitely less than 2.

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So we could just simplify it to
or x is less than negative 5.

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So the solutions to this
problem is x could be greater

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than 2 or x could be
less than negative 5.

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And so let's just think
about how that looks

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on the number line.

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So if 2 is here, x could
be greater than 2.

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So it's all of these numbers.

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And if this is negative 5--
I shouldn't have done it

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so close to the bottom.

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x is less the negative 5.

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So these are the numbers
that satisfy this equation.

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And I'll leave it up to you
to try out to see that

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they actually work.

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Let's try another one
and hopefully, I can

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confuse you even more.

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Let's say I have minus x
times 2x minus 14 is greater

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than or equal to 24.

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Well, the first thing we want
to do is just manipulate this

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so it looks in the
standard form.

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So we get negative 2x squared
plus 14x-- I'm just

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distributing the minus x-- is
greater than or equal to 24.

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I don't like any coefficient it
front of my x squared term,

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so let's divide both sides of
this equation by negative 2.

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So we get x squared--
we divided by negative

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2-- minus 7x.

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And remember, when you divide
by a negative number you switch

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the sign on the inequality, or
you switch the direction

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of the inequality.

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So we're dividing by negative
2, so we switched it.

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We went from greater than
or equal to, to less

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than or equal to.

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And then 24 divided by
negative 2 is minus 12.

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And now we can just bring this
minus 12 onto the left-hand

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side of the equation.

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Add 12 to both sides.

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We get x squared minus 7x plus
12 is less than or equal to 0.

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And then we can just factor
that and we get, what is that?

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It's x minus 3 times x minus 4
is less than or equal to 0.

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So now we know that when we
multiply these two terms

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we get a negative number.

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So that means that these
expressions have to be

00:07:29.350 --> 00:07:31.780
of different signs.

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Does that make sense?

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If I tell you I have two
number and I multiply them,

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I get a negative number.

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You know that they have to
be of different signs.

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So we know that either x minus
3 is less than or equal to 0

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and x minus 4 is greater
than or equal to 0.

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So that's one case.

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And the other case is x minus 3
is greater than or equal to 0,

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which means x minus
3 is positive.

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And x minus 4 is less
than or equal to 0--

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oh, I went to the edge.

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So let's solve this and
hopefully it'll simplify more.

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So this just says that x is
less than or equal to 3.

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And this says x is greater
than or equal to 4.

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So both of these things have to
be true. x has to be less than

00:08:20.915 --> 00:08:26.030
or equal to 3 and x has to be
greater than or equal to 4.

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Well, let me ask
you a question.

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Can something be both less than
or equal to 3 and greater

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than or equal to 4?

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Well, no.

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So we know that this
situation can't happen.

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There's no numbers that's
less than or equal to 3 and

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greater than or equal to 4.

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So let's look at
this situation.

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This says x is greater than
or equal to 3 and x is

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less than or equal to 4.

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Can this happen?

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Sure.

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That just means that x is
some number between 3 and 4.

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If we were to draw this on the
number line, we would get--

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if this is 3, this is 4.

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And it's greater than or
equal to so we fill it in.

00:09:06.750 --> 00:09:09.250
And less than or equal
to so we'd fill it in.

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And it would be any number
between 3 and 4 would

00:09:11.580 --> 00:09:12.485
satisfy this equation.

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And I'll leave it up
to you to try it out.

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I know this is confusing at
first, and this is actually

00:09:17.240 --> 00:09:19.160
something that they normally
don't teach really well, I

00:09:19.160 --> 00:09:21.420
think, in most high schools
until 10th or 11th grade.

00:09:21.420 --> 00:09:25.510
But just think about you're
multiplying two expressions.

00:09:25.510 --> 00:09:27.630
If the answer is negative
then they must be

00:09:27.630 --> 00:09:28.390
of different signs.

00:09:28.390 --> 00:09:31.360
If the answer is positive
they must be the same sign.

00:09:31.360 --> 00:09:33.020
And then you just work
through the logic.

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And you say, well, no number
can be less than 3 and greater

00:09:35.700 --> 00:09:37.900
than 4, so this doesn't apply.

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And then you do this side
and you're like, oh, this

00:09:39.490 --> 00:09:40.410
situation does work.

00:09:40.410 --> 00:09:42.580
It's any number
between 3 and 4.

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Hopefully that gives you
a sense of how to do

00:09:44.090 --> 00:09:45.110
these type of problems.

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I'll let you do the
exercises now.

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Have fun.


Algebra: Solving Inequalities

WEBVTT
Kind: captions
Language: en

00:00:01.100 --> 00:00:03.775
Welcome to the presentation
on solving inequalities,

00:00:03.775 --> 00:00:06.890
or I guess you call them
algebra inequalities.

00:00:06.890 --> 00:00:09.120
So let's get started.

00:00:09.120 --> 00:00:14.500
If I were to tell you that,
well, let's just say x is

00:00:14.500 --> 00:00:17.840
greater than 5, right?

00:00:17.840 --> 00:00:22.700
So x could be 5.01, it could be
5.5, it could be a million.

00:00:22.700 --> 00:00:26.680
It just can't be 4, or 3, or 0,
or negative 8, and actually,

00:00:26.680 --> 00:00:28.500
just for convenience,
let's actually draw that

00:00:28.500 --> 00:00:30.970
on the number line.

00:00:30.970 --> 00:00:33.260
That's the number line.

00:00:33.260 --> 00:00:38.270
And if this is 5, x can't be
equal to 5, so we draw a big

00:00:38.270 --> 00:00:41.400
circle here, and then we would
color in all the values

00:00:41.400 --> 00:00:42.060
that x could be.

00:00:42.060 --> 00:00:46.510
So x could be just the small--
it could be 5.0000001.

00:00:46.510 --> 00:00:49.110
It just has to be a little bit
bigger than 5, and any of those

00:00:49.110 --> 00:00:51.310
would satisfy it, right?

00:00:51.310 --> 00:00:53.660
So let's just write some
numbers that satisfy.

00:00:53.660 --> 00:00:56.120
6 would satisfy it, 10
would satisfy it, 100

00:00:56.120 --> 00:00:58.060
would satisfy it.

00:00:58.060 --> 00:01:01.640
Now, if I were to multiply, or
I guess divide, both sides of

00:01:01.640 --> 00:01:05.700
this, I guess we could say,
equation or this inequality

00:01:05.700 --> 00:01:10.000
by negative 1, I want to
understand what happens.

00:01:10.000 --> 00:01:18.360
So what's the relation between
negative x and negative 5?

00:01:18.360 --> 00:01:21.500
When I say what's the relation,
is it greater than or is

00:01:21.500 --> 00:01:24.410
it less than negative 5?

00:01:24.410 --> 00:01:27.530
Well, 6 is a value
that works for x.

00:01:27.530 --> 00:01:33.170
So negative 6, is that greater
than or less than negative 5?

00:01:33.170 --> 00:01:36.810
Well, negative 6 is less
than negative 5, right?

00:01:36.810 --> 00:01:40.665
So let me draw the
number line here.

00:01:40.665 --> 00:01:44.622
If we have negative 5 here, and
let's just draw a circle around

00:01:44.622 --> 00:01:47.450
it because we know it's not
going to be equal to negative 5

00:01:47.450 --> 00:01:49.770
because we're deciding between
greater than or less than.

00:01:49.770 --> 00:01:51.500
So we're saying 6 works for x.

00:01:51.500 --> 00:01:55.580
So negative 6 is here, right?

00:01:55.580 --> 00:01:58.880
So negative 6 is less than
negative 5, so is negative 10,

00:01:58.880 --> 00:02:02.330
so is negative 100, so is
negative a million, right?

00:02:02.330 --> 00:02:07.920
So it turns out that negative
x is less than negative 5.

00:02:07.920 --> 00:02:11.540
And this is really all you have
to remember when you are

00:02:11.540 --> 00:02:13.940
working with inequalities
in algebra.

00:02:13.940 --> 00:02:17.260
Inequalities, you can treat
them just the way-- a greater

00:02:17.260 --> 00:02:19.050
than or less than sign, you
could treat them exactly the

00:02:19.050 --> 00:02:20.930
way you would treat
an equal sign.

00:02:20.930 --> 00:02:26.320
The only difference is, if you
multiply or divide both sides

00:02:26.320 --> 00:02:30.290
of the equation by a negative
number, you swap it.

00:02:30.290 --> 00:02:31.850
That's all you
have to remember.

00:02:31.850 --> 00:02:33.290
Let's do some problems,
and hopefully, that'll

00:02:33.290 --> 00:02:34.370
hit the point home.

00:02:34.370 --> 00:02:37.230
And if you ever forget, you
just have to try-- you just

00:02:37.230 --> 00:02:39.770
remember this: if x is greater
than 5, well, then negative

00:02:39.770 --> 00:02:40.840
x is less than negative 5.

00:02:40.840 --> 00:02:42.480
And keep trying out numbers.

00:02:42.480 --> 00:02:45.090
That's what's going to give
you the best intuition.

00:02:45.090 --> 00:02:47.650
Let's do some problems.

00:02:47.650 --> 00:02:55.120
So if I said that 3x plus 2 is,
let's say, less than or equal

00:02:55.120 --> 00:02:57.820
to 1-- well, this is a pretty
easy equation to solve.

00:02:57.820 --> 00:03:01.540
We just say 3x-- let's subtract
2 from both sides, and when you

00:03:01.540 --> 00:03:04.490
add or subtract, you don't do
anything to the inequality.

00:03:04.490 --> 00:03:09.180
So if you subtract 2 from both
sides, you get 3x is less than

00:03:09.180 --> 00:03:13.150
or equal to negative 1, right?

00:03:13.150 --> 00:03:16.730
And then, now we're going
to divide both sides by 3.

00:03:16.730 --> 00:03:21.690
We get x is less than or equal
to negative 1/3, right?

00:03:21.690 --> 00:03:23.020
And notice, we didn't
change anything because

00:03:23.020 --> 00:03:28.470
we divided both sides
by a positive 3, right?

00:03:28.470 --> 00:03:30.270
We could have actually
done this equation in a

00:03:30.270 --> 00:03:31.250
slightly different way.

00:03:31.250 --> 00:03:34.240
What if we subtracted
1 from both sides?

00:03:34.240 --> 00:03:36.270
So this is another
way of solving it.

00:03:36.270 --> 00:03:41.780
What if we said 3x plus 1 is
less than or equal to 0, right?

00:03:41.780 --> 00:03:44.820
I just subtracted 1 from both
sides, and now let's subtract

00:03:44.820 --> 00:03:45.920
3x from both sides.

00:03:45.920 --> 00:03:51.010
And we get 1 is less than or
equal to minus 3x, right?

00:03:51.010 --> 00:03:54.180
I subtracted 3x from here, so I
have to subtract 3x from here.

00:03:54.180 --> 00:03:56.480
Now, I would have to
divide both sides by a

00:03:56.480 --> 00:03:58.360
negative number, right?

00:03:58.360 --> 00:04:01.780
Because I'm going to divide
both sides by negative 3.

00:04:01.780 --> 00:04:06.170
So I get negative 1/3 on this
side, and based on what we had

00:04:06.170 --> 00:04:08.060
just learned, since we're
dividing by a negative number,

00:04:08.060 --> 00:04:10.350
we want to swap the
inequality, right?

00:04:10.350 --> 00:04:11.580
It was less than or equal,
now it's going to be

00:04:11.580 --> 00:04:14.840
greater than or equal to x.

00:04:14.840 --> 00:04:16.480
Now, did we get the same
answer when we did it

00:04:16.480 --> 00:04:18.520
both-- two different ways?

00:04:18.520 --> 00:04:22.310
Here, we got x is less than or
equal to negative 1/3, and

00:04:22.310 --> 00:04:25.440
here we got negative 1/3 is
greater than or equal to x.

00:04:25.440 --> 00:04:27.110
Well, that's the same answer,
right? x is less than or

00:04:27.110 --> 00:04:28.760
equal to negative 1/3.

00:04:28.760 --> 00:04:30.530
And that's-- I always
find that to be the cool

00:04:30.530 --> 00:04:31.210
thing about algebra.

00:04:31.210 --> 00:04:33.500
You can tackle the problem in a
bunch of different ways, and

00:04:33.500 --> 00:04:35.170
you should always get to the
right answer as long as

00:04:35.170 --> 00:04:37.940
you, I guess, do it right.

00:04:37.940 --> 00:04:40.840
Let's do a couple
more problems.

00:04:40.840 --> 00:04:42.680
Oh, let's erase this thing.

00:04:42.680 --> 00:04:43.710
There you go.

00:04:43.710 --> 00:04:45.450
I'll do a slightly harder one.

00:04:45.450 --> 00:04:56.760
Let's say negative 8x plus 7
is greater than 5x plus 2.

00:04:56.760 --> 00:05:00.030
Let's subtract 5x
from both sides.

00:05:00.030 --> 00:05:05.540
Negative 13x plus 7
is greater than 2.

00:05:05.540 --> 00:05:07.300
Now, we could subtract
7 from both sides.

00:05:07.300 --> 00:05:12.780
Negative 13x is
greater than minus 5.

00:05:12.780 --> 00:05:14.180
Now, we're going to divide
both sides of this

00:05:14.180 --> 00:05:17.050
equation by negative 13.

00:05:17.050 --> 00:05:18.720
Well, very easy.

00:05:18.720 --> 00:05:23.120
It's just x, and on this
side, negative 5 divided by

00:05:23.120 --> 00:05:24.990
negative 13 is 5/13, right?

00:05:24.990 --> 00:05:26.200
The negatives cancel out.

00:05:26.200 --> 00:05:31.382
And since we divided by a
negative, we switch the sign.

00:05:31.382 --> 00:05:33.750
x is less than 5/13.

00:05:33.750 --> 00:05:35.620
And once again, just like the
beginning, if you don't believe

00:05:35.620 --> 00:05:36.930
me, try out some numbers.

00:05:36.930 --> 00:05:39.460
And I remember when I first
learned this, I didn't believe

00:05:39.460 --> 00:05:41.780
the teacher, so I did try out
numbers, and that's how I got

00:05:41.780 --> 00:05:43.690
convinced that it
actually works.

00:05:43.690 --> 00:05:47.020
When you multiply or divide
both sides of this equation

00:05:47.020 --> 00:05:50.600
by a negative sign, you
swap the inequality.

00:05:50.600 --> 00:05:53.510
And remember, that's only when
you multiply or divide, not

00:05:53.510 --> 00:05:55.700
when you add or subtract.

00:05:55.700 --> 00:05:58.630
I think that should give
you a good idea of how

00:05:58.630 --> 00:05:59.350
to do these problems.

00:05:59.350 --> 00:06:01.160
There's really not
much new here.

00:06:01.160 --> 00:06:05.360
You do an inequality-- or I
guess you could call this an

00:06:05.360 --> 00:06:08.450
inequality equation-- you do it
exactly the same way you do a

00:06:08.450 --> 00:06:12.090
normal linear equation, the
only difference being is if you

00:06:12.090 --> 00:06:15.270
multiply or you divide both
sides of the equation by a

00:06:15.270 --> 00:06:19.250
negative number, then you
swap the inequality.

00:06:19.250 --> 00:06:22.810
I think you're ready now to
try some practice problems.

00:06:22.810 --> 00:06:24.300
Have fun.


Why borrowing works

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:05.080
Welcome to the presentation on
why, not how, borrowing works.

00:00:05.080 --> 00:00:06.750
And I think this is very
important because a lot of

00:00:06.750 --> 00:00:11.500
people who even know math
fairly well or have an advanced

00:00:11.500 --> 00:00:15.630
degree still aren't completely
sure on why borrowing works.

00:00:15.630 --> 00:00:17.710
That's the focus of
this presentation.

00:00:17.710 --> 00:00:20.500
Let's say I have the
subtraction problem

00:00:20.500 --> 00:00:23.450
1,000-- that's a 0.

00:00:23.450 --> 00:00:31.840
1,005 minus 616.

00:00:31.840 --> 00:00:34.050
What I'm going to do is I'm
going to write the same problem

00:00:34.050 --> 00:00:35.300
in a slightly different way.

00:00:35.300 --> 00:00:37.640
We could call this
the expanded form.

00:00:37.640 --> 00:00:40.080
1,005-- what I'm going to do
is I'm going to separate

00:00:40.080 --> 00:00:42.330
the digits out into
their respective places.

00:00:42.330 --> 00:00:49.720
So that is equal to 1,000
plus let's say zero 100's

00:00:49.720 --> 00:00:53.550
plus zero 10's plus 5.

00:00:53.550 --> 00:00:56.830
1,005 is just 1,000
plus 0 plus 0 plus 5.

00:00:56.830 --> 00:01:00.340
And then that's minus 616.

00:01:00.340 --> 00:01:08.910
So that's minus 600
minus 10 minus 6.

00:01:08.910 --> 00:01:13.440
616 could be rewritten
as 600 plus 10 plus 6.

00:01:13.440 --> 00:01:15.150
And I put a minus there
because we're subtracting

00:01:15.150 --> 00:01:16.130
the whole thing.

00:01:16.130 --> 00:01:18.940
So let's do this problem.

00:01:18.940 --> 00:01:23.220
Well, if you're familiar with
how you borrow is, this 5 is

00:01:23.220 --> 00:01:25.940
less than this 6, so we have to
somehow make this 5 a bigger

00:01:25.940 --> 00:01:28.220
number so that we could
subtract the 6 from it.

00:01:28.220 --> 00:01:31.060
Well, we know from traditional
borrowing that we have to

00:01:31.060 --> 00:01:33.890
borrow 1 from someplace and
make this it into a 15.

00:01:33.890 --> 00:01:36.240
But what I want to see
actually, is understand where

00:01:36.240 --> 00:01:38.530
that 1 or actually where
that 10 comes from.

00:01:38.530 --> 00:01:41.090
Because if you're turning this
5 into a 15 you actually

00:01:41.090 --> 00:01:43.090
have to add 10 to it.

00:01:43.090 --> 00:01:45.550
Well, if we look at this top
number, the only place that

00:01:45.550 --> 00:01:49.970
a 10 could come from is
here, is from this 1,000.

00:01:49.970 --> 00:01:52.950
But what we're going to do
since this is the 1,000's

00:01:52.950 --> 00:01:57.040
place, instead of borrowing 10
from here, which would make it

00:01:57.040 --> 00:01:59.480
kind of a very messy problem,
I'm going to borrow

00:01:59.480 --> 00:02:02.430
1,000 from here.

00:02:02.430 --> 00:02:03.960
I'm going to get
rid of this 1,000.

00:02:03.960 --> 00:02:08.110
And I have a 1,000 that
I took from this 1,000.

00:02:08.110 --> 00:02:12.740
I have 1,000 now that
I can distribute into

00:02:12.740 --> 00:02:14.790
these 3 buckets.

00:02:14.790 --> 00:02:17.360
Into the 100's, 10's
and 1's buckets.

00:02:17.360 --> 00:02:21.270
Well, we need 10 here,
so let's put 10 here.

00:02:21.270 --> 00:02:24.320
So it's 10 plus 5
is equal to 15.

00:02:24.320 --> 00:02:25.040
We got our 15.

00:02:27.820 --> 00:02:32.030
If we took 10 from the 1,000
then we have 990 left.

00:02:32.030 --> 00:02:37.960
So we could put 900
here and 90 here.

00:02:37.960 --> 00:02:41.250
Notice, we just said-- so we
had 1,000 and we just rewrote

00:02:41.250 --> 00:02:44.040
it as 900 plus 90 plus 10.

00:02:44.040 --> 00:02:45.670
And we added this 10 to this 5.

00:02:45.670 --> 00:02:47.670
And now we could do this
subtraction just how we

00:02:47.670 --> 00:02:49.110
would do a normal problem.

00:02:49.110 --> 00:02:52.710
15 minus 6 is 9.

00:02:52.710 --> 00:02:56.300
90 minus 10 is 80.

00:02:56.300 --> 00:03:00.730
900 minus 600 is 300.

00:03:00.730 --> 00:03:06.550
So 300 plus 80 plus 9 is 389.

00:03:06.550 --> 00:03:08.940
And let's see how we would have
done it traditionally and make

00:03:08.940 --> 00:03:12.790
sure that it would have kind of
translated into the same way.

00:03:12.790 --> 00:03:15.210
Well, the way I teach it and I
don't know if this is actually

00:03:15.210 --> 00:03:20.440
the traditional way of teaching
borrowing, is I say, OK, I need

00:03:20.440 --> 00:03:23.480
to turn this 5 into a 15.

00:03:23.480 --> 00:03:25.220
So I have to borrow
a 1 from someplace.

00:03:25.220 --> 00:03:26.840
Well, we know from this side of
the problem that we actually

00:03:26.840 --> 00:03:28.630
borrowed a 10 because that's
why it turned to 15.

00:03:28.630 --> 00:03:30.540
If we're going to borrow
1, I'd say, well, can I

00:03:30.540 --> 00:03:31.660
borrow the 1 from the 0?

00:03:31.660 --> 00:03:32.100
No.

00:03:32.100 --> 00:03:33.730
Can I borrow the 1 from this 0?

00:03:33.730 --> 00:03:34.440
No.

00:03:34.440 --> 00:03:36.630
I could borrow it from
here, but I'm borrowing

00:03:36.630 --> 00:03:38.980
it from 100, right?

00:03:38.980 --> 00:03:43.110
So 100 minus 1 is 99.

00:03:43.110 --> 00:03:44.500
So that's the how I do it.

00:03:44.500 --> 00:03:47.560
And I say 15 minus 6 is 9.

00:03:47.560 --> 00:03:49.370
9 minus 1 is 8.

00:03:49.370 --> 00:03:51.570
And 9 minus 6 is 300.

00:03:51.570 --> 00:03:55.600
So this way that I just did it
is clearly faster and, I guess

00:03:55.600 --> 00:03:57.780
you could say it's easier, but
a lot of people might say, well

00:03:57.780 --> 00:03:59.200
Sal, that looks like a
little bit of magic.

00:03:59.200 --> 00:04:02.160
You just took that 5, put a 1
on it, and then you borrowed

00:04:02.160 --> 00:04:04.600
a 1 from this 100 here.

00:04:04.600 --> 00:04:07.080
But really, what I
did is right here.

00:04:07.080 --> 00:04:12.600
I took 1,000 from this 1
and I redistributed that

00:04:12.600 --> 00:04:17.050
1,000 amongst the 100's,
10's, and 1's place.

00:04:17.050 --> 00:04:18.010
Let me do another example.

00:04:18.010 --> 00:04:19.810
I think it might make it a
little bit more clearer

00:04:19.810 --> 00:04:22.050
of why borrowing works.

00:04:25.130 --> 00:04:26.820
Let me do a simpler problem.

00:04:26.820 --> 00:04:29.210
I actually started off with a
problem that tends to confuse

00:04:29.210 --> 00:04:30.670
the most number of people.

00:04:30.670 --> 00:04:45.290
Let's say I had 732 minus-- Let
me do a fairly simple one.

00:04:45.290 --> 00:04:46.573
Minus 23.

00:04:49.150 --> 00:04:50.820
Sometimes those 3's
just come out weird.

00:04:50.820 --> 00:04:55.120
Well, we just learned that's
the same thing as 700 plus

00:04:55.120 --> 00:05:03.240
30 plus 2 minus 20 minus 3.

00:05:03.240 --> 00:05:07.150
Well, we see this 2, 2 is less
than 3, so we can't subtract.

00:05:07.150 --> 00:05:09.050
Wouldn't it be great if we
could get a 10 from someplace?

00:05:09.050 --> 00:05:10.870
We could get a 10 from here.

00:05:10.870 --> 00:05:16.960
We make this into 20 and add
the 10 to the 2 and we get 12.

00:05:16.960 --> 00:05:21.940
And notice, 700 plus 20
plus 12 is still 732.

00:05:21.940 --> 00:05:24.420
So we really didn't change
the number up top at all.

00:05:24.420 --> 00:05:28.520
We just redistributed its
quantity amongst the

00:05:28.520 --> 00:05:29.200
different places.

00:05:29.200 --> 00:05:30.260
And now we're ready
to subtract.

00:05:30.260 --> 00:05:32.340
12 minus 3 is 9.

00:05:32.340 --> 00:05:36.610
20 minus 20 is 0 and then you
just bring down the 700.

00:05:36.610 --> 00:05:42.190
You get 700 plus 0 plus 9,
which is the same thing as 709.

00:05:42.190 --> 00:05:45.260
And that's the reason why
this borrowing will work.

00:05:45.260 --> 00:05:47.380
Well, we say, oh, let's
borrow 1 from the 3.

00:05:47.380 --> 00:05:48.290
Makes it a 2.

00:05:48.290 --> 00:05:49.750
This becomes a 12.

00:05:49.750 --> 00:05:52.200
And then we subtract.

00:05:52.200 --> 00:05:54.520
9 0 7.

00:05:54.520 --> 00:05:57.370
Let's do another
problem, one last one.

00:05:57.370 --> 00:05:59.160
And once again, you don't
have to do it this way.

00:05:59.160 --> 00:06:00.850
You don't have to every
time you do a subtraction

00:06:00.850 --> 00:06:01.480
problem do it this way.

00:06:01.480 --> 00:06:03.520
Although if you ever get
confused, you can do it this

00:06:03.520 --> 00:06:05.700
way and you won't make a
mistake, and you'll actually

00:06:05.700 --> 00:06:06.990
understand what you're doing.

00:06:06.990 --> 00:06:08.930
But if you're on a test and you
have to do things really fast

00:06:08.930 --> 00:06:10.860
you should do it the
conventional way.

00:06:10.860 --> 00:06:13.570
But it takes a lot of practice
to make sure you never are

00:06:13.570 --> 00:06:15.710
doing something improper.

00:06:15.710 --> 00:06:16.390
And that's the problem.

00:06:16.390 --> 00:06:18.280
People learn just the rules,
and then they forget the

00:06:18.280 --> 00:06:19.980
rules, and then they
forgot how to do it.

00:06:19.980 --> 00:06:22.620
If you learn what you're doing,
you'll never really forget it

00:06:22.620 --> 00:06:26.230
because it should make
some sense to you.

00:06:26.230 --> 00:06:28.000
Let's do another problem.

00:06:28.000 --> 00:06:35.970
If I had 512 minus 38.

00:06:35.970 --> 00:06:38.350
Well, let's keep doing it
that way I just showed you.

00:06:38.350 --> 00:06:45.020
That's the same thing
as 500 plus 10 plus

00:06:45.020 --> 00:06:51.080
2 minus 30 minus 8.

00:06:51.080 --> 00:06:52.130
Well, 2 is less than 8.

00:06:52.130 --> 00:06:53.290
I need a 10 from someplace.

00:06:53.290 --> 00:06:55.290
Well, one option we can
do is we can just get

00:06:55.290 --> 00:06:56.600
the 10 from here.

00:06:56.600 --> 00:06:58.720
So then that becomes 0.

00:06:58.720 --> 00:07:00.220
And then this will become a 12.

00:07:00.220 --> 00:07:05.370
Notice that 500 plus 0 plus
12, same thing as 512 still.

00:07:05.370 --> 00:07:06.270
So we could subtract.

00:07:06.270 --> 00:07:09.770
12 minus 8 is 4.

00:07:09.770 --> 00:07:14.860
But here we see this 0 is less
than 30, so we can't subtract.

00:07:14.860 --> 00:07:17.290
But we can borrow from the 500.

00:07:17.290 --> 00:07:22.930
Well, all we need is 100, so if
we turn this into 100 then we

00:07:22.930 --> 00:07:25.260
took the 100 from the 500.

00:07:25.260 --> 00:07:28.070
This becomes 400.

00:07:28.070 --> 00:07:31.430
I just rewrote 500
as 400 plus 100.

00:07:31.430 --> 00:07:32.340
Now I can subtract.

00:07:32.340 --> 00:07:35.510
100 minus 30 is 70.

00:07:35.510 --> 00:07:38.790
Bring down the 400.

00:07:38.790 --> 00:07:42.560
And this is the
same thing as 474.

00:07:42.560 --> 00:07:44.490
And the way you learn how to do
it in school is you say, oh,

00:07:44.490 --> 00:07:47.815
well, 2 is less than 8,
so let me borrow the 1.

00:07:47.815 --> 00:07:48.930
It becomes 12.

00:07:48.930 --> 00:07:50.890
This becomes a 0.

00:07:50.890 --> 00:07:56.120
0 is less than 3, so let
me borrow 1 from this 5.

00:07:56.120 --> 00:07:57.140
Make this 4.

00:07:57.140 --> 00:07:58.710
This becomes 10.

00:07:58.710 --> 00:08:01.270
So then you say
12 minus 8 is 4.

00:08:01.270 --> 00:08:05.780
10 minus 3 is 7 and
you bring down the 4.

00:08:05.780 --> 00:08:09.400
Hopefully what I've done here
will give you an intuition

00:08:09.400 --> 00:08:10.510
of why borrowing works.

00:08:10.510 --> 00:08:12.760
And this is something that
actually I didn't quite

00:08:12.760 --> 00:08:16.690
understand until a while after
I learned how to borrow.

00:08:16.690 --> 00:08:19.650
And if you learned this, you'll
realize that what you're doing

00:08:19.650 --> 00:08:21.260
here isn't really magic.

00:08:21.260 --> 00:08:23.780
And hopefully, you'll never
forget what you're actually

00:08:23.780 --> 00:08:25.400
doing and you can always
kind of think about what's

00:08:25.400 --> 00:08:28.900
fundamentally happening to
the numbers when you borrow.

00:08:28.900 --> 00:08:31.580
I hope you found that useful.

00:08:31.580 --> 00:08:32.400
Talk to later.

00:08:32.400 --> 00:08:33.700
Bye.


Advanced ratio problems

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:04.020
Welcome to the presentation on
more advanced ratio problems.

00:00:04.020 --> 00:00:06.240
Let's get started
with some problems.

00:00:06.240 --> 00:00:11.870
So let's say that I have a
class, and then-- oh the pen is

00:00:11.870 --> 00:00:17.180
messed up --OK, so in the class
the total number of

00:00:17.180 --> 00:00:21.840
students is 57.

00:00:21.840 --> 00:00:26.850
And I would also tell you
that the ratio of girls

00:00:26.850 --> 00:00:34.450
to boys is equal to 4:15.

00:00:34.450 --> 00:00:36.425
So now this the interesting
part, so far it

00:00:36.425 --> 00:00:37.530
doesn't look to tough.

00:00:37.530 --> 00:00:42.450
My question is how many boys
need to leave the room, so how

00:00:42.450 --> 00:01:02.760
many boys need to leave for the
ratio of girls to

00:01:02.760 --> 00:01:10.960
boys to be 4:11.

00:01:10.960 --> 00:01:12.590
This is fascinating.

00:01:12.590 --> 00:01:15.100
So, a good place to start is
just to figure out how many

00:01:15.100 --> 00:01:17.460
girls and how many boys there
are in this classroom.

00:01:17.460 --> 00:01:19.650
And we already learned how to
do that in the introduction

00:01:19.650 --> 00:01:21.130
to ratio problems.

00:01:21.130 --> 00:01:27.140
We know that the girls plus the
boys is equal to 57, right,

00:01:27.140 --> 00:01:29.550
because there are 57
kids in the room.

00:01:29.550 --> 00:01:31.810
And we also know, just
multiplying-- taking this

00:01:31.810 --> 00:01:36.400
equation --and multiplying both
sides by b, we also know that

00:01:36.400 --> 00:01:44.030
the girls are equal to 4/15
times the boys, right?

00:01:44.030 --> 00:01:47.040
And then we can just substitute
that back into this equation,

00:01:47.040 --> 00:01:58.180
and then we get 4/15b plus b is
equal to 57, is the same thing

00:01:58.180 --> 00:02:04.110
as 19/15b is equal to 57.

00:02:06.910 --> 00:02:10.210
Let me clean this
up a little bit.

00:02:10.210 --> 00:02:14.630
That's separate, and
then let me go here.

00:02:14.630 --> 00:02:23.820
And we say b is equal to 57--
oh, woops --it's actually 57

00:02:23.820 --> 00:02:27.930
times 15 all of that
over 19, right?

00:02:27.930 --> 00:02:31.430
I just multiply both
sides by 15/19.

00:02:31.430 --> 00:02:36.260
So 57 divided by 19 is 3.

00:02:36.260 --> 00:02:40.520
So b is equal to 45.

00:02:40.520 --> 00:02:44.240
And we know there're a total of
57 kids in the class-- g plus

00:02:44.240 --> 00:02:48.220
b is 57 --so we know that
there are 12 girls, right?

00:02:48.220 --> 00:02:50.370
57 minus 45.

00:02:50.370 --> 00:02:51.110
Good.

00:02:51.110 --> 00:02:54.240
So now we know that the
current boys and girls

00:02:54.240 --> 00:02:58.850
are 45 boys and 12 girls.

00:02:58.850 --> 00:02:59.870
So let's write that down.

00:02:59.870 --> 00:03:07.520
So there's 12 girls
and 45 boys.

00:03:07.520 --> 00:03:11.010
Now, the question says, how
many boys need to leave for

00:03:11.010 --> 00:03:13.550
the ratio of girls to
boys equals 4 4:11?

00:03:13.550 --> 00:03:15.690
So this is the number of
girls right now, 12,

00:03:15.690 --> 00:03:17.170
this is the number boys.

00:03:17.170 --> 00:03:21.770
Let's say x is a number of boys
that need to leave the room.

00:03:21.770 --> 00:03:26.950
So if x boys leave the room the
new ratio will be 12 girls to

00:03:26.950 --> 00:03:31.720
the 45 boys minus the x
boys that leave, right?

00:03:31.720 --> 00:03:34.490
If that confuses you, sit and
look at that for a second.

00:03:34.490 --> 00:03:38.010
We start off with 12 girls
and 45 boys in the room.

00:03:38.010 --> 00:03:40.260
And we're saying x boys are
going to leave, so the new

00:03:40.260 --> 00:03:43.820
ratio is going to
be 12:45 minus x.

00:03:43.820 --> 00:03:47.090
And we know from this part of
the problem that that new

00:03:47.090 --> 00:03:51.420
ratio is going to equal 4:11.

00:03:51.420 --> 00:03:54.120
There, we just set up a
equation with one unknown

00:03:54.120 --> 00:03:56.050
and we can solve for x.

00:03:56.050 --> 00:03:57.130
I hope that doesn't
confuse you much.

00:03:57.130 --> 00:03:59.090
All we did is we figured out
how many boys, how many

00:03:59.090 --> 00:04:00.380
girls are in the room now.

00:04:00.380 --> 00:04:02.900
We said x is the number
boys that need to leave.

00:04:02.900 --> 00:04:06.770
And we said the new ratio is
going to be girls to the new

00:04:06.770 --> 00:04:09.280
number of boys, which is 45
minus x, and that's going to

00:04:09.280 --> 00:04:11.400
be equal to the new ratio.

00:04:11.400 --> 00:04:13.070
So let's solve for x.

00:04:13.070 --> 00:04:19.230
Well, 12 times 11 is
what that's 132.

00:04:19.230 --> 00:04:30.500
132 is equal to 4 times
45, 160, 180, minus 4x.

00:04:30.500 --> 00:04:33.320
And then if you solve for x, I
think you know how to do this

00:04:33.320 --> 00:04:42.270
right now, and we can say minus
4x is equal to minus 48.

00:04:42.270 --> 00:04:44.890
x is equal to 12.

00:04:44.890 --> 00:04:45.810
There we solved it.

00:04:45.810 --> 00:04:50.960
So we say that if 12 boys left
the room, the new ratio of

00:04:50.960 --> 00:04:53.430
girls to boys would be 4:11.

00:04:53.430 --> 00:04:54.620
And does that make sense?

00:04:54.620 --> 00:04:57.810
Well if 12 boys left the room,
then the new ratio of girls to

00:04:57.810 --> 00:05:03.440
boys would be 12:33, right?

00:05:03.440 --> 00:05:06.630
Because 45 minus 12 is 33.

00:05:06.630 --> 00:05:08.870
And that's the same thing
as if you divide the

00:05:08.870 --> 00:05:10.020
top and bottom by 3.

00:05:10.020 --> 00:05:11.930
That's 4:11.

00:05:11.930 --> 00:05:13.150
So there, we got it right.

00:05:13.150 --> 00:05:16.200
So what looked like a very hard
problem actually wasn't so bad

00:05:16.200 --> 00:05:21.940
when you just sit down and
work through the algebra.

00:05:21.940 --> 00:05:22.930
Let's do another problem.

00:05:27.930 --> 00:05:35.470
Let's say --this thing
sometimes malfunctions --OK,

00:05:35.470 --> 00:05:41.630
let's say that the ratio of
apples to bananas in a

00:05:41.630 --> 00:05:46.550
basket is equal to 5:19.

00:05:46.550 --> 00:06:03.160
And when we add 23 bananas the
ratio of apples to bananas--

00:06:03.160 --> 00:06:06.330
and actually let's write it
right now, we now have 23

00:06:06.330 --> 00:06:17.370
bananas more --is
equal to 10:61.

00:06:17.370 --> 00:06:20.820
So the question is, what
is the total amount of

00:06:20.820 --> 00:06:21.730
fruit in the basket?

00:06:26.760 --> 00:06:33.020
Amount of fruit-- --ah,
that's so messy --after

00:06:33.020 --> 00:06:33.815
adding the bananas.

00:06:40.790 --> 00:06:42.780
So I actually gave you a
hint just when I wrote

00:06:42.780 --> 00:06:43.700
down initial problem.

00:06:43.700 --> 00:06:48.580
We're saying the ratio of a to
b-- so let a equal the number

00:06:48.580 --> 00:06:52.710
of apples, and b equal the
number bananas --so the ratio

00:06:52.710 --> 00:06:55.450
of apples to bananas
equals 5:19.

00:06:55.450 --> 00:07:01.080
When I add 23 bananas, now the
new ratio's going to be the

00:07:01.080 --> 00:07:04.460
number of apples to b plus 23.

00:07:04.460 --> 00:07:06.600
The new ratio is 10:61.

00:07:06.600 --> 00:07:07.540
So how do we solve this?

00:07:07.540 --> 00:07:10.080
Well, once again we have two
equations and two unknowns.

00:07:10.080 --> 00:07:12.630
We know that-- I guess let's
take this equation first,

00:07:12.630 --> 00:07:18.200
because it's a little more
complicated --we know if we

00:07:18.200 --> 00:07:32.890
cross multiply that 61a is
equal to 10b plus 230 and if we

00:07:32.890 --> 00:07:36.980
divide both sides by 61, we
know that a is equal to

00:07:36.980 --> 00:07:45.830
10/61b plus 230/61.

00:07:45.830 --> 00:07:46.670
Right?

00:07:46.670 --> 00:07:50.070
And we could take this equation
and multiply both sides by b

00:07:50.070 --> 00:07:54.850
and we could say that
a is equal to 5/19b.

00:07:54.850 --> 00:07:55.810
Right?

00:07:55.810 --> 00:07:58.440
Well both of these are equal
to a so we could set them

00:07:58.440 --> 00:07:59.800
equal to each other.

00:07:59.800 --> 00:08:24.550
And you get 5/19b is equal to
10/61b plus 230 times 61.

00:08:24.550 --> 00:08:26.070
And we solve for b.

00:08:26.070 --> 00:08:28.640
While this might seem
complicated to you at first,

00:08:28.640 --> 00:08:31.110
but it's just a basic
linear equation.

00:08:31.110 --> 00:08:33.100
And for the sake of time,
because I only have 2 minutes

00:08:33.100 --> 00:08:33.960
left in this youtube.

00:08:33.960 --> 00:08:39.500
I'm just going to solve for b,
and you get b is equal to 38.

00:08:39.500 --> 00:08:44.700
if b is equal to 38, we know
that the initial ratio is 5:19.

00:08:44.700 --> 00:08:45.620
So that's pretty easy.

00:08:45.620 --> 00:08:53.250
We just say a is equal to 5/19
times 38 is equal to 10.

00:08:53.250 --> 00:08:56.540
So the initial number of apples
was 10, the initial number

00:08:56.540 --> 00:08:58.840
bananas is 38, right?

00:08:58.840 --> 00:09:03.420
So initially we started off
with 48 pieces of fruit, and

00:09:03.420 --> 00:09:09.180
then we're going to add 23
more pieces of fruit, right?

00:09:09.180 --> 00:09:14.660
And 43 plus 28, that's
what, 71 pieces of fruit.

00:09:14.660 --> 00:09:16.250
So, let me review real
quick what we said.

00:09:16.250 --> 00:09:20.430
We said the ratio of apples
to bananas is 5:19.

00:09:20.430 --> 00:09:21.990
That's a is the number
of apples, b is the

00:09:21.990 --> 00:09:23.380
number of bananas.

00:09:23.380 --> 00:09:27.470
When I add 23 bananas, I now
have b plus 23 bananas, the

00:09:27.470 --> 00:09:31.520
new ratio of apples to total
number bananas is 10:61.

00:09:31.520 --> 00:09:33.700
And I just used both
of these equations.

00:09:33.700 --> 00:09:35.610
Two equations and two unknowns.

00:09:35.610 --> 00:09:38.830
Solved for a, and then
substituted, and

00:09:38.830 --> 00:09:39.920
I solved for b.

00:09:39.920 --> 00:09:40.740
Nothing fancy here.

00:09:40.740 --> 00:09:42.120
I know there's a lot of
fractions here, but if you

00:09:42.120 --> 00:09:44.770
just work through this, the
fractions actually work out.

00:09:44.770 --> 00:09:46.750
And I was able to
solve for a and b.

00:09:46.750 --> 00:09:51.120
Add the 23 pieces and I got
71 total pieces of fruit.

00:09:51.120 --> 00:09:53.190
I think you're now ready
to try some of the more

00:09:53.190 --> 00:09:54.980
difficult ratio problems.

00:09:54.980 --> 00:09:56.490
Have fun!


Introduction to Ratios

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:03.022
Welcome to the
presentation on ratios.

00:00:03.022 --> 00:00:05.580
Now, I'm going to start just
giving you a definition

00:00:05.580 --> 00:00:07.770
of ratios, and this I
got from Wikipedia.

00:00:07.770 --> 00:00:11.420
"A ratio is a quantity that
denotes the proportional amount

00:00:11.420 --> 00:00:14.980
of magnitude of one quantity
relative to another." So I'm

00:00:14.980 --> 00:00:16.600
going to tell you from the
beginning that I think a ratio

00:00:16.600 --> 00:00:18.870
is something that's easier to
understand than to give a

00:00:18.870 --> 00:00:20.580
definition for because I
don't think that Wikipedia

00:00:20.580 --> 00:00:22.090
definition is that useful.

00:00:22.090 --> 00:00:24.490
Let me give you some examples.

00:00:24.490 --> 00:00:29.210
If there are-- let's say
there are-- let me make

00:00:29.210 --> 00:00:30.640
this pen size is right.

00:00:30.640 --> 00:00:39.480
Let's say there 10 boys
and 2 girls in a class.

00:00:39.480 --> 00:00:47.270
So the ratio of boys to girls
would be 10:2 or 10/2.

00:00:47.270 --> 00:00:49.070
Those are two different
ways of writing it.

00:00:49.070 --> 00:00:51.030
And we know from fractions
that that's also the same

00:00:51.030 --> 00:00:57.720
thing as 5:1 or 5/1.

00:00:57.720 --> 00:00:59.320
We want to keep the 1 there
because we know that it's a

00:00:59.320 --> 00:01:01.520
ratio of one thing
to another thing.

00:01:01.520 --> 00:01:02.560
So what does that mean?

00:01:02.560 --> 00:01:06.240
All that means is that for
every 5 boys there's 1 girl.

00:01:06.240 --> 00:01:10.240
And so if we told you that the
ratio of boys to girls in a

00:01:10.240 --> 00:01:16.290
room is 5:1, and we told you
that there are-- let's say that

00:01:16.290 --> 00:01:23.300
we told you that there are 100
girls, then we'd know that,

00:01:23.300 --> 00:01:26.290
well, for every 1 of those
girls, there's 5 boys, so

00:01:26.290 --> 00:01:28.860
that means that there'd
be 500 boys, right?

00:01:31.440 --> 00:01:34.795
Or you could also look at that
as the ratio of boys to girls

00:01:34.795 --> 00:01:39.320
is 500/100, which equals 5/1.

00:01:41.820 --> 00:01:43.730
And this is the typical way
that a ratio is written:

00:01:43.730 --> 00:01:47.690
500:100 of boys girls.

00:01:47.690 --> 00:01:50.620
Now, let me ask you a couple
of questions based on that.

00:01:50.620 --> 00:01:53.340
I think you get
the general idea.

00:01:53.340 --> 00:02:04.380
If I told you that the ratio
of, let's say, red balls to

00:02:04.380 --> 00:02:15.190
green balls in a bag is 2:3.

00:02:15.190 --> 00:02:33.330
And then if I also told you
that there are 40 red balls,

00:02:33.330 --> 00:02:36.750
how many green balls are there?

00:02:36.750 --> 00:02:40.730
Well, what we say is the ratio
of red balls to green balls--

00:02:40.730 --> 00:02:45.240
so we know that there are 40
red balls, and then we want to

00:02:45.240 --> 00:02:47.970
solve for the number of green
balls, that that

00:02:47.970 --> 00:02:49.770
is equal to 2/3.

00:02:52.360 --> 00:02:53.970
And then we could
just solve this.

00:02:53.970 --> 00:02:55.440
We just cross multiply.

00:02:55.440 --> 00:03:04.325
40 times 3 is 120 is equal to
2g, and then we just solve.

00:03:04.325 --> 00:03:08.190
We just say g equals 60.

00:03:08.190 --> 00:03:10.560
And there's an easier way of
doing this kind of in your

00:03:10.560 --> 00:03:13.270
head, and this is the algebraic
way that'll always work.

00:03:13.270 --> 00:03:15.440
But you could also just say--
let me write this a little bit.

00:03:15.440 --> 00:03:17.690
This is a 3 down here.

00:03:17.690 --> 00:03:20.570
You can also say, well, to get
from 2 to 40, you have to

00:03:20.570 --> 00:03:26.005
multiply by 20, so to get from
3 to g, I'm also going

00:03:26.005 --> 00:03:28.170
to multiply by 20.

00:03:28.170 --> 00:03:30.090
And so 3 times 20 is 60.

00:03:30.090 --> 00:03:31.270
That's another way to do it.

00:03:31.270 --> 00:03:33.110
A lot of you might actually
find it more intuitive

00:03:33.110 --> 00:03:33.900
just to think about it.

00:03:33.900 --> 00:03:38.510
Well, if for every 2 red balls,
there are 3 green balls.

00:03:38.510 --> 00:03:43.310
Then if there are 40 red balls,
then it makes sense that there

00:03:43.310 --> 00:03:47.640
would be 60 green balls because
for every 20, there'd be 30,

00:03:47.640 --> 00:03:49.020
for every 40, there'd be 60.

00:03:49.020 --> 00:03:52.940
I hope I'm not completely
confusing you.

00:03:52.940 --> 00:03:54.070
Let me give you
another example.

00:03:54.070 --> 00:04:11.790
Let's say the ratio of boys
to girls is equal to 2 to 7.

00:04:11.790 --> 00:04:20.950
And if I were tell you that the
total class has 180 kids in it,

00:04:20.950 --> 00:04:23.050
can we figure out how many
boys and girls there

00:04:23.050 --> 00:04:24.620
are in the class?

00:04:24.620 --> 00:04:26.110
Well, let's think about it.

00:04:26.110 --> 00:04:36.000
Well, we know that the boys to
girls is equal to 2/7, and we

00:04:36.000 --> 00:04:42.750
also know that the boys plus
girls is equal to 180.

00:04:42.750 --> 00:04:45.330
So here, we have a system of
two equations and two unknowns.

00:04:45.330 --> 00:04:47.380
And you could actually, if
you really think about it,

00:04:47.380 --> 00:04:49.210
you could actually solve
this without algebra.

00:04:49.210 --> 00:04:51.920
But I'll show you the algebraic
way, because when problems get

00:04:51.920 --> 00:04:54.590
complicated, this'll
always work.

00:04:54.590 --> 00:04:56.730
So what we can do is we
can do substitution.

00:04:56.730 --> 00:05:03.220
We know that b is
equal to 2/7g, right?

00:05:03.220 --> 00:05:07.650
I just multiplied both sides
of this equation by g.

00:05:07.650 --> 00:05:11.570
It cancels out there, and then
times g, and you get this.

00:05:11.570 --> 00:05:14.940
And then we can just substitute
that back in for b.

00:05:14.940 --> 00:05:24.620
So then we have 2/7g
plus g is equal to 180.

00:05:24.620 --> 00:05:31.290
And what's 2/7g plus--
we could 1g or 7/7g?

00:05:31.290 --> 00:05:33.900
Well, you could do the
fraction, but it's 2/7 plus 1

00:05:33.900 --> 00:05:41.420
is the same thing as-- that's
equal to 2/7 plus 7/7, right,

00:05:41.420 --> 00:05:45.430
because that's just
1g is equal to 180.

00:05:45.430 --> 00:05:47.670
And I'm jumping around on the
chalkboard on purpose to

00:05:47.670 --> 00:05:51.170
intentionally confuse you.

00:05:51.170 --> 00:05:52.550
OK, this is where I am.

00:05:52.550 --> 00:05:55.400
So 2/7 plus 7/7g equals 180.

00:05:55.400 --> 00:06:00.580
So we have 9/7g
is equal to 180.

00:06:00.580 --> 00:06:02.640
And then we just multiply
both sides times the

00:06:02.640 --> 00:06:07.560
reciprocal of 7/9.

00:06:07.560 --> 00:06:07.930
Oops!

00:06:07.930 --> 00:06:08.490
That's not a g.

00:06:08.490 --> 00:06:09.570
That's a 9.

00:06:09.570 --> 00:06:12.965
Once, again, an intentional
device to confuse you.

00:06:12.965 --> 00:06:23.450
These cancel out, and you
get g equals 180 times 7/9.

00:06:23.450 --> 00:06:27.420
Well, 180 divided by 9,
this is just 20, right?

00:06:27.420 --> 00:06:30.650
So g is equal to 140.

00:06:30.650 --> 00:06:34.250
So if there's 140 girls in
the room, how many boys

00:06:34.250 --> 00:06:34.850
are there going to be?

00:06:34.850 --> 00:06:39.060
Well, we know that the whole
class has 180 people, and we

00:06:39.060 --> 00:06:41.960
know b plus g is 180, so
there's going to be-- the

00:06:41.960 --> 00:06:45.610
boys are equal to 40.

00:06:45.610 --> 00:06:50.420
And this is really about as
difficult as I guess we could

00:06:50.420 --> 00:06:52.650
say basic ratio problems get.

00:06:52.650 --> 00:06:54.280
There's nothing really
difficult about ratios.

00:06:54.280 --> 00:06:57.710
They're just representing, for
every amount of one thing, how

00:06:57.710 --> 00:06:59.350
much do you have of
the other thing?

00:06:59.350 --> 00:07:04.170
And then you can use that
ratio if you have some other

00:07:04.170 --> 00:07:06.840
information in terms of how
many total people there are, or

00:07:06.840 --> 00:07:09.000
how many total objects there
are, or how much of one object

00:07:09.000 --> 00:07:11.810
there is, you can use that to
figure out how much of the

00:07:11.810 --> 00:07:16.110
other object there is or how
many total objects there are.

00:07:16.110 --> 00:07:19.840
I think you're now ready to try
some of the ratio problems, and

00:07:19.840 --> 00:07:22.230
I'm going to do another
presentation on what I would

00:07:22.230 --> 00:07:25.260
consider slightly more
advanced ratio problems.

00:07:25.260 --> 00:07:27.330
So have fun.


Averages

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:03.510
Welcome to the
presentation on averages.

00:00:03.510 --> 00:00:05.980
Averages is probably a concept
that you've already used

00:00:05.980 --> 00:00:08.020
before, maybe not in
a mathematical way.

00:00:08.020 --> 00:00:11.790
But people will talk in terms
of, the average voter wants a

00:00:11.790 --> 00:00:14.750
politician to do this, or the
average student in a class

00:00:14.750 --> 00:00:15.760
wants to get out early.

00:00:15.760 --> 00:00:17.760
So you're probably already
familiar with the

00:00:17.760 --> 00:00:18.760
concept of an average.

00:00:18.760 --> 00:00:21.230
And you probably already
intuitively knew that an

00:00:21.230 --> 00:00:27.710
average is just a number that
represents the different values

00:00:27.710 --> 00:00:29.290
that a group could have.

00:00:29.290 --> 00:00:32.970
But it can represent that as
one number as opposed to giving

00:00:32.970 --> 00:00:34.350
all the different values.

00:00:34.350 --> 00:00:36.010
And let's give a couple of
examples of how to compute

00:00:36.010 --> 00:00:38.070
an average, and you might
already know how to do this.

00:00:38.070 --> 00:00:47.150
So let's say I had the
numbers 1, 3, 5, and 20.

00:00:47.150 --> 00:00:51.980
And I asked you, what is the
average of these four numbers?

00:00:51.980 --> 00:00:54.380
Well, what we do is,
we literally just

00:00:54.380 --> 00:00:55.280
add up the numbers.

00:00:55.280 --> 00:00:57.410
And then divide by the
number of numbers we have.

00:00:57.410 --> 00:01:01.220
So we say 1 plus 3 is 4.

00:01:01.220 --> 00:01:01.960
So let me write that.

00:01:01.960 --> 00:01:10.630
1 plus 3 plus 5 plus 20 equals,
let's see, 1 plus 3 is 4.

00:01:10.630 --> 00:01:12.190
4 plus 5 is 9.

00:01:12.190 --> 00:01:15.310
9 plus 20 is 29.

00:01:15.310 --> 00:01:18.430
And we had 4 numbers;
one, two, three, four.

00:01:18.430 --> 00:01:22.210
So 4 goes into 29.

00:01:22.210 --> 00:01:26.300
And it goes, 7, 7, 28.

00:01:26.300 --> 00:01:29.900
And then we have 10, I didn't
have to do that decimal

00:01:29.900 --> 00:01:31.260
there, oh well.

00:01:31.260 --> 00:01:33.845
2, 8, 25.

00:01:36.910 --> 00:01:40.880
So 4 goes into 29 7.25 times.

00:01:40.880 --> 00:01:49.590
So the average of these four
numbers is equal to 7.25.

00:01:49.590 --> 00:01:54.000
And that might make sense to
you because 7.25 is someplace

00:01:54.000 --> 00:01:55.700
in between these numbers.

00:01:55.700 --> 00:02:00.450
And we can kind of view this,
7.25, as one way to represent

00:02:00.450 --> 00:02:03.090
these four numbers without
having to list these

00:02:03.090 --> 00:02:03.990
four numbers.

00:02:03.990 --> 00:02:06.480
There are other representations
you'll learn later on.

00:02:06.480 --> 00:02:07.960
Like the mode.

00:02:07.960 --> 00:02:12.700
You'll also the mean, which
we'll talk about later,

00:02:12.700 --> 00:02:14.610
is actually the same
thing as the average.

00:02:14.610 --> 00:02:16.680
But the average is just one
number that you can use to

00:02:16.680 --> 00:02:19.400
represent a set of numbers.

00:02:19.400 --> 00:02:22.730
So let's do some problems
which I think are going to

00:02:22.730 --> 00:02:26.180
be close to your heart.

00:02:26.180 --> 00:02:31.830
Let's say on the first four
tests of an exam, I got a--

00:02:31.830 --> 00:02:39.690
let's see, I got an 80, an 81.

00:02:39.690 --> 00:02:45.250
An 87, and an 88.

00:02:45.250 --> 00:02:47.580
What's my average in
the class so far?

00:02:47.580 --> 00:02:50.890
Well, all I have to do is
add up these four numbers.

00:02:50.890 --> 00:03:00.790
So I say, 80 plus 81
plus 87 plus 88.

00:03:00.790 --> 00:03:02.370
Well, zero plus 1 is 1.

00:03:02.370 --> 00:03:06.190
1 plus 7 is 8.

00:03:06.190 --> 00:03:07.490
8 plus 8 is 16.

00:03:07.490 --> 00:03:10.770
I just ran eight miles,
so I'm a bit tired.

00:03:10.770 --> 00:03:13.310
And, 4/8, so that's 32.

00:03:13.310 --> 00:03:14.210
Plus 1 is 33.

00:03:16.950 --> 00:03:20.750
And now we divide
this number by 4.

00:03:20.750 --> 00:03:25.300
4 goes into 336.

00:03:25.300 --> 00:03:27.000
Goes into 33, 8 times.

00:03:27.000 --> 00:03:28.900
8 times 4 is 32.

00:03:31.850 --> 00:03:34.000
33 minus 32 is 1, 16.

00:03:34.000 --> 00:03:34.460
4.

00:03:34.460 --> 00:03:39.750
So the average is equal to 84.

00:03:39.750 --> 00:03:43.220
So depending on what school you
go to that's either a B or a C.

00:03:43.220 --> 00:03:49.250
So, so far my average after the
first four exams is an 84.

00:03:49.250 --> 00:03:51.740
Now let's make this a
little bit more difficult.

00:03:51.740 --> 00:04:00.150
We know that the average
after four exams, at four

00:04:00.150 --> 00:04:04.980
exams, is equal to 84.

00:04:04.980 --> 00:04:09.200
If I were to ask you what do I
have to get on the next test to

00:04:09.200 --> 00:04:16.010
average an 88, to average
an 88 in the class.

00:04:20.310 --> 00:04:23.490
So let's say that x is what
I get on the next test.

00:04:28.180 --> 00:04:31.990
So now what we can say is, is
that the first four exams, I

00:04:31.990 --> 00:04:36.910
could either list out the
first four exams that I took.

00:04:36.910 --> 00:04:38.950
Or I already know
what the average is.

00:04:38.950 --> 00:04:42.210
So I know the sum of the
first four exams is

00:04:42.210 --> 00:04:45.470
going to 4 times 84.

00:04:45.470 --> 00:04:51.420
And now I want to add the, what
I get on the 5th exam, x.

00:04:51.420 --> 00:04:55.640
And I'm going to divide
that by all five exams.

00:04:55.640 --> 00:04:58.770
So in other words, this
number is the average

00:04:58.770 --> 00:05:00.220
of my first five exams.

00:05:00.220 --> 00:05:02.510
We just figured out the average
of the first four exams.

00:05:02.510 --> 00:05:06.620
But now, we sum up the
first four exams here.

00:05:06.620 --> 00:05:09.220
We add what I got on the fifth
exam, and then we divide it

00:05:09.220 --> 00:05:11.940
by 5, because now we're
averaging five exams.

00:05:11.940 --> 00:05:16.030
And I said that I need to
get in an 88 in the class.

00:05:16.030 --> 00:05:18.230
And now we solve for x.

00:05:18.230 --> 00:05:20.080
Let me make some space here.

00:05:22.710 --> 00:05:25.080
So, 5 times 88 is, let's see.

00:05:25.080 --> 00:05:31.110
5 times 80 is 400, so it's 440.

00:05:31.110 --> 00:05:36.030
440 equals 4 times 84,
we just saw that, is

00:05:36.030 --> 00:05:39.910
320 plus 16 is 336.

00:05:39.910 --> 00:05:44.210
336 plus x is equal to 440.

00:05:44.210 --> 00:05:47.200
Well, it turns out if you
subtract 336 from both sides,

00:05:47.200 --> 00:05:51.550
you get x is equal to 104.

00:05:51.550 --> 00:05:55.560
So unless you have a exam that
has some bonus problems on it,

00:05:55.560 --> 00:06:00.760
it's probably impossible for
you to get ah an 88 average in

00:06:00.760 --> 00:06:03.180
the class after just
the next exam.

00:06:03.180 --> 00:06:05.380
You'd have to get 104
on that next exam.

00:06:05.380 --> 00:06:06.810
And let's just look
at what we just did.

00:06:06.810 --> 00:06:10.380
We said, after 4
exams we had an 84.

00:06:10.380 --> 00:06:14.530
What do I have to get on that
next exam to average an 88

00:06:14.530 --> 00:06:17.090
in the class after 5 exams?

00:06:17.090 --> 00:06:19.920
And that's what we solved
for when we got x.

00:06:19.920 --> 00:06:23.670
Now, let's ask
another question.

00:06:23.670 --> 00:06:29.830
I said after four exams,
after four exams, I

00:06:29.830 --> 00:06:32.080
had an 84 average.

00:06:34.670 --> 00:06:39.050
If I said that there are 6
exams in the class, and the

00:06:39.050 --> 00:06:42.170
highest score I could get on an
exam is 100, what is the

00:06:42.170 --> 00:06:44.730
highest average I can finish in
the class if I were to really

00:06:44.730 --> 00:06:48.030
study hard and get 100
on the next 2 exams?

00:06:48.030 --> 00:06:51.250
Well, once again, what we'll
want to do is assume we get

00:06:51.250 --> 00:06:54.520
100 on the next 2 exams
and then take the average.

00:06:54.520 --> 00:06:56.890
So we'll have to
solve all 6 exams.

00:06:56.890 --> 00:06:58.820
So we're going to have the
average of 6, so in the

00:06:58.820 --> 00:07:01.000
denominator we're
going to have 6.

00:07:01.000 --> 00:07:05.370
The first four exams, the sum,
as we already learned, is 4

00:07:05.370 --> 00:07:07.710
exams times the 84 average.

00:07:07.710 --> 00:07:09.590
And this dot is just times.

00:07:09.590 --> 00:07:12.140
Plus, and there's going to
be 2 more exams, right?

00:07:12.140 --> 00:07:13.740
Because there's 6
exams in the class.

00:07:13.740 --> 00:07:15.550
And I'm going to
get 100 in each.

00:07:15.550 --> 00:07:18.260
So that's 200.

00:07:18.260 --> 00:07:19.230
And what's this average?

00:07:19.230 --> 00:07:23.050
Well, 4 times 84, we
already said, is 336.

00:07:23.050 --> 00:07:27.250
Plus 200 over 6.

00:07:27.250 --> 00:07:31.090
So that's 536 over 6.

00:07:31.090 --> 00:07:33.840
6 goes into 5 36.

00:07:33.840 --> 00:07:36.580
I don't know if if I gave
myself enough space.

00:07:36.580 --> 00:07:41.190
But 6 goes into 53, 8 times.

00:07:41.190 --> 00:07:42.990
48.

00:07:42.990 --> 00:07:44.940
56.

00:07:44.940 --> 00:07:46.640
9 times.

00:07:46.640 --> 00:07:49.850
9 times 6 is 54.

00:07:49.850 --> 00:07:54.730
6 minus is 20 6 goes into--
so we'll see it's actually

00:07:54.730 --> 00:07:57.630
89.333333, goes on forever.

00:07:57.630 --> 00:07:59.770
So 89.3 repeating.

00:07:59.770 --> 00:08:03.950
So no matter how hard I try in
this class, the best I can do.

00:08:03.950 --> 00:08:05.970
Because I only have two exams
left, even if I were to get

00:08:05.970 --> 00:08:07.740
100 on the next two exams.

00:08:07.740 --> 00:08:13.090
I can finish the class
with an 89.333 average.

00:08:13.090 --> 00:08:16.180
Hopefully, I think some of
this might have been a little

00:08:16.180 --> 00:08:16.990
bit of a review for you.

00:08:16.990 --> 00:08:19.170
You already had kind of a
sense of what an average is.

00:08:19.170 --> 00:08:22.340
And hopefully these last two
problems not only taught you

00:08:22.340 --> 00:08:25.100
how to do some algebra problems
involving average, but they'll

00:08:25.100 --> 00:08:27.830
also help you figure out how
well you have to do on your

00:08:27.830 --> 00:08:30.350
exams to get an A in
your math class.

00:08:30.350 --> 00:08:33.310
I think you're now ready
for the average module.

00:08:33.310 --> 00:08:34.860
Have fun.


Introduction to the quadratic equation

WEBVTT
Kind: captions
Language: en

00:00:01.010 --> 00:00:04.520
Welcome to the presentation on
using the quadratic equation.

00:00:04.520 --> 00:00:06.730
So the quadratic equation,
it sounds like something

00:00:06.730 --> 00:00:07.810
very complicated.

00:00:07.810 --> 00:00:09.930
And when you actually first see
the quadratic equation, you'll

00:00:09.930 --> 00:00:11.590
say, well, not only does it
sound like something

00:00:11.590 --> 00:00:13.110
complicated, but it is
something complicated.

00:00:13.110 --> 00:00:14.930
But hopefully you'll see,
over the course of this

00:00:14.930 --> 00:00:16.580
presentation, that it's
actually not hard to use.

00:00:16.580 --> 00:00:19.040
And in a future presentation
I'll actually show you

00:00:19.040 --> 00:00:21.300
how it was derived.

00:00:21.300 --> 00:00:24.810
So, in general, you've already
learned how to factor a

00:00:24.810 --> 00:00:25.810
second degree equation.

00:00:25.810 --> 00:00:30.910
You've learned that if I
had, say, x squared minus

00:00:30.910 --> 00:00:40.340
x, minus 6, equals 0.

00:00:40.340 --> 00:00:42.970
If I had this equation. x
squared minus x minus x equals

00:00:42.970 --> 00:00:48.720
zero, that you could factor
that as x minus 3 and

00:00:48.720 --> 00:00:52.210
x plus 2 equals 0.

00:00:52.210 --> 00:00:54.955
Which either means that
x minus 3 equals 0 or

00:00:54.955 --> 00:00:57.073
x plus 2 equals 0.

00:00:57.073 --> 00:01:03.512
So x minus 3 equals 0
or x plus 2 equals 0.

00:01:03.512 --> 00:01:08.500
So, x equals 3 or negative 2.

00:01:08.500 --> 00:01:17.980
And, a graphical representation
of this would be, if I had the

00:01:17.980 --> 00:01:26.150
function f of x is equal to
x squared minus x minus 6.

00:01:26.150 --> 00:01:28.760
So this axis is
the f of x axis.

00:01:28.760 --> 00:01:32.670
You might be more familiar with
the y axis, and for the purpose

00:01:32.670 --> 00:01:34.780
of this type of problem,
it doesn't matter.

00:01:34.780 --> 00:01:36.270
And this is the x axis.

00:01:36.270 --> 00:01:40.430
And if I were to graph this
equation, x squared minus x,

00:01:40.430 --> 00:01:42.380
minus 6, it would look
something like this.

00:01:42.380 --> 00:01:50.130
A bit like -- this is f
of x equals minus 6.

00:01:50.130 --> 00:01:52.900
And the graph will kind of
do something like this.

00:01:57.150 --> 00:02:00.030
Go up, it will keep going
up in that direction.

00:02:00.030 --> 00:02:03.150
And know it goes through minus
6, because when x equals 0,

00:02:03.150 --> 00:02:05.110
f of x is equal to minus 6.

00:02:05.110 --> 00:02:07.800
So I know it goes
through this point.

00:02:07.800 --> 00:02:11.520
And I know that when f of x is
equal to 0, so f of x is equal

00:02:11.520 --> 00:02:14.960
to 0 along the x axis, right?

00:02:14.960 --> 00:02:16.600
Because this is 1.

00:02:16.600 --> 00:02:17.870
This is 0.

00:02:17.870 --> 00:02:19.160
This is negative 1.

00:02:19.160 --> 00:02:21.510
So this is where f of x
is equal to 0, along

00:02:21.510 --> 00:02:23.420
this x axis, right?

00:02:23.420 --> 00:02:29.210
And we know it equals 0 at the
points x is equal to 3 and

00:02:29.210 --> 00:02:32.330
x is equal to minus 2.

00:02:32.330 --> 00:02:34.360
That's actually what
we solved here.

00:02:34.360 --> 00:02:36.440
Maybe when we were doing the
factoring problems we didn't

00:02:36.440 --> 00:02:38.940
realize graphically
what we were doing.

00:02:38.940 --> 00:02:42.070
But if we said that f of x is
equal to this function, we're

00:02:42.070 --> 00:02:43.270
setting that equal to 0.

00:02:43.270 --> 00:02:44.820
So we're saying this
function, when does

00:02:44.820 --> 00:02:48.220
this function equal 0?

00:02:48.220 --> 00:02:49.390
When is it equal to 0?

00:02:49.390 --> 00:02:51.720
Well, it's equal to 0 at
these points, right?

00:02:51.720 --> 00:02:55.360
Because this is where
f of x is equal to 0.

00:02:55.360 --> 00:02:57.490
And then what we were doing
when we solved this by

00:02:57.490 --> 00:03:01.970
factoring is, we figured out,
the x values that made f of x

00:03:01.970 --> 00:03:04.160
equal to 0, which is
these two points.

00:03:04.160 --> 00:03:06.740
And, just a little terminology,
these are also called

00:03:06.740 --> 00:03:09.860
the zeroes, or the
roots, of f of x.

00:03:12.470 --> 00:03:14.810
Let's review that a little bit.

00:03:14.810 --> 00:03:23.700
So, if I had something like f
of x is equal to x squared plus

00:03:23.700 --> 00:03:29.550
4x plus 4, and I asked you,
where are the zeroes, or

00:03:29.550 --> 00:03:31.770
the roots, of f of x.

00:03:31.770 --> 00:03:33.970
That's the same thing as
saying, where does f of x

00:03:33.970 --> 00:03:36.300
interject intersect the x axis?

00:03:36.300 --> 00:03:38.210
And it intersects the
x axis when f of x is

00:03:38.210 --> 00:03:39.440
equal to 0, right?

00:03:39.440 --> 00:03:42.120
If you think about the
graph I had just drawn.

00:03:42.120 --> 00:03:45.720
So, let's say if f of x is
equal to 0, then we could

00:03:45.720 --> 00:03:51.860
just say, 0 is equal to x
squared plus 4x plus 4.

00:03:51.860 --> 00:03:53.940
And we know, we could just
factor that, that's x

00:03:53.940 --> 00:03:57.080
plus 2 times x plus 2.

00:03:57.080 --> 00:04:07.090
And we know that it's equal
to 0 at x equals minus 2.

00:04:10.170 --> 00:04:13.940
x equals minus 2.

00:04:13.940 --> 00:04:18.270
Well, that's a little
-- x equals minus 2.

00:04:18.270 --> 00:04:22.380
So now, we know how to find
the 0's when the the actual

00:04:22.380 --> 00:04:24.560
equation is easy to factor.

00:04:24.560 --> 00:04:27.500
But let's do a situation where
the equation is actually

00:04:27.500 --> 00:04:28.850
not so easy to factor.

00:04:32.120 --> 00:04:39.750
Let's say we had f of x
is equal to minus 10x

00:04:39.750 --> 00:04:45.380
squared minus 9x plus 1.

00:04:45.380 --> 00:04:47.580
Well, when I look at this, even
if I were to divide it by 10 I

00:04:47.580 --> 00:04:48.650
would get some fractions here.

00:04:48.650 --> 00:04:53.130
And it's very hard to imagine
factoring this quadratic.

00:04:53.130 --> 00:04:54.860
And that's what's actually
called a quadratic equation, or

00:04:54.860 --> 00:04:57.580
this second degree polynomial.

00:04:57.580 --> 00:04:59.600
But let's set it -- So we're
trying to solve this.

00:04:59.600 --> 00:05:02.420
Because we want to find
out when it equals 0.

00:05:02.420 --> 00:05:07.130
Minus 10x squared
minus 9x plus 1.

00:05:07.130 --> 00:05:09.090
We want to find out what
x values make this

00:05:09.090 --> 00:05:11.260
equation equal to zero.

00:05:11.260 --> 00:05:13.730
And here we can use a tool
called a quadratic equation.

00:05:13.730 --> 00:05:15.625
And now I'm going to give you
one of the few things in math

00:05:15.625 --> 00:05:18.030
that's probably a good
idea to memorize.

00:05:18.030 --> 00:05:21.330
The quadratic equation says
that the roots of a quadratic

00:05:21.330 --> 00:05:24.810
are equal to -- and let's say
that the quadratic equation is

00:05:24.810 --> 00:05:31.900
a x squared plus b
x plus c equals 0.

00:05:31.900 --> 00:05:35.790
So, in this example,
a is minus 10.

00:05:35.790 --> 00:05:39.940
b is minus 9, and c is 1.

00:05:39.940 --> 00:05:48.040
The formula is the roots x
equals negative b plus or minus

00:05:48.040 --> 00:05:58.060
the square root of b squared
minus 4 times a times c,

00:05:58.060 --> 00:06:00.230
all of that over 2a.

00:06:00.230 --> 00:06:02.843
I know that looks complicated,
but the more you use it, you'll

00:06:02.843 --> 00:06:04.400
see it's actually not that bad.

00:06:04.400 --> 00:06:07.720
And this is a good
idea to memorize.

00:06:07.720 --> 00:06:10.730
So let's apply the quadratic
equation to this equation

00:06:10.730 --> 00:06:12.670
that we just wrote down.

00:06:12.670 --> 00:06:15.260
So, I just said -- and look,
the a is just the coefficient

00:06:15.260 --> 00:06:18.610
on the x term, right?

00:06:18.610 --> 00:06:20.300
a is the coefficient on
the x squared term.

00:06:20.300 --> 00:06:23.570
b is the coefficient on the x
term, and c is the constant.

00:06:23.570 --> 00:06:25.100
So let's apply it
tot this equation.

00:06:25.100 --> 00:06:26.250
What's b?

00:06:26.250 --> 00:06:28.700
Well, b is negative 9.

00:06:28.700 --> 00:06:29.970
We could see here.

00:06:29.970 --> 00:06:33.980
b is negative 9, a
is negative 10.

00:06:33.980 --> 00:06:34.970
c is 1.

00:06:34.970 --> 00:06:36.090
Right?

00:06:36.090 --> 00:06:42.350
So if b is negative 9 -- so
let's say, that's negative 9.

00:06:42.350 --> 00:06:49.260
Plus or minus the square
root of negative 9 squared.

00:06:49.260 --> 00:06:49.810
Well, that's 81.

00:06:53.140 --> 00:06:56.940
Minus 4 times a.

00:06:56.940 --> 00:06:59.760
a is minus 10.

00:06:59.760 --> 00:07:03.240
Minus 10 times c, which is 1.

00:07:03.240 --> 00:07:05.110
I know this is messy,
but hopefully you're

00:07:05.110 --> 00:07:06.470
understanding it.

00:07:06.470 --> 00:07:09.560
And all of that over 2 times a.

00:07:09.560 --> 00:07:14.050
Well, a is minus 10, so
2 times a is minus 20.

00:07:14.050 --> 00:07:14.990
So let's simplify that.

00:07:14.990 --> 00:07:19.410
Negative times negative
9, that's positive 9.

00:07:19.410 --> 00:07:26.460
Plus or minus the
square root of 81.

00:07:26.460 --> 00:07:30.660
We have a negative 4
times a negative 10.

00:07:30.660 --> 00:07:31.870
This is a minus 10.

00:07:31.870 --> 00:07:33.280
I know it's very messy,
I really apologize

00:07:33.280 --> 00:07:34.380
for that, times 1.

00:07:34.380 --> 00:07:39.410
So negative 4 times negative
10 is 40, positive 40.

00:07:39.410 --> 00:07:41.040
Positive 40.

00:07:41.040 --> 00:07:46.070
And then we have all of
that over negative 20.

00:07:46.070 --> 00:07:48.300
Well, 81 plus 40 is 121.

00:07:48.300 --> 00:07:52.330
So this is 9 plus or
minus the square root

00:07:52.330 --> 00:07:58.290
of 121 over minus 20.

00:07:58.290 --> 00:08:01.620
Square root of 121 is 11.

00:08:01.620 --> 00:08:03.170
So I'll go here.

00:08:03.170 --> 00:08:06.184
Hopefully you won't lose
track of what I'm doing.

00:08:06.184 --> 00:08:13.720
So this is 9 plus or
minus 11, over minus 20.

00:08:13.720 --> 00:08:19.090
And so if we said 9 plus 11
over minus 20, that is 9

00:08:19.090 --> 00:08:22.540
plus 11 is 20, so this
is 20 over minus 20.

00:08:22.540 --> 00:08:23.730
Which equals negative 1.

00:08:23.730 --> 00:08:24.900
So that's one root.

00:08:24.900 --> 00:08:28.260
That's 9 plus -- because
this is plus or minus.

00:08:28.260 --> 00:08:33.790
And the other root would be 9
minus 11 over negative 20.

00:08:33.790 --> 00:08:37.720
Which equals minus
2 over minus 20.

00:08:37.720 --> 00:08:40.700
Which equals 1 over 10.

00:08:40.700 --> 00:08:42.690
So that's the other root.

00:08:42.690 --> 00:08:48.950
So if we were to graph this
equation, we would see that it

00:08:48.950 --> 00:08:52.640
actually intersects the x axis.

00:08:52.640 --> 00:08:57.770
Or f of x equals 0 at the
point x equals negative

00:08:57.770 --> 00:09:01.690
1 and x equals 1/10.

00:09:01.690 --> 00:09:04.080
I'm going to do a lot more
examples in part 2, because I

00:09:04.080 --> 00:09:06.100
think, if anything, I might
have just confused

00:09:06.100 --> 00:09:08.120
you with this one.

00:09:08.120 --> 00:09:11.680
So, I'll see you in the
part 2 of using the

00:09:11.680 --> 00:09:12.150
quadratic equation.


Quadratic equation part 2

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:03.590
Welcome to part two of
the presentation on

00:00:03.590 --> 00:00:05.660
quadratic equations.

00:00:05.660 --> 00:00:08.470
Well, I think I thoroughly
confused you the last time

00:00:08.470 --> 00:00:11.170
around, so let me see if I
can fix that a bit by doing

00:00:11.170 --> 00:00:12.770
several more examples.

00:00:12.770 --> 00:00:15.430
So let's just start with
a review of what the

00:00:15.430 --> 00:00:16.380
quadratic equation is.

00:00:16.380 --> 00:00:19.650
The quadratic equation says, if
I'm trying to solve the

00:00:19.650 --> 00:00:31.590
equation Ax squared plus Bx
plus C equals 0, then the

00:00:31.590 --> 00:00:35.440
solution or the solutions
because there's usually two

00:00:35.440 --> 00:00:38.970
times that it intersects the
x-axis, or two solutions for

00:00:38.970 --> 00:00:47.610
this equation is x equals minus
B plus or minus the square root

00:00:47.610 --> 00:00:56.390
of B squared minus
4 times A times C.

00:00:56.390 --> 00:01:00.270
And all of that over 2A.

00:01:00.270 --> 00:01:02.040
So let's do a problem and
hopefully this should make

00:01:02.040 --> 00:01:02.690
a little more sense.

00:01:02.690 --> 00:01:04.620
That's a 2 on the bottom.

00:01:04.620 --> 00:01:13.890
So let's say I had the equation
minus 9x squared minus

00:01:13.890 --> 00:01:19.950
9x plus 6 equals 0.

00:01:19.950 --> 00:01:22.230
So in this example what's A?

00:01:22.230 --> 00:01:25.410
Well, A is the coefficient
on the x squared term.

00:01:25.410 --> 00:01:29.820
The x squared term is here,
the coefficient is minus 9.

00:01:29.820 --> 00:01:30.620
So let's write that.

00:01:30.620 --> 00:01:34.120
A equals minus 9.

00:01:34.120 --> 00:01:35.400
What does B equal?

00:01:35.400 --> 00:01:39.180
B is the coefficient on the x
term, so that's this term here.

00:01:39.180 --> 00:01:43.220
So B is also equal to minus 9.

00:01:43.220 --> 00:01:47.140
And C is the constant term,
which in this example is 6.

00:01:47.140 --> 00:01:49.550
So C is equal to 6.

00:01:49.550 --> 00:01:52.070
Now we just substitute these
values into the actual

00:01:52.070 --> 00:01:53.260
quadratic equation.

00:01:53.260 --> 00:01:59.600
So negative B, so it's
negative times negative 9.

00:01:59.600 --> 00:02:00.780
That's B.

00:02:00.780 --> 00:02:08.110
Plus or minus the square root
of B squared, so that's 81.

00:02:08.110 --> 00:02:08.390
Right?

00:02:08.390 --> 00:02:10.030
Negative 9 squared.

00:02:10.030 --> 00:02:14.720
Minus 4 times negative 9.

00:02:14.720 --> 00:02:16.140
That's A.

00:02:16.140 --> 00:02:19.480
Times C, which is 6.

00:02:19.480 --> 00:02:23.950
And all of that over 2
times negative 9, which

00:02:23.950 --> 00:02:25.630
is minus 18, right?

00:02:25.630 --> 00:02:26.720
2 times negative 9-- 2A.

00:02:29.230 --> 00:02:33.760
Let's try to simplify
this up here.

00:02:33.760 --> 00:02:37.930
Well, negative negative
9, that's positive 9.

00:02:37.930 --> 00:02:46.480
Plus or minus the
square root of 81.

00:02:46.480 --> 00:02:47.900
Let's see.

00:02:47.900 --> 00:02:50.270
This is negative 4
times negative 9.

00:02:50.270 --> 00:02:53.470
Negative 4 times negative
9 is positive 36.

00:02:53.470 --> 00:02:58.310
And then positive 36
times 6 is-- let's see.

00:02:58.310 --> 00:03:01.330
30 times 6 is 180.

00:03:01.330 --> 00:03:07.890
And then 180 plus
another 36 is 216.

00:03:07.890 --> 00:03:10.980
Plus 216, is that right?

00:03:10.980 --> 00:03:14.490
180 plus 36 is 216.

00:03:14.490 --> 00:03:16.840
All of that over 2A.

00:03:16.840 --> 00:03:19.570
2A we already said is minus 19.

00:03:19.570 --> 00:03:20.740
So we simplify that more.

00:03:20.740 --> 00:03:28.090
That's 9 plus or minus the
square root 81 plus 216.

00:03:28.090 --> 00:03:30.400
That's 80 plus 217.

00:03:30.400 --> 00:03:38.040
That's 297.

00:03:38.040 --> 00:03:41.900
And all of that over minus 18.

00:03:41.900 --> 00:03:45.020
Now, this is actually-- the
hardest part with the quadratic

00:03:45.020 --> 00:03:47.720
equation is oftentimes just
simplifying this expression.

00:03:47.720 --> 00:03:50.860
We have to figure out if we
can simplify this radical.

00:03:50.860 --> 00:03:53.090
Well, let's see.

00:03:53.090 --> 00:03:56.490
One way to figure out if a
number is divisible by 9 is to

00:03:56.490 --> 00:03:58.320
actually add up the digits
and see if the digits

00:03:58.320 --> 00:03:59.260
are divisible by 9.

00:03:59.260 --> 00:03:59.950
In this case, it is.

00:03:59.950 --> 00:04:02.510
2 plus 9 plus 7 is equal to 18.

00:04:02.510 --> 00:04:04.600
So let's see how many
times 9 goes into that.

00:04:04.600 --> 00:04:07.150
I'll do it on the side here; I
don't want to be too messy.

00:04:07.150 --> 00:04:09.450
9 goes into 2 97.

00:04:13.630 --> 00:04:16.190
3 times 27.

00:04:16.190 --> 00:04:19.040
27-- it goes 33 times, right?

00:04:19.040 --> 00:04:24.290
So this is the same thing as 9
plus or minus the square root

00:04:24.290 --> 00:04:31.110
of 9 times 33 over minus 18.

00:04:31.110 --> 00:04:32.470
And 9 is a perfect square.

00:04:32.470 --> 00:04:34.650
That's why I actually wanted to
see if 9 would work because

00:04:34.650 --> 00:04:36.390
that's the only way I could get
it out of the radical, if

00:04:36.390 --> 00:04:37.390
it's a perfect square.

00:04:37.390 --> 00:04:40.410
As you learned in that exponent
rules number one module.

00:04:40.410 --> 00:04:46.140
So this is equal to 9 plus
or minus 3 times the square

00:04:46.140 --> 00:04:53.230
root of 33, and all of
that over minus 18.

00:04:53.230 --> 00:04:54.570
We're almost done.

00:04:54.570 --> 00:04:57.840
We can actually simplify it
because 9, 3, and minus 18

00:04:57.840 --> 00:05:00.650
are all divisible by 3.

00:05:00.650 --> 00:05:02.270
Let's divide everything by 3.

00:05:02.270 --> 00:05:14.370
3 plus or minus the square
root of 33 over minus 6.

00:05:14.370 --> 00:05:15.610
And we are done.

00:05:15.610 --> 00:05:17.010
So as you see, the hardest
thing with the quadratic

00:05:17.010 --> 00:05:20.110
equation is often just
simplifying the expression.

00:05:20.110 --> 00:05:22.750
But what we've said, I know you
might have lost track-- we did

00:05:22.750 --> 00:05:27.120
all this math-- is we said,
this equation: minus 9x

00:05:27.120 --> 00:05:30.550
squared minus 9x plus 6.

00:05:30.550 --> 00:05:34.200
Now we found two x values that
would satisfy this equation

00:05:34.200 --> 00:05:35.970
and make it equal to 0.

00:05:35.970 --> 00:05:39.830
One x value is x equals
3 plus the square root

00:05:39.830 --> 00:05:42.100
of 33 over minus 6.

00:05:42.100 --> 00:05:45.860
And the second value is
3 minus the square root

00:05:45.860 --> 00:05:50.160
of 33 over minus 6.

00:05:50.160 --> 00:05:52.250
And you might want to
think about why we have

00:05:52.250 --> 00:05:53.370
that plus or minus.

00:05:53.370 --> 00:05:55.490
We have that plus or minus
because a square root could

00:05:55.490 --> 00:05:59.550
actually be a positive
or a negative number.

00:05:59.550 --> 00:06:02.180
Let's do another problem.

00:06:02.180 --> 00:06:05.890
Hopefully this one will
be a little bit simpler.

00:06:09.210 --> 00:06:16.780
Let's say I wanted to
solve minus 8x squared

00:06:16.780 --> 00:06:21.000
plus 5x plus 9.

00:06:21.000 --> 00:06:23.150
Now I'm going to assume that
you've memorized the quadratic

00:06:23.150 --> 00:06:25.310
equation because that's
something you should do.

00:06:25.310 --> 00:06:26.630
Or you should write it
down on a piece of paper.

00:06:26.630 --> 00:06:31.630
But the quadratic equation is
negative B-- So b is 5, right?

00:06:31.630 --> 00:06:34.160
We're trying to solve that
equal to 0, so negative B.

00:06:34.160 --> 00:06:39.790
So negative 5, plus or minus
the square root of B squared-

00:06:39.790 --> 00:06:44.030
that's 5 squared, 25.

00:06:44.030 --> 00:06:50.470
Minus 4 times A,
which is minus 8.

00:06:50.470 --> 00:06:53.820
Times C, which is 9.

00:06:53.820 --> 00:06:56.400
And all of that over 2 times A.

00:06:56.400 --> 00:07:00.320
Well, A is minus 8, so all
of that is over minus 16.

00:07:00.320 --> 00:07:04.090
So let's simplify this
expression up here.

00:07:04.090 --> 00:07:09.440
Well, that's equal to
minus 5 plus or minus

00:07:09.440 --> 00:07:13.630
the square root of 25.

00:07:13.630 --> 00:07:14.620
Let's see.

00:07:14.620 --> 00:07:18.220
4 times 8 is 32 and the
negatives cancel out, so

00:07:18.220 --> 00:07:21.520
that's positive 32 times 9.

00:07:21.520 --> 00:07:24.480
Positive 32 times 9, let's see.

00:07:24.480 --> 00:07:26.720
30 times 9 is 270.

00:07:26.720 --> 00:07:31.110
It's 288.

00:07:31.110 --> 00:07:31.570
I think.

00:07:31.570 --> 00:07:31.800
Right?

00:07:36.130 --> 00:07:37.490
288.

00:07:37.490 --> 00:07:40.590
We have all of that
over minus 16.

00:07:40.590 --> 00:07:42.560
Now simplify it more.

00:07:42.560 --> 00:07:47.760
Minus 5 plus or minus the
square root-- 25 plus

00:07:47.760 --> 00:07:51.340
288 is 313 I believe.

00:07:56.950 --> 00:08:00.230
And all of that over minus 16.

00:08:00.230 --> 00:08:03.430
And I think, I'm not 100% sure,
although I'm pretty sure.

00:08:03.430 --> 00:08:04.570
I haven't checked it.

00:08:04.570 --> 00:08:10.370
That 313 can't be factored
into a product of a perfect

00:08:10.370 --> 00:08:11.690
square and another number.

00:08:11.690 --> 00:08:13.670
In fact, it actually
might be a prime number.

00:08:13.670 --> 00:08:15.600
That's something that you
might want to check out.

00:08:15.600 --> 00:08:18.200
So if that is the case and
we've got it in completely

00:08:18.200 --> 00:08:21.840
simplified form, and we say
there are two solutions, two

00:08:21.840 --> 00:08:24.940
x values that will make
this equation true.

00:08:24.940 --> 00:08:30.750
One of them is x is equal
to minus 5 plus the square

00:08:30.750 --> 00:08:35.830
root of 313 over minus 16.

00:08:35.830 --> 00:08:44.110
And the other one is x is equal
to minus 5 minus the square

00:08:44.110 --> 00:08:49.660
root of 313 over minus 16.

00:08:49.660 --> 00:08:51.760
Hopefully those two examples
will give you a good

00:08:51.760 --> 00:08:53.940
sense of how to use the
quadratic equation.

00:08:53.940 --> 00:08:55.860
I might add some more modules.

00:08:55.860 --> 00:08:58.230
And then, once you master this,
I'll actually teach you how to

00:08:58.230 --> 00:09:00.370
solve quadratic equations when
you actually get a negative

00:09:00.370 --> 00:09:01.910
number under the radical.

00:09:01.910 --> 00:09:03.140
Very interesting.

00:09:03.140 --> 00:09:06.760
Anyway, I hope you can do the
module now and maybe I'll add a

00:09:06.760 --> 00:09:10.370
few more presentations because
this isn't the easiest module.

00:09:10.370 --> 00:09:11.840
But I hope you have fun.

00:09:11.840 --> 00:09:13.140
Bye.