Discriminant of quadratic equations

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:03.470
I think we've had some pretty
good exposure to the quadratic

00:00:03.470 --> 00:00:07.550
formula, but just in case you
haven't memorized it yet, let

00:00:07.550 --> 00:00:08.760
me write it down again.

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So let's say we have a quadratic
equation of the

00:00:10.770 --> 00:00:18.180
form, ax squared plus bx,
plus c is equal to 0.

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The quadratic formula, which we
proved in the last video,

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says that the solutions to this
equation are x is equal

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to negative b plus or minus the
square root of b squared,

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minus 4ac, all of
that over 2a.

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Now, in this video, rather than
just giving a bunch of

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examples of substituting in the
a's, the b's, and the c's,

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I want to talk a little bit
about this part of the

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quadratic formula, this
part right there.

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The b squared minus 4ac.

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And we've seen it in a couple of
the problems we've done as

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examples, that this kind of
determines what our solution

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is going to look like.

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If, for example, b squared minus
4ac is greater than 0,

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we're going to have two
solutions, right?

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The square root of some
positive number that's

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non-zero, there's going to be
a positive and negative

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version of it-- we're always
going to have a b over 2a or

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negative b over 2a-- so you're
going to have negative b plus

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that positive square root, and
a negative b minus that

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positive square root,
all over 2a.

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So if the discriminant is
greater 0, then that tells us

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that we have two solutions.

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Now I just used a word, and
that word is discriminant.

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And all that is referring
to is this part of

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the quadratic formula.

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That right there-- let me do it
in a different color-- this

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right here is the discriminant
of the quadratic equation

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right here.

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And you just have to remember,
it's the part that's under the

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radical sign of the
quadratic formula.

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And that's why it matters,
because if this is greater

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than 0, you're having a positive
square root, and

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you'll have the positive and
negative version of it, you'll

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have two solutions.

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Now, what happens if b squared
minus 4ac is equal to 0?

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If this is equal to 0-- if you
take b squared minus 4, times

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a, times c, and that's equal to
0-- that tells us that this

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part of the quadratic formula
is going to be 0, and the

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square root of 0 is just 0.

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And then, actually, your only
solution is going to be x is

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going to be equal to
negative b over 2a.

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Or another way to think
about it is you

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only have one solution.

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So if the discriminant
is equal to 0, you

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only have one solution.

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And that solution is actually
going to be the vertex, or the

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x-coordinate of the vertex,
because you're going to have a

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parabola that just touches the
x-axis like that, just touches

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there, or just touches like
that, just touches at exactly

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one point, when b squared
minus 4ac is equal to 0.

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And then the last situation
is if b squared minus 4ac

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is less than 0.

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Then over here, you're going to
get a negative number under

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the radical.

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And we saw an example of
that in the last video.

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If we're dealing with real
numbers, we can't take a

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square root of a negative
number, so this means that we

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have no real solutions.

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In the future, you're going to
see that we will have complex

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solutions, but if we're dealing
with real numbers we

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have no real solution.

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Because this makes no sense.

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The square of a negative number,
at least it makes no

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sense in the real numbers.

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And then there's more
you can think about.

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If we do have a positive
discriminant, if b squared

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minus 4ac is positive, we can
think about whether the

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solutions are going to
be rational or not.

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If this is 2, then we're going
to have the square root of 2

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in our answer, it's going to
be an irrational answer, or

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our solutions are going
to be irrational.

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If b squared minus 4ac is 16,
we know that's a perfect

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square, you take the square
root of a perfect square,

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we're going to have
a rational answer.

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Anyway, with all of that talk,
let's do some examples,

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because I think that's
what makes all

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of these ideas tangible.

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So let's say I have the equation
negative x squared

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plus 3x, minus 6
is equal to 0.

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And all I'm concerned about is
I just want to know a little

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bit about what kinds of
solutions this has.

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I don't want to necessarily
even solve for x.

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So if you're in a situation like
that, I can just look at

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the discriminant.

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I can just look at b
squared minus 4ac.

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So the discriminant here is
what? b squared is 9 minus 4,

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times a-- negative 1-- times
c, which is negative 6.

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So what is this equal to?

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This negative and that negative
cancel out, but we

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still have that negative
out there, so it's 9

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minus 4, times 6.

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This is 9 minus 24, which
is less than 0.

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So we're going to have a number
smaller than 0 under

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the radical.

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So we have no real solutions.

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That was this scenario
right here.

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And so this graph is going to
point downwards, because we

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have a negative sign there,
so it probably looks like

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something like that.

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If that's the x-axis, the
graph is dipping down.

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Its vertex is below the x-axis
and it's downward-opening, so

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it never intersects
the x-axis.

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We have no real solutions.

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Let's do another one.

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Let's say I have-- I'll do this
one in pink-- let's say I

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have the equation, 5x squared
is equal to 6x.

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Well, let's put this in the
form that we're used to.

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So let's subtract 6x from both
sides, and we get 5x squared

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minus 6x is equal to 0.

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And let's calculate
the discriminant.

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So, we want to get b squared.

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b squared is negative
6 squared minus 4,

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times a, times c.

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Well, where is the c here?

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There is no c here.

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There's a plus 0 that I'm
not writing here.

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There's no c.

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So in this situation,
c is equal to 0.

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There is no c in
that equation.

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So times 0.

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So that all cancels out.

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Negative 6 squared
is positive 36.

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The discriminant is positive.

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You'd have a positive 36 under
the radical right there, so

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not only is it positive, it's
also a perfect square.

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So this tells me that I'm going
to have two solutions.

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So I'm going to have
two real solutions.

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And not only are they're going
to be real, but I also know

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they're going to be rational,
because I have the

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square root of 36.

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The square root of 36 is
positive or negative 6.

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I don't end up with an
irrational number here, so two

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real solutions that
are also rational.

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This is this scenario
right there.

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And you could also have
irrational in this scenario,

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so it's this [? here ?]

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plus the irrational.

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Let's do a couple more, just
to get really warmed.

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Let's say I have 41x squared
minus 31x, minus

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52 is equal to 0.

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Once again, I just want to
think about what type of

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solution I might be
dealing with.

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So b squared minus 4ac.

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b squared.

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Negative the 31 squared minus
4, times a, times 41, times

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c-- times negative 52.

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So what do I have here?

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This is going to be a
positive 31 squared.

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The negative times
the negative,

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these are both positive.

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So I'm going to have
a positive, right?

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This is the same thing as 31
squared, plus-- this is a

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positive number right here, I
mean, we could calculate it,

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but it's 4 times 41, times 52.

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All I care about is my
discriminant is positive.

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It is greater than 0,
so that means I

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have two real solutions.

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And we could think about whether
this is some type of

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perfect square.

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I don't know.

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I'm not going to do it here.

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That would take a little
bit of computation.

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So we know they're real, we
don't know if they're rational

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or irrational solutions.

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Let's do one more of these.

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Let's say I have x squared
minus 8x, plus

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16 is equal to 0.

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Once again, let's look
at the discriminant.

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b squared, that's negative 8
squared minus 4, times a,

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which is 1, times
c, which is 16.

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This is equal to 64 minus
64, which is equal to 0.

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So we only have one solution,
and by definition it's going

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to be rational.

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I mean, you could actually
look at it right here.

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It's x minus 4, times x
minus 4 is equal to 0.

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The one solution is x
equal to positive 4.

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And when I say by definition of
the quadratic formula, you

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look there, if this is a 0,
all you're left with is

00:09:45.390 --> 00:09:48.230
negative b over 2a, which is
definitely going to be

00:09:48.230 --> 00:09:51.110
rational, assuming you have a,
b, and c are, of course,

00:09:51.110 --> 00:09:52.480
rational numbers.

00:09:52.480 --> 00:09:54.410
Anyway, hopefully you
found that useful.

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It's a quick way.

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You don't have to go all the way
to solving the solution,

00:09:57.470 --> 00:10:00.080
you just want to have to say
what types of solutions or how

00:10:00.080 --> 00:10:02.910
many solutions, how many real
solutions, or inspect whether

00:10:02.910 --> 00:10:04.120
they're real or rational.

00:10:04.120 --> 00:10:07.320
The discriminant can be kind
of a useful shortcut.

00:10:07.320 --> 00:10:10.640
And I also think it makes you
kind of appreciate the parts

00:10:10.640 --> 00:10:13.590
of the quadratic formula
a little bit better.


Proof of quadratic formula

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.170
In the last video, I told you
that if you had a quadratic

00:00:03.170 --> 00:00:10.020
equation of the form ax squared
plus bx, plus c is

00:00:10.020 --> 00:00:14.770
equal to zero, you could use the
quadratic formula to find

00:00:14.770 --> 00:00:16.800
the solutions to
this equation.

00:00:16.800 --> 00:00:19.590
And the quadratic
formula was x.

00:00:19.590 --> 00:00:24.780
The solutions would be equal
to negative b plus or minus

00:00:24.780 --> 00:00:30.800
the square root of b squared
minus 4ac, all

00:00:30.800 --> 00:00:32.950
of that over 2a.

00:00:32.950 --> 00:00:34.240
And we learned how to use it.

00:00:34.240 --> 00:00:38.040
You literally just substitute
the numbers a for a, b for b,

00:00:38.040 --> 00:00:41.660
c for c, and then it gives you
two answers, because you have

00:00:41.660 --> 00:00:43.830
a plus or a minus right there.

00:00:43.830 --> 00:00:45.000
What I want to do
in this video is

00:00:45.000 --> 00:00:46.380
actually prove it to you.

00:00:46.380 --> 00:00:49.930
Prove that using, essentially
completing the square, I can

00:00:49.930 --> 00:00:53.540
get from that to that
right over there.

00:00:53.540 --> 00:00:56.190
So the first thing I want to
do, so that I can start

00:00:56.190 --> 00:00:59.600
completing the square from this
point right here, is--

00:00:59.600 --> 00:01:02.635
let me rewrite the equation
right here-- so we have ax--

00:01:02.635 --> 00:01:08.340
let me do it in a different
color-- I have ax squared plus

00:01:08.340 --> 00:01:12.920
bx, plus c is equal to 0.

00:01:12.920 --> 00:01:15.060
So the first I want to do is
divide everything by a, so I

00:01:15.060 --> 00:01:16.790
just have a 1 out here
as a coefficient.

00:01:16.790 --> 00:01:22.160
So you divide everything by a,
you get x squared plus b over

00:01:22.160 --> 00:01:28.030
ax, plus c over a, is equal
to 0 over a, which

00:01:28.030 --> 00:01:30.530
is still just 0.

00:01:30.530 --> 00:01:33.620
Now we want to-- well, let me
get the c over a term on to

00:01:33.620 --> 00:01:35.880
the right-hand side,
so let's subtract c

00:01:35.880 --> 00:01:37.710
over a from both sides.

00:01:37.710 --> 00:01:45.850
And we get x squared plus b over
a x, plus-- well, I'll

00:01:45.850 --> 00:01:47.950
just leave it blank there,
because this is gone now; we

00:01:47.950 --> 00:01:53.250
subtracted it from both sides--
is equal to negative c

00:01:53.250 --> 00:01:56.580
over a I left a space
there so that we can

00:01:56.580 --> 00:01:58.740
complete the square.

00:01:58.740 --> 00:02:01.030
And you saw in the completing
the square video, you

00:02:01.030 --> 00:02:04.800
literally just take 1/2 of this
coefficient right here

00:02:04.800 --> 00:02:06.060
and you square it.

00:02:06.060 --> 00:02:08.990
So what is b over
a divided by 2?

00:02:08.990 --> 00:02:11.800
Or what is 1/2 times b over a?

00:02:11.800 --> 00:02:17.610
Well, that is just b over 2a,
and, of course, we are going

00:02:17.610 --> 00:02:19.110
to square it.

00:02:19.110 --> 00:02:21.520
You take 1/2 of this
and you square it.

00:02:21.520 --> 00:02:23.500
That's what we do in completing
a square, so that

00:02:23.500 --> 00:02:28.770
we can turn this into the
perfect square of a binomial.

00:02:28.770 --> 00:02:33.570
Now, of course, we cannot just
add the b over 2a squared to

00:02:33.570 --> 00:02:34.280
the left-hand side.

00:02:34.280 --> 00:02:36.130
We have to add it
to both sides.

00:02:36.130 --> 00:02:43.750
So you have a plus b over 2a
squared there as well.

00:02:43.750 --> 00:02:45.310
Now what happens?

00:02:45.310 --> 00:02:50.040
Well, this over here, this
expression right over here,

00:02:50.040 --> 00:02:59.250
this is the exact same thing as
x plus b over 2a squared.

00:02:59.250 --> 00:03:02.120
And if you don't believe me, I'm
going to multiply it out.

00:03:02.120 --> 00:03:10.030
That x plus b over 2a squared
is x plus b over 2a, times x

00:03:10.030 --> 00:03:15.830
plus b over 2a. x times
x is x squared.

00:03:15.830 --> 00:03:21.960
x times b over 2a is
plus b over 2ax.

00:03:21.960 --> 00:03:27.560
You have b over 2a times x,
which is another b over 2ax,

00:03:27.560 --> 00:03:31.580
and then you have b over 2a
times b over 2a, that is plus

00:03:31.580 --> 00:03:34.660
b over 2a squared.

00:03:34.660 --> 00:03:38.180
That and this are the same
thing, because these two

00:03:38.180 --> 00:03:43.910
middle terms, b over 2a plus b
over 2a, that's the same thing

00:03:43.910 --> 00:03:50.200
as 2b over 2ax, which is the
same thing as b over ax.

00:03:50.200 --> 00:03:55.520
So this simplifies to x squared
plus b over ax, plus b

00:03:55.520 --> 00:03:59.960
over 2a squared, which is
exactly what we have written

00:03:59.960 --> 00:04:00.720
right there.

00:04:00.720 --> 00:04:03.580
That was the whole point of
adding this term to both

00:04:03.580 --> 00:04:06.340
sides, so it becomes
a perfect square.

00:04:06.340 --> 00:04:08.970
So the left-hand side
simplifies to this.

00:04:08.970 --> 00:04:13.810
The right-hand side, maybe
not quite as simple.

00:04:13.810 --> 00:04:16.490
Maybe we'll leave it the
way it is right now.

00:04:16.490 --> 00:04:18.160
Actually, let's simplify
it a little bit.

00:04:18.160 --> 00:04:21.339
So the right-hand side,
we can rewrite it.

00:04:21.339 --> 00:04:24.960
This is going to be equal
to-- well, this is

00:04:24.960 --> 00:04:26.090
going to be b squared.

00:04:26.090 --> 00:04:29.220
I'll write that term first. This
is b-- let me do it in

00:04:29.220 --> 00:04:31.340
green so we can follow along.

00:04:31.340 --> 00:04:33.250
So that term right there
can be written as b

00:04:33.250 --> 00:04:37.210
squared over 4a square.

00:04:37.210 --> 00:04:38.420
And what's this term?

00:04:38.420 --> 00:04:40.270
What would that become?

00:04:40.270 --> 00:04:43.730
This would become-- in order
to have 4a squared as the

00:04:43.730 --> 00:04:45.640
denominator, we have to multiply
the numerator and the

00:04:45.640 --> 00:04:48.280
denominator by 4a.

00:04:48.280 --> 00:04:51.590
So this term right
here will become

00:04:51.590 --> 00:04:58.480
minus 4ac over 4a squared.

00:04:58.480 --> 00:05:00.970
And you can verify for yourself
that that is the same

00:05:00.970 --> 00:05:01.550
thing as that.

00:05:01.550 --> 00:05:03.790
I just multiplied the
numerator and the

00:05:03.790 --> 00:05:05.400
denominator by 4a.

00:05:05.400 --> 00:05:08.400
In fact, the 4's cancel out and
then this a cancels out

00:05:08.400 --> 00:05:09.930
and you just have a c over a.

00:05:09.930 --> 00:05:12.730
So these, this and that
are equivalent.

00:05:12.730 --> 00:05:16.460
I just switched which I write
first. And you might already

00:05:16.460 --> 00:05:21.980
be seeing the beginnings of the
quadratic formula here.

00:05:21.980 --> 00:05:24.700
So this I can rewrite.

00:05:24.700 --> 00:05:25.760
This I can rewrite.

00:05:25.760 --> 00:05:29.220
The right-hand side, right
here, I can rewrite as b

00:05:29.220 --> 00:05:37.060
squared minus 4ac, all of
that over 4a squared.

00:05:37.060 --> 00:05:38.150
This is looking very close.

00:05:38.150 --> 00:05:41.800
Notice, b squared minus 4ac,
it's already appearing.

00:05:41.800 --> 00:05:43.470
We don't have a square root yet,
but we haven't taken the

00:05:43.470 --> 00:05:44.720
square root of both
sides of this

00:05:44.720 --> 00:05:46.670
equation, so let's do that.

00:05:46.670 --> 00:05:49.720
So if you take the square root
of both sides, the left-hand

00:05:49.720 --> 00:05:53.260
side will just become x plus--
let me scroll down a little

00:05:53.260 --> 00:05:59.900
bit-- x plus b over 2a is going
to be equal to the plus

00:05:59.900 --> 00:06:02.670
or minus square root
of this thing.

00:06:02.670 --> 00:06:04.460
And the square root of this
is the square root of the

00:06:04.460 --> 00:06:06.360
numerator over the square
root of the denominator.

00:06:06.360 --> 00:06:11.040
So it's going to be the plus or
minus the square root of b

00:06:11.040 --> 00:06:18.690
squared minus 4ac over the
square root of 4a squared.

00:06:18.690 --> 00:06:20.690
Now, what is the square
root of 4a squared?

00:06:20.690 --> 00:06:23.820
It is 2a, right?

00:06:23.820 --> 00:06:25.390
2a squared is 4a squared.

00:06:28.580 --> 00:06:30.530
The square root of this
is that right here.

00:06:30.530 --> 00:06:34.260
So to go from here to here, I
just took the square root of

00:06:34.260 --> 00:06:36.650
both sides of this equation.

00:06:36.650 --> 00:06:38.860
Now, this is looking very
close to the quadratic.

00:06:38.860 --> 00:06:41.990
We have a b squared minus
4ac over 2a, now we just

00:06:41.990 --> 00:06:45.070
essentially have to subtract
this b over 2a from both sides

00:06:45.070 --> 00:06:46.740
of the equation and
we're done.

00:06:46.740 --> 00:06:48.210
So let's do that.

00:06:48.210 --> 00:06:51.640
So if you subtract the b over
2a from both sides of this

00:06:51.640 --> 00:06:52.980
equation, what do you get?

00:06:52.980 --> 00:07:00.080
You get x is equal to negative
b over 2a, plus or minus the

00:07:00.080 --> 00:07:07.000
square root of b squared minus
4ac over 2a, common

00:07:07.000 --> 00:07:07.960
denominator.

00:07:07.960 --> 00:07:10.640
So this is equal
to negative b.

00:07:10.640 --> 00:07:14.500
Let me do this in a new color.

00:07:14.500 --> 00:07:15.370
So it's orange.

00:07:15.370 --> 00:07:21.980
Negative b plus or minus the
square root of b squared minus

00:07:21.980 --> 00:07:27.970
4ac, all of that over 2a.

00:07:27.970 --> 00:07:29.370
And we are done!

00:07:29.370 --> 00:07:33.630
By completing the square with
just general coefficients in

00:07:33.630 --> 00:07:36.890
front of our a, b and c, we
were able to derive the

00:07:36.890 --> 00:07:38.600
quadratic formula.

00:07:38.600 --> 00:07:40.360
Just like that.

00:07:40.360 --> 00:07:43.920
Hopefully you found that as
entertaining as I did.


Solving quadratic equations by completing the square

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:02.930
In this video, I'm going to show
you a technique called

00:00:02.930 --> 00:00:09.310
completing the square.

00:00:09.310 --> 00:00:14.490
And what's neat about this is
that this will work for any

00:00:14.490 --> 00:00:18.210
quadratic equation, and it's
actually the basis for the

00:00:18.210 --> 00:00:18.750
quadratic formula.

00:00:18.750 --> 00:00:21.990
And in the next video or the
video after that I'll prove

00:00:21.990 --> 00:00:25.630
the quadratic formula using
completing the square.

00:00:25.630 --> 00:00:28.450
But before we do that, we
need to understand even

00:00:28.450 --> 00:00:29.470
what it's all about.

00:00:29.470 --> 00:00:32.070
And it really just builds off
of what we did in the last

00:00:32.070 --> 00:00:33.880
video, where we solved
quadratics

00:00:33.880 --> 00:00:36.130
using perfect squares.

00:00:36.130 --> 00:00:39.900
So let's say I have the
quadratic equation x squared

00:00:39.900 --> 00:00:44.880
minus 4x is equal to 5.

00:00:44.880 --> 00:00:47.490
And I put this big space
here for a reason.

00:00:47.490 --> 00:00:49.680
In the last video, we saw
that these can be pretty

00:00:49.680 --> 00:00:53.200
straightforward to solve if
the left-hand side is a

00:00:53.200 --> 00:00:56.500
perfect square.

00:00:56.500 --> 00:00:59.050
You see, completing the square
is all about making the

00:00:59.050 --> 00:01:01.900
quadratic equation into a
perfect square, engineering

00:01:01.900 --> 00:01:05.190
it, adding and subtracting from
both sides so it becomes

00:01:05.190 --> 00:01:05.970
a perfect square.

00:01:05.970 --> 00:01:07.710
So how can we do that?

00:01:07.710 --> 00:01:10.130
Well, in order for this
left-hand side to be a perfect

00:01:10.130 --> 00:01:12.990
square, there has to be
some number here.

00:01:12.990 --> 00:01:17.510
There has to be some number here
that if I have my number

00:01:17.510 --> 00:01:20.910
squared I get that number, and
then if I have two times my

00:01:20.910 --> 00:01:22.890
number I get negative 4.

00:01:22.890 --> 00:01:24.750
Remember that, and I
think it'll become

00:01:24.750 --> 00:01:27.700
clear with a few examples.

00:01:27.700 --> 00:01:35.230
I want x squared minus 4x plus
something to be equal to x

00:01:35.230 --> 00:01:37.740
minus a squared.

00:01:37.740 --> 00:01:41.010
We don't know what a
is just yet, but we

00:01:41.010 --> 00:01:42.110
know a couple of things.

00:01:42.110 --> 00:01:46.180
When I square things-- so this
is going to be x squared minus

00:01:46.180 --> 00:01:49.330
2a plus a squared.

00:01:49.330 --> 00:01:53.640
So if you look at this pattern
right here, that has to be--

00:01:53.640 --> 00:01:59.880
sorry, x squared minus 2ax--
this right here has to be 2ax.

00:01:59.880 --> 00:02:03.530
And this right here would
have to be a squared.

00:02:03.530 --> 00:02:07.690
So this number, a is going to
be half of negative 4, a has

00:02:07.690 --> 00:02:10.370
to be negative 2, right?

00:02:10.370 --> 00:02:13.570
Because 2 times a is going
to be negative 4.

00:02:13.570 --> 00:02:18.330
a is negative 2, and if a is
negative 2, what is a squared?

00:02:18.330 --> 00:02:21.550
Well, then a squared is going
to be positive 4.

00:02:21.550 --> 00:02:24.220
And this might look all
complicated to you right now,

00:02:24.220 --> 00:02:25.910
but I'm showing you
the rationale.

00:02:25.910 --> 00:02:29.080
You literally just look at this
coefficient right here,

00:02:29.080 --> 00:02:32.670
and you say, OK, well what's
half of that coefficient?

00:02:32.670 --> 00:02:35.920
Well, half of that coefficient
is negative 2.

00:02:35.920 --> 00:02:40.230
So we could say a is equal to
negative 2-- same idea there--

00:02:40.230 --> 00:02:41.720
and then you square it.

00:02:41.720 --> 00:02:44.100
You square a, you
get positive 4.

00:02:44.100 --> 00:02:46.540
So we add positive 4 here.

00:02:46.540 --> 00:02:47.630
Add a 4.

00:02:47.630 --> 00:02:50.990
Now, from the very first
equation we ever did, you

00:02:50.990 --> 00:02:55.240
should know that you can never
do something to just one side

00:02:55.240 --> 00:02:55.900
of the equation.

00:02:55.900 --> 00:02:58.700
You can't add 4 to just one
side of the equation.

00:02:58.700 --> 00:03:02.710
If x squared minus 4x was equal
to 5, then when I add 4

00:03:02.710 --> 00:03:04.720
it's not going to be
equal to 5 anymore.

00:03:04.720 --> 00:03:07.950
It's going to be equal
to 5 plus 4.

00:03:07.950 --> 00:03:11.430
We added 4 on the left-hand side
because we wanted this to

00:03:11.430 --> 00:03:12.435
be a perfect square.

00:03:12.435 --> 00:03:15.210
But if you add something to the
left-hand side, you've got

00:03:15.210 --> 00:03:17.320
to add it to the right-hand
side.

00:03:17.320 --> 00:03:20.630
And now, we've gotten ourselves
to a problem that's

00:03:20.630 --> 00:03:23.410
just like the problems we
did in the last video.

00:03:23.410 --> 00:03:25.960
What is this left-hand side?

00:03:25.960 --> 00:03:27.000
Let me rewrite the
whole thing.

00:03:27.000 --> 00:03:33.020
We have x squared minus 4x
plus 4 is equal to 9 now.

00:03:33.020 --> 00:03:35.380
All we did is add 4 to both
sides of the equation.

00:03:35.380 --> 00:03:39.070
But we added 4 on purpose so
that this left-hand side

00:03:39.070 --> 00:03:41.080
becomes a perfect square.

00:03:41.080 --> 00:03:41.760
Now what is this?

00:03:41.760 --> 00:03:45.340
What number when I multiply it
by itself is equal to 4 and

00:03:45.340 --> 00:03:47.770
when I add it to itself I'm
equal to negative 2?

00:03:47.770 --> 00:03:49.000
Well, we already answered
that question.

00:03:49.000 --> 00:03:50.040
It's negative 2.

00:03:50.040 --> 00:03:55.310
So we get x minus 2 times
x minus 2 is equal to 9.

00:03:55.310 --> 00:03:59.350
Or we could have skipped this
step and written x minus 2

00:03:59.350 --> 00:04:02.990
squared is equal to 9.

00:04:02.990 --> 00:04:07.280
And then you take the square
root of both sides, you get x

00:04:07.280 --> 00:04:10.840
minus 2 is equal to
plus or minus 3.

00:04:10.840 --> 00:04:16.870
Add 2 to both sides, you get x
is equal to 2 plus or minus 3.

00:04:16.870 --> 00:04:22.440
That tells us that x could be
equal to 2 plus 3, which is 5.

00:04:22.440 --> 00:04:28.960
Or x could be equal to 2 minus
3, which is negative 1.

00:04:28.960 --> 00:04:30.650
And we are done.

00:04:30.650 --> 00:04:31.840
Now I want to be very clear.

00:04:31.840 --> 00:04:34.300
You could have done this without
completing the square.

00:04:34.300 --> 00:04:37.640
We could've started off
with x squared minus

00:04:37.640 --> 00:04:39.850
4x is equal to 5.

00:04:39.850 --> 00:04:42.970
We could have subtracted 5 from
both sides and gotten x

00:04:42.970 --> 00:04:47.160
squared minus 4x minus
5 is equal to 0.

00:04:47.160 --> 00:04:51.940
And you could say, hey, if I
have a negative 5 times a

00:04:51.940 --> 00:04:56.190
positive 1, then their product
is negative 5 and their sum is

00:04:56.190 --> 00:04:57.000
negative 4.

00:04:57.000 --> 00:05:00.800
So I could say this is x
minus 5 times x plus

00:05:00.800 --> 00:05:02.480
1 is equal to 0.

00:05:02.480 --> 00:05:06.810
And then we would say that x is
equal to 5 or x is equal to

00:05:06.810 --> 00:05:07.700
negative 1.

00:05:07.700 --> 00:05:10.350
And in this case, this actually
probably would have

00:05:10.350 --> 00:05:13.450
been a faster way to
do the problem.

00:05:13.450 --> 00:05:16.140
But the neat thing about the
completing the square is it

00:05:16.140 --> 00:05:17.770
will always work.

00:05:17.770 --> 00:05:21.580
It'll always work no matter what
the coefficients are or

00:05:21.580 --> 00:05:23.385
no matter how crazy
the problem is.

00:05:23.385 --> 00:05:25.400
And let me prove it to you.

00:05:25.400 --> 00:05:28.440
Let's do one that traditionally
would have been

00:05:28.440 --> 00:05:31.140
a pretty painful problem if
we just tried to do it by

00:05:31.140 --> 00:05:36.200
factoring, especially if we
did it using grouping or

00:05:36.200 --> 00:05:37.020
something like that.

00:05:37.020 --> 00:05:45.070
Let's say we had 10x squared
minus 30x minus

00:05:45.070 --> 00:05:47.530
8 is equal to 0.

00:05:47.530 --> 00:05:50.060
Now, right from the get-go, you
could say, hey look, we

00:05:50.060 --> 00:05:53.280
could maybe divide
both sides by 2.

00:05:53.280 --> 00:05:54.800
That does simplify
a little bit.

00:05:54.800 --> 00:05:56.450
Let's divide both sides by 2.

00:05:56.450 --> 00:06:02.150
So if you divide everything
by 2, what do you get?

00:06:02.150 --> 00:06:11.990
We get 5x squared minus 15x
minus 4 is equal to 0.

00:06:11.990 --> 00:06:14.540
But once again, now we have this
crazy 5 in front of this

00:06:14.540 --> 00:06:16.810
coefficent and we would have to
solve it by grouping which

00:06:16.810 --> 00:06:20.410
is a reasonably painful
process.

00:06:20.410 --> 00:06:23.410
But we can now go straight to
completing the square, and to

00:06:23.410 --> 00:06:27.500
do that I'm now going to divide
by 5 to get a 1 leading

00:06:27.500 --> 00:06:28.870
coefficient here.

00:06:28.870 --> 00:06:31.660
And you're going to see why this
is different than what

00:06:31.660 --> 00:06:33.010
we've traditionally done.

00:06:33.010 --> 00:06:35.730
So if I divide this whole thing
by 5, I could have just

00:06:35.730 --> 00:06:38.050
divided by 10 from the get-go
but I wanted to go to this the

00:06:38.050 --> 00:06:40.030
step first just to show
you that this really

00:06:40.030 --> 00:06:41.800
didn't give us much.

00:06:41.800 --> 00:06:43.660
Let's divide everything by 5.

00:06:43.660 --> 00:06:52.693
So if you divide everything by
5, you get x squared minus 3x

00:06:52.693 --> 00:06:58.720
minus 4/5 is equal to 0.

00:06:58.720 --> 00:07:02.020
So, you might say, hey, why did
we ever do that factoring

00:07:02.020 --> 00:07:02.630
by grouping?

00:07:02.630 --> 00:07:06.140
If we can just always divide by
this leading coefficient,

00:07:06.140 --> 00:07:07.220
we can get rid of that.

00:07:07.220 --> 00:07:09.840
We can always turn this into a 1
or a negative 1 if we divide

00:07:09.840 --> 00:07:10.910
by the right number.

00:07:10.910 --> 00:07:14.410
But notice, by doing that we
got this crazy 4/5 here.

00:07:14.410 --> 00:07:17.630
So this is super hard to do
just using factoring.

00:07:17.630 --> 00:07:19.500
You'd have to say, what two
numbers when I take the

00:07:19.500 --> 00:07:22.100
product is equal to
negative 4/5?

00:07:22.100 --> 00:07:25.210
It's a fraction and when I take
their sum, is equal to

00:07:25.210 --> 00:07:26.140
negative 3?

00:07:26.140 --> 00:07:29.310
This is a hard problem
with factoring.

00:07:29.310 --> 00:07:36.860
This is hard using factoring.

00:07:36.860 --> 00:07:42.080
So, the best thing to do is to
use completing the square.

00:07:42.080 --> 00:07:44.720
So let's think a little bit
about how we can turn this

00:07:44.720 --> 00:07:45.950
into a perfect square.

00:07:45.950 --> 00:07:48.080
What I like to do-- and you'll
see this done some ways and

00:07:48.080 --> 00:07:50.040
I'll show you both ways because
you'll see teachers do

00:07:50.040 --> 00:07:53.880
it both ways-- I like to get
the 4/5 on the other side.

00:07:53.880 --> 00:07:56.900
So let's add 4/5 to both
sides of this equation.

00:07:56.900 --> 00:07:59.980
You don't have to do it this
way, but I like to get the 4/5

00:07:59.980 --> 00:08:01.160
out of the way.

00:08:01.160 --> 00:08:04.010
And then what do we get
if we add 4/5 to both

00:08:04.010 --> 00:08:05.250
sides of this equation?

00:08:05.250 --> 00:08:08.350
The left-hand hand side of the
equation just becomes x

00:08:08.350 --> 00:08:11.800
squared minus 3x,
no 4/5 there.

00:08:11.800 --> 00:08:13.660
I'm going to leave a little
bit of space.

00:08:13.660 --> 00:08:17.790
And that's going to
be equal to 4/5.

00:08:17.790 --> 00:08:19.990
Now, just like the last problem,
we want to turn this

00:08:19.990 --> 00:08:23.350
left-hand side into the perfect
square of a binomial.

00:08:23.350 --> 00:08:24.740
How do we do that?

00:08:24.740 --> 00:08:28.360
Well, we say, well, what number
times 2 is equal to

00:08:28.360 --> 00:08:30.110
negative 3?

00:08:30.110 --> 00:08:32.309
So some number times
2 is negative 3.

00:08:32.309 --> 00:08:35.330
Or we essentially just take
negative 3 and divide it by 2,

00:08:35.330 --> 00:08:37.370
which is negative 3/2.

00:08:37.370 --> 00:08:39.554
And then we square
negative 3/2.

00:08:39.554 --> 00:08:44.840
So in the example, we'll
say a is negative 3/2.

00:08:44.840 --> 00:08:48.380
And if we square negative
3/2, what do we get?

00:08:48.380 --> 00:08:54.100
We get positive 9/4.

00:08:54.100 --> 00:08:56.810
I just took half of this
coefficient, squared it, got

00:08:56.810 --> 00:08:58.010
positive 9/4.

00:08:58.010 --> 00:09:00.720
The whole purpose of doing that
is to turn this left-hand

00:09:00.720 --> 00:09:02.920
side into a perfect square.

00:09:02.920 --> 00:09:05.530
Now, anything you do to one side
of the equation, you've

00:09:05.530 --> 00:09:06.600
got to do to the other side.

00:09:06.600 --> 00:09:11.030
So we added a 9/4 here, let's
add a 9/4 over there.

00:09:11.030 --> 00:09:13.850
And what does our
equation become?

00:09:13.850 --> 00:09:22.530
We get x squared minus 3x plus
9/4 is equal to-- let's see if

00:09:22.530 --> 00:09:24.460
we can get a common
denominator.

00:09:24.460 --> 00:09:29.120
So, 4/5 is the same
thing as 16/20.

00:09:29.120 --> 00:09:31.880
Just multiply the numerator
and denominator by 4.

00:09:31.880 --> 00:09:33.820
Plus over 20.

00:09:33.820 --> 00:09:36.960
9/4 is the same thing
if you multiply the

00:09:36.960 --> 00:09:42.150
numerator by 5 as 45/20.

00:09:42.150 --> 00:09:44.970
And so what is 16 plus 45?

00:09:44.970 --> 00:09:47.020
You see, this is kind of getting
kind of hairy, but

00:09:47.020 --> 00:09:48.930
that's the fun, I guess, of

00:09:48.930 --> 00:09:50.380
completing the square sometimes.

00:09:50.380 --> 00:09:53.420
16 plus 45.

00:09:53.420 --> 00:09:55.780
See that's 55, 61.

00:09:55.780 --> 00:09:59.750
So this is equal to 61/20.

00:09:59.750 --> 00:10:02.680
So let me just rewrite it.

00:10:02.680 --> 00:10:09.480
x squared minus 3x plus
9/4 is equal to 61/20.

00:10:09.480 --> 00:10:11.030
Crazy number.

00:10:11.030 --> 00:10:13.630
Now this, at least on
the left hand side,

00:10:13.630 --> 00:10:15.970
is a perfect square.

00:10:15.970 --> 00:10:21.610
This is the same thing as
x minus 3/2 squared.

00:10:21.610 --> 00:10:24.200
And it was by design.

00:10:24.200 --> 00:10:27.590
Negative 3/2 times negative
3/2 is positive 9/4.

00:10:27.590 --> 00:10:32.790
Negative 3/2 plus negative 3/2
is equal to negative 3.

00:10:32.790 --> 00:10:37.960
So this squared is
equal to 61/20.

00:10:37.960 --> 00:10:43.090
We can take the square root of
both sides and we get x minus

00:10:43.090 --> 00:10:47.820
3/2 is equal to the positive
or the negative

00:10:47.820 --> 00:10:53.320
square root of 61/20.

00:10:53.320 --> 00:10:57.640
And now, we can add 3/2 to both
sides of this equation

00:10:57.640 --> 00:11:03.600
and you get x is equal to
positive 3/2 plus or minus the

00:11:03.600 --> 00:11:07.300
square root of 61/20.

00:11:07.300 --> 00:11:09.290
And this is a crazy number and
it's hopefully obvious you

00:11:09.290 --> 00:11:11.430
would not have been able to-- at
least I would not have been

00:11:11.430 --> 00:11:15.250
able to-- get to this number
just by factoring.

00:11:15.250 --> 00:11:17.260
And if you want their actual
values, you can get your

00:11:17.260 --> 00:11:18.510
calculator out.

00:11:20.620 --> 00:11:22.510
And then let me clear
all of this.

00:11:25.950 --> 00:11:28.760
And 3/2-- let's do the plus
version first. So we want to

00:11:28.760 --> 00:11:33.710
do 3 divided by 2 plus the
second square root.

00:11:33.710 --> 00:11:35.050
We want to pick that little
yellow square root.

00:11:35.050 --> 00:11:46.480
So the square root of 61 divided
by 20, which is 3.24.

00:11:46.480 --> 00:11:52.760
This crazy 3.2464, I'll
just write 3.246.

00:11:52.760 --> 00:12:02.230
So this is approximately equal
to 3.246, and that was just

00:12:02.230 --> 00:12:03.110
the positive version.

00:12:03.110 --> 00:12:06.710
Let's do the subtraction
version.

00:12:06.710 --> 00:12:09.180
So we can actually put our
entry-- if you do second and

00:12:09.180 --> 00:12:11.535
then entry, that we want that
little yellow entry, that's

00:12:11.535 --> 00:12:12.465
why I pressed the
second button.

00:12:12.465 --> 00:12:16.130
So I press enter, it puts in
what we just put, we can just

00:12:16.130 --> 00:12:23.400
change the positive or the
addition to a subtraction and

00:12:23.400 --> 00:12:27.970
you get negative 0.246.

00:12:27.970 --> 00:12:33.800
So you get negative 0.246.

00:12:33.800 --> 00:12:38.200
And you can actually verify
that these satisfy our

00:12:38.200 --> 00:12:39.360
original equation.

00:12:39.360 --> 00:12:42.050
Our original equation
was up here.

00:12:42.050 --> 00:12:43.840
Let me just verify
for one of them.

00:12:47.400 --> 00:12:50.130
So the second answer on your
graphing calculator is the

00:12:50.130 --> 00:12:51.760
last answer you use.

00:12:51.760 --> 00:12:54.160
So if you use a variable answer,
that's this number

00:12:54.160 --> 00:12:55.160
right here.

00:12:55.160 --> 00:13:00.090
So if I have my answer squared--
I'm using answer

00:13:00.090 --> 00:13:02.380
represents negative 0.24.

00:13:02.380 --> 00:13:11.975
Answer squared minus 3 times
answer minus 4/5-- 4 divided

00:13:11.975 --> 00:13:16.030
by 5-- it equals--.

00:13:16.030 --> 00:13:18.490
And this just a little
bit of explanation.

00:13:18.490 --> 00:13:21.860
This doesn't store the entire
number, it goes up to some

00:13:21.860 --> 00:13:22.880
level of precision.

00:13:22.880 --> 00:13:24.910
It stores some number
of digits.

00:13:24.910 --> 00:13:28.930
So when it calculated it using
this stored number right here,

00:13:28.930 --> 00:13:32.240
it got 1 times 10 to
the negative 14.

00:13:32.240 --> 00:13:34.980
So that is 0.0000.

00:13:34.980 --> 00:13:37.100
So that's 13 zeroes
and then a 1.

00:13:37.100 --> 00:13:38.870
A decimal, then 13
zeroes and a 1.

00:13:38.870 --> 00:13:41.060
So this is pretty much 0.

00:13:41.060 --> 00:13:43.550
Or actually, if you got the
exact answer right here, if

00:13:43.550 --> 00:13:46.480
you went through an infinite
level of precision here, or

00:13:46.480 --> 00:13:49.050
maybe if you kept it in this
radical form, you would get

00:13:49.050 --> 00:13:52.390
that it is indeed equal to 0.

00:13:52.390 --> 00:13:55.300
So hopefully you found that
helpful, this whole notion of

00:13:55.300 --> 00:13:56.160
completing the square.

00:13:56.160 --> 00:13:58.670
Now we're going to extend it
to the actual quadratic

00:13:58.670 --> 00:14:01.510
formula that we can use, we
can essentially just plug

00:14:01.510 --> 00:14:03.610
things into to solve any
quadratic equation.


Solving quadratic equations by square roots

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:03.190
In this video, I'm going to do
several examples of quadratic

00:00:03.190 --> 00:00:06.820
equations that are really of a
special form, and it's really

00:00:06.820 --> 00:00:10.150
a bit of warm-up for the next
video that we're going to do

00:00:10.150 --> 00:00:11.670
on completing the square.

00:00:11.670 --> 00:00:13.910
So let me show you what
I'm talking about.

00:00:13.910 --> 00:00:20.960
So let's say I have 4x
plus 1 squared, minus

00:00:20.960 --> 00:00:24.330
8 is equal to 0.

00:00:24.330 --> 00:00:26.740
Now, based on everything we've
done so far, you might be

00:00:26.740 --> 00:00:29.980
tempted to multiply this out,
then subtract 8 from the

00:00:29.980 --> 00:00:33.380
constant you get out here, and
then try to factor it.

00:00:33.380 --> 00:00:37.600
And then you're going to have
x minus something, times x

00:00:37.600 --> 00:00:43.220
minus something else
is equal to 0.

00:00:43.220 --> 00:00:45.300
And you're going to say, oh, one
of these must be equal to

00:00:45.300 --> 00:00:47.990
0, so x could be that or that.

00:00:47.990 --> 00:00:50.760
We're not going to do that this
time, because you might

00:00:50.760 --> 00:00:52.810
see something interesting
here.

00:00:52.810 --> 00:00:56.090
We can solve this without
factoring it.

00:00:56.090 --> 00:00:57.050
And how do we do that?

00:00:57.050 --> 00:00:59.950
Well, what happens if
we add 8 to both

00:00:59.950 --> 00:01:02.070
sides of this equation?

00:01:02.070 --> 00:01:06.480
Then the left-hand side of the
equation becomes 4x plus 1

00:01:06.480 --> 00:01:09.240
squared, and these
8's cancel out.

00:01:09.240 --> 00:01:13.130
The right-hand becomes
just a positive 8.

00:01:13.130 --> 00:01:15.780
Now, what can we do to both
sides of this equation?

00:01:15.780 --> 00:01:18.090
And this is just kind
of straight, vanilla

00:01:18.090 --> 00:01:19.170
equation-solving.

00:01:19.170 --> 00:01:21.930
This isn't any kind of
fancy factoring.

00:01:21.930 --> 00:01:28.700
We can take the square root of
both sides of this equation.

00:01:28.700 --> 00:01:29.600
We could take the square root.

00:01:29.600 --> 00:01:34.410
So 4x plus 1-- I'm just taking
the square root of both sides.

00:01:34.410 --> 00:01:36.840
You take the square root of both
sides, and, of course,

00:01:36.840 --> 00:01:39.740
you want to take the positive
and the negative square root,

00:01:39.740 --> 00:01:43.390
because 4x plus 1 could be the
positive square root of 8, or

00:01:43.390 --> 00:01:45.400
it could be the negative
square root of 8.

00:01:45.400 --> 00:01:50.100
So 4x plus 1 is equal to the
positive or negative

00:01:50.100 --> 00:01:51.860
square root of 8.

00:01:51.860 --> 00:01:55.450
Instead of 8, let me write
8 as 4 times 2.

00:01:55.450 --> 00:01:58.350
We all know that's what 8 is,
and obviously the square root

00:01:58.350 --> 00:02:02.820
of 4x plus 1 squared
is 4x plus 1.

00:02:02.820 --> 00:02:09.500
So we get 4x plus 1 is equal
to-- we can factor out the 4,

00:02:09.500 --> 00:02:12.840
or the square root of 4, which
is 2-- is equal to the plus or

00:02:12.840 --> 00:02:17.250
minus times 2 times the square
root of 2, right?

00:02:17.250 --> 00:02:19.460
Square root of 4 times square
root of 2 is the same thing as

00:02:19.460 --> 00:02:23.290
square root of 4 times the
square root of 2, plus or

00:02:23.290 --> 00:02:26.850
minus the square root of 4
is that 2 right there.

00:02:26.850 --> 00:02:29.970
Now, it might look like a really
bizarro equation, with

00:02:29.970 --> 00:02:32.340
this plus or minus 2 times
the square of 2,

00:02:32.340 --> 00:02:34.020
but it really isn't.

00:02:34.020 --> 00:02:36.010
These are actually two numbers
here, and we're actually

00:02:36.010 --> 00:02:38.210
simultaneously solving
two equations.

00:02:38.210 --> 00:02:43.980
We could write this as 4x plus
1 is equal to the positive 2,

00:02:43.980 --> 00:02:52.360
square root of 2, or 4x plus 1
is equal to negative 2 times

00:02:52.360 --> 00:02:53.510
the square root of 2.

00:02:53.510 --> 00:02:56.110
This one statement is equivalent
to this right here,

00:02:56.110 --> 00:02:59.230
because we have this plus or
minus here, this or statement.

00:02:59.230 --> 00:03:01.710
Let me solve all of these
simultaneously.

00:03:01.710 --> 00:03:05.360
So if I subtract 1 from
both sides of this

00:03:05.360 --> 00:03:07.030
equation, what do I have?

00:03:07.030 --> 00:03:10.330
On the left-hand side, I'm
just left with 4x.

00:03:10.330 --> 00:03:13.330
On the right-hand side, I
have-- you can't really

00:03:13.330 --> 00:03:15.260
mathematically, I mean, you
could do them if you had a

00:03:15.260 --> 00:03:19.210
calculator, but I'll just leave
it as negative 1 plus or

00:03:19.210 --> 00:03:23.570
minus the square root, or 2
times the square root of 2.

00:03:23.570 --> 00:03:25.200
That's what 4x is equal to.

00:03:25.200 --> 00:03:28.560
If we did it here, as two
separate equations, same idea.

00:03:28.560 --> 00:03:32.715
Subtract 1 from both sides of
this equation, you get 4x is

00:03:32.715 --> 00:03:36.310
equal to negative 1 plus 2,
times the square root of 2.

00:03:36.310 --> 00:03:39.150
This equation, subtract
1 from both sides.

00:03:39.150 --> 00:03:44.090
4x is equal to negative
1 minus 2, times the

00:03:44.090 --> 00:03:45.560
square root of 2.

00:03:45.560 --> 00:03:52.970
This statement right here is
completely equivalent to these

00:03:52.970 --> 00:03:53.880
two statements.

00:03:53.880 --> 00:03:57.590
Now, last step, we just have to
divide both sides by 4, so

00:03:57.590 --> 00:04:02.770
you divide both sides by 4,
and you get x is equal to

00:04:02.770 --> 00:04:06.620
negative 1 plus or minus
2, times the square

00:04:06.620 --> 00:04:09.570
root of 2, over 4.

00:04:09.570 --> 00:04:11.670
Now, this statement is
completely equivalent to

00:04:11.670 --> 00:04:16.550
dividing each of these by 4,
and you get x is equal to

00:04:16.550 --> 00:04:21.470
negative 1 plus 2, times the
square root 2, over 4.

00:04:21.470 --> 00:04:22.890
This is one solution.

00:04:22.890 --> 00:04:27.270
And then the other solution is x
is equal to negative 1 minus

00:04:27.270 --> 00:04:31.370
2 roots of 2, all
of that over 4.

00:04:31.370 --> 00:04:35.340
That statement and these two
statements are equivalent.

00:04:35.340 --> 00:04:38.230
And if you want, I encourage you
to-- let's substitute one

00:04:38.230 --> 00:04:41.280
of these back in, just so you
feel confident that something

00:04:41.280 --> 00:04:45.960
as bizarro as one of these
expressions can be a solution

00:04:45.960 --> 00:04:48.970
to a nice, vanilla-looking
equation like this.

00:04:48.970 --> 00:04:50.870
So let's substitute
it back in.

00:04:50.870 --> 00:04:58.290
4 times x, or 4 times negative
1, plus 2 root 2, over 4, plus

00:04:58.290 --> 00:05:03.450
1 squared, minus 8
is equal to 0.

00:05:03.450 --> 00:05:07.350
Now, these 4's cancel out, so
you're left with negative 1

00:05:07.350 --> 00:05:12.260
plus 2 roots 2, plus
1, squared, minus

00:05:12.260 --> 00:05:14.410
8 is equal to 0.

00:05:14.410 --> 00:05:17.410
This negative 1 and this
positive 1 cancel out, so

00:05:17.410 --> 00:05:21.690
you're left with 2 roots
of 2 squared, minus

00:05:21.690 --> 00:05:24.840
8 is equal to 0.

00:05:24.840 --> 00:05:28.680
And then what are you
going to have here?

00:05:28.680 --> 00:05:33.460
So when you square this, you
get 4 times 2, minus 8 is

00:05:33.460 --> 00:05:38.060
equal to 0, which is true.

00:05:38.060 --> 00:05:40.300
8 minus 8 is equal to 0.

00:05:40.300 --> 00:05:42.740
And if you try this one out,
you're going to get the exact

00:05:42.740 --> 00:05:44.180
same answer.

00:05:44.180 --> 00:05:45.220
Let's do another
one like this.

00:05:45.220 --> 00:05:46.960
And remember, these are special
forms where we have

00:05:46.960 --> 00:05:51.470
squares of binomials
in our expression.

00:05:51.470 --> 00:05:53.620
And we're going to see that the
entire quadratic formula

00:05:53.620 --> 00:05:55.870
is actually derived from a
notion like this, because you

00:05:55.870 --> 00:06:00.250
can actually turn any, you can
turn any, quadratic equation

00:06:00.250 --> 00:06:05.050
into a perfect square equalling
something else.

00:06:05.050 --> 00:06:06.525
We'll see that two
videos from now.

00:06:06.525 --> 00:06:08.260
But let's get a little
warmed up just seeing

00:06:08.260 --> 00:06:09.410
this type of thing.

00:06:09.410 --> 00:06:16.650
So let's say you have x squared
minus 10x, plus 25 is

00:06:16.650 --> 00:06:18.680
equal to 9.

00:06:18.680 --> 00:06:20.920
Now, once again your
temptation-- and it's not a

00:06:20.920 --> 00:06:23.960
bad temptation-- would be to
subtract 9 from both sides, so

00:06:23.960 --> 00:06:27.320
you get a 0 on the right-hand
side, but before you do that,

00:06:27.320 --> 00:06:31.050
just inspect this really fast.
And say, hey, is this just

00:06:31.050 --> 00:06:34.140
maybe a perfect square
of a binomial?

00:06:34.140 --> 00:06:38.150
And you see-- well, what two
numbers when I multiply them I

00:06:38.150 --> 00:06:41.760
get positive 25, and when I add
them I get negative 10?

00:06:41.760 --> 00:06:44.380
And hopefully negative
5 jumps out at you.

00:06:44.380 --> 00:06:52.060
So this expression right here is
x minus 5, times x minus 5.

00:06:52.060 --> 00:06:56.495
So this left-hand side can be
written as x minus 5 squared,

00:06:56.495 --> 00:06:59.910
and the right-hand
side is still 9.

00:06:59.910 --> 00:07:01.540
And I want to really
emphasize.

00:07:01.540 --> 00:07:03.860
I don't want this to ruin all of
the training you've gotten

00:07:03.860 --> 00:07:05.280
on factoring so far.

00:07:05.280 --> 00:07:08.260
We can only do this when this
is a perfect square.

00:07:08.260 --> 00:07:12.210
If you got, like, x minus 3,
times x plus 4, and that would

00:07:12.210 --> 00:07:15.120
be equal to 9, that would
be a dead end.

00:07:15.120 --> 00:07:16.580
You wouldn't be able to
really do anything

00:07:16.580 --> 00:07:17.490
constructive with that.

00:07:17.490 --> 00:07:21.080
Only because this is a perfect
square, can we now say x minus

00:07:21.080 --> 00:07:23.640
5 squared is equal to 9, and
now we can take the square

00:07:23.640 --> 00:07:25.060
root of both sides.

00:07:25.060 --> 00:07:31.280
So we could say that x minus 5
is equal to plus or minus 3.

00:07:31.280 --> 00:07:36.050
Add 5 to both sides of this
equation, you get x is equal

00:07:36.050 --> 00:07:43.050
to 5 plus or minus 3, or x is
equal to-- what's 5 plus 3?

00:07:43.050 --> 00:07:49.780
Well, x could be 8 or x could be
equal to 5 minus 3, or x is

00:07:49.780 --> 00:07:51.340
equal to 2.

00:07:51.340 --> 00:07:54.890
Now, we could have done this
equation, this quadratic

00:07:54.890 --> 00:07:58.070
equation, the traditional
way, the way you were

00:07:58.070 --> 00:07:59.160
tempted to do it.

00:07:59.160 --> 00:08:01.600
What happens if you subtract
9 from both

00:08:01.600 --> 00:08:03.230
sides of this equation?

00:08:03.230 --> 00:08:07.650
You'll get x squared
minus 10x.

00:08:07.650 --> 00:08:09.820
And what's 25 minus 9?

00:08:09.820 --> 00:08:16.040
25 minus 9 is 16, and that
would be equal to 0.

00:08:16.040 --> 00:08:18.970
And here, this would be just a
traditional factoring problem,

00:08:18.970 --> 00:08:21.400
the type that we've seen
in the last few videos.

00:08:21.400 --> 00:08:23.710
What two numbers, when you take
their product, you get

00:08:23.710 --> 00:08:29.020
positive 16, and when you sum
them you get negative 10?

00:08:29.020 --> 00:08:31.800
And maybe negative
8 and negative 2

00:08:31.800 --> 00:08:33.390
jump into your brain.

00:08:33.390 --> 00:08:38.954
So we get x minus 8, times
x minus 2 is equal to 0.

00:08:38.954 --> 00:08:43.710
And so x could be equal to 8
or x could be equal to 2.

00:08:43.710 --> 00:08:46.450
That's the fun thing about
algebra: you can do things in

00:08:46.450 --> 00:08:49.670
two completely different ways,
but as long as you do them in

00:08:49.670 --> 00:08:52.210
algebraically-valid ways,
you're not going to get

00:08:52.210 --> 00:08:55.170
different answers.

00:08:55.170 --> 00:08:57.330
And on some level, if you
recognize this, this is easier

00:08:57.330 --> 00:08:58.950
because you didn't have to do
that little game in your head,

00:08:58.950 --> 00:09:01.130
in terms of, oh, what two
numbers, when you multiply

00:09:01.130 --> 00:09:02.840
them you get 16, and when you
add them you get negative 10?

00:09:02.840 --> 00:09:05.340
Here, you just said, OK, this is
x minus 5-- oh, I guess you

00:09:05.340 --> 00:09:06.020
did have to do it.

00:09:06.020 --> 00:09:10.020
You had to say, oh, 5 times 5
is 25, and negative 10 is

00:09:10.020 --> 00:09:11.160
negative 5 plus negative 5.

00:09:11.160 --> 00:09:14.050
So I take that back, you still
have to do that little

00:09:14.050 --> 00:09:15.490
game in your head.

00:09:15.490 --> 00:09:18.700
So let's do another one.

00:09:18.700 --> 00:09:22.880
Let's do one more of these, just
to really get ourselves

00:09:22.880 --> 00:09:25.230
nice and warmed up here.

00:09:25.230 --> 00:09:38.140
So, let's say we have x squared
plus 18x, plus 81 is

00:09:38.140 --> 00:09:41.470
equal to 1.

00:09:41.470 --> 00:09:42.940
So once again, we can
do it in two ways.

00:09:42.940 --> 00:09:46.580
We could subtract 1 from both
sides, or we could recognize

00:09:46.580 --> 00:09:53.140
that this is x plus
9, times x plus 9.

00:09:53.140 --> 00:09:58.010
This right here, 9 times 9
is 81, 9 plus 9 is 18.

00:09:58.010 --> 00:10:01.210
So we can write our equation
as x plus 9

00:10:01.210 --> 00:10:03.760
squared is equal to 1.

00:10:03.760 --> 00:10:06.940
Take the square root of both
sides, you get x plus 9 is

00:10:06.940 --> 00:10:09.240
equal to plus or minus
the square root of 1,

00:10:09.240 --> 00:10:10.910
which is just 1.

00:10:10.910 --> 00:10:15.140
So x is equal to-- subtract 9
from both sides-- negative 9

00:10:15.140 --> 00:10:16.866
plus or minus 1.

00:10:16.866 --> 00:10:21.060
And that means that x could be
equal to-- negative 9 plus 1

00:10:21.060 --> 00:10:26.620
is negative 8, or x could be
equal to-- negative 9 minus 1,

00:10:26.620 --> 00:10:28.930
which is negative 10.

00:10:28.930 --> 00:10:31.560
And once again, you could have
done this the traditional way.

00:10:31.560 --> 00:10:33.730
You could have subtracted 1 from
both sides and you would

00:10:33.730 --> 00:10:41.240
have gotten x squared plus 18x,
plus 80 is equal to 0.

00:10:41.240 --> 00:10:44.370
And you'd say, hey, gee, 8 times
10 is 80, 8 plus 10 is

00:10:44.370 --> 00:10:50.850
18, so you get x plus 8, times
x plus 10 is equal to 0.

00:10:50.850 --> 00:10:54.640
And then you'd get x could be
equal to negative 8, or x

00:10:54.640 --> 00:10:57.360
could be equal to negative 10.

00:10:57.360 --> 00:10:58.710
That was good warm up.

00:10:58.710 --> 00:11:02.250
Now, I think we're ready to
tackle completing the square.


Solving Quadratic Equations by Graphing

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:02.480
What I want to do in this video
is to actually use a

00:00:02.480 --> 00:00:05.320
graphing calculator, get some
experience on the calculator

00:00:05.320 --> 00:00:08.310
itself, for graphing some of
these quadratics and actually

00:00:08.310 --> 00:00:10.540
figuring out the roots
visually by

00:00:10.540 --> 00:00:12.030
looking at those graphs.

00:00:12.030 --> 00:00:17.410
So the first equation I want to
find the x-intercept for is

00:00:17.410 --> 00:00:25.880
y is equal to negative 3x
squared plus 4x minus 1.

00:00:25.880 --> 00:00:28.875
So we're just going to go
straight to the calculator.

00:00:28.875 --> 00:00:31.850
We'll get our TI-85 out here.

00:00:31.850 --> 00:00:35.160
And there are obviously better
things you can use, especially

00:00:35.160 --> 00:00:38.020
on your computer, that are going
to be much more visually

00:00:38.020 --> 00:00:41.400
interesting and in color and
even faster, but many of us

00:00:41.400 --> 00:00:43.380
have a TI-85 at our disposal.

00:00:43.380 --> 00:00:46.180
The TI-85 hasn't changed much
actually since I was in high

00:00:46.180 --> 00:00:48.710
school, or the TI-83 or whatever
else there is out

00:00:48.710 --> 00:00:49.690
there, TI-92.

00:00:49.690 --> 00:00:52.210
And this at least will give you
a sense of how to use it,

00:00:52.210 --> 00:00:55.040
because this is what you have,
and how to analyze the graph

00:00:55.040 --> 00:00:56.370
once you are able to graph it.

00:00:56.370 --> 00:01:00.000
So let's-- I already turned it
on-- so let's input this graph

00:01:00.000 --> 00:01:01.160
right here.

00:01:01.160 --> 00:01:06.555
So we click on GRAPH and then
we'll click on-- this F1 tells

00:01:06.555 --> 00:01:10.070
us to pick this option right
here, this y of x, right?

00:01:10.070 --> 00:01:12.670
So they're saying y is a
function of x, y is indeed a

00:01:12.670 --> 00:01:13.980
function of x.

00:01:13.980 --> 00:01:19.140
So let's click there, and then
that's going to be negative

00:01:19.140 --> 00:01:22.150
3x-- let's see where
is the variable--

00:01:22.150 --> 00:01:23.390
there's the x variable.

00:01:23.390 --> 00:01:31.330
x squared plus 4x, minus 1.

00:01:31.330 --> 00:01:32.890
There is our function.

00:01:32.890 --> 00:01:33.580
Now let's graph it.

00:01:33.580 --> 00:01:36.530
So we want to pick this
choice up here.

00:01:36.530 --> 00:01:38.390
So to pick this second choice--
so notice, if you

00:01:38.390 --> 00:01:41.110
want to pick any of these
choices on the bottom row, you

00:01:41.110 --> 00:01:43.630
can just punch in
these numbers.

00:01:43.630 --> 00:01:45.960
But if you want to pick the
choice that's above, you could

00:01:45.960 --> 00:01:49.040
press 2nd, and then press that,
and then 2nd tells you

00:01:49.040 --> 00:01:51.500
to pick the thing that's
right above that.

00:01:51.500 --> 00:01:54.390
so I'll pick the thing in
the yellow, essentially.

00:01:54.390 --> 00:01:56.150
So I'll do 2nd.

00:01:56.150 --> 00:01:58.340
And I'm selecting the
GRAPH option.

00:02:00.980 --> 00:02:03.520
When I was in high school, this
was all very fancy stuff,

00:02:03.520 --> 00:02:06.260
but now compared to pretty much
most cell phones, this is

00:02:06.260 --> 00:02:07.050
pretty archaic.

00:02:07.050 --> 00:02:09.150
But it gets the job done.

00:02:09.150 --> 00:02:11.940
Now, let's zoom in, because
I really can't see

00:02:11.940 --> 00:02:13.430
the 0's that well.

00:02:13.430 --> 00:02:17.100
So let me zoom here.

00:02:17.100 --> 00:02:21.660
I like to use the box, we zoom
in on a certain area that I

00:02:21.660 --> 00:02:22.195
can define.

00:02:22.195 --> 00:02:25.420
So, I'll select the box, and
then I select where I want to

00:02:25.420 --> 00:02:28.390
start the box for using
the arrow keys.

00:02:28.390 --> 00:02:33.660
Press ENTER, that tells it to
start the box, and then I can

00:02:33.660 --> 00:02:35.820
zoom in using the box.

00:02:35.820 --> 00:02:38.130
I define the box, and
when I have a box I

00:02:38.130 --> 00:02:39.890
like, I press ENTER.

00:02:39.890 --> 00:02:45.670
And it'll redefine the range
of x and y values that it

00:02:45.670 --> 00:02:47.050
calculates the graphing
function.

00:02:47.050 --> 00:02:51.500
Now, I want to figure out what
these x-intercepts are.

00:02:51.500 --> 00:02:53.330
So I'm going to use the
trace function.

00:02:53.330 --> 00:02:54.620
And the trace function
is just going to

00:02:54.620 --> 00:02:56.820
walk along this graph.

00:02:56.820 --> 00:03:01.550
So let me hit 2nd, trace.

00:03:01.550 --> 00:03:04.450
And now I'm walking along this
graph as I go to the right.

00:03:04.450 --> 00:03:06.800
It just increments
the x-value.

00:03:06.800 --> 00:03:11.590
We're getting closer and closer
to y is equal to 0.

00:03:11.590 --> 00:03:13.810
And it looks like it happens
right at about 1.

00:03:13.810 --> 00:03:18.120
Right as I cross 1, I'm crossing
y is equal to 0.

00:03:18.120 --> 00:03:21.260
y is a very small positive
number here, and then it goes

00:03:21.260 --> 00:03:25.040
to becoming a less small, but
still a small negative number,

00:03:25.040 --> 00:03:26.220
right as we cross 1.

00:03:26.220 --> 00:03:30.100
I'm going to evaluate it
directly at x is equal to 1.

00:03:30.100 --> 00:03:32.560
But, I think that's a pretty
good approximation, and we can

00:03:32.560 --> 00:03:34.350
even try it out in
our equation.

00:03:34.350 --> 00:03:35.490
Let's try it out.

00:03:35.490 --> 00:03:40.860
If x is equal to 1-- let
me write it this way.

00:03:40.860 --> 00:03:44.960
y of 1, if we write y is a
function of 1, this is going

00:03:44.960 --> 00:03:49.680
to be equal to negative 3 times
1 squared, plus 4 times

00:03:49.680 --> 00:03:51.200
1, minus 1.

00:03:51.200 --> 00:03:56.350
So this is equal to negative
3 plus 4, minus 1,

00:03:56.350 --> 00:03:58.550
which is indeed 0.

00:03:58.550 --> 00:04:04.710
So the point x is equal to 1, y
is equal to 0 is definitely

00:04:04.710 --> 00:04:07.320
on the graph, and this is
one of our x-intercepts.

00:04:07.320 --> 00:04:10.250
Let's see if we can figure out
the other one using our

00:04:10.250 --> 00:04:12.310
calculator.

00:04:12.310 --> 00:04:14.230
So let's see, we need to figure
out this other point

00:04:14.230 --> 00:04:14.720
right here.

00:04:14.720 --> 00:04:16.839
Let's get the zoom box going.

00:04:16.839 --> 00:04:20.730
Actually, let's try just
to trace there.

00:04:20.730 --> 00:04:22.650
Scrolling all the way.

00:04:22.650 --> 00:04:28.950
And it looks like it's
around 0.33.

00:04:28.950 --> 00:04:32.040
I suspect it's close to 1/3,
but let me zoom in a little

00:04:32.040 --> 00:04:32.490
bit better.

00:04:32.490 --> 00:04:35.770
So let me click GRAPH again,
and click on zoom.

00:04:35.770 --> 00:04:38.320
I'm going to use
the box again.

00:04:38.320 --> 00:04:41.840
And then let me do the top left
corner of the bottom, and

00:04:41.840 --> 00:04:46.240
zoom in really narrowly on
that point right there.

00:04:46.240 --> 00:04:49.015
Really, really zoom in.

00:04:49.015 --> 00:04:51.640
So it's around 0.33 that
I want to zoom in.

00:04:51.640 --> 00:04:53.970
So that's going to be the
beginning of my box, and let

00:04:53.970 --> 00:05:02.390
me go right below the y-axis and
then go right above 0.33.

00:05:02.390 --> 00:05:05.640
And so it's really
zoomed in on that

00:05:05.640 --> 00:05:06.500
y-intercept right there.

00:05:06.500 --> 00:05:08.900
So now if I trace it, I really
should be able to figure out

00:05:08.900 --> 00:05:09.720
what that value is.

00:05:09.720 --> 00:05:14.950
So let me do 2nd, and trace,
and keep scrolling.

00:05:14.950 --> 00:05:18.600
So it's a little bit
more than 0.33.

00:05:18.600 --> 00:05:20.410
It looks like it's really
approaching

00:05:20.410 --> 00:05:22.980
0.3333333, which is 1/3.

00:05:22.980 --> 00:05:25.520
It looks like it's
0.3 repeating.

00:05:25.520 --> 00:05:27.730
Then we're getting to very,
very small y-values.

00:05:27.730 --> 00:05:32.750
So it looks like 1/3, so let's
try out 1/3 and see if we got

00:05:32.750 --> 00:05:34.390
the right answer.

00:05:34.390 --> 00:05:43.500
So, if we have y of 1/3 is equal
to negative 3 times 1/3

00:05:43.500 --> 00:05:48.800
squared, plus 4, times
1/3, minus 1.

00:05:48.800 --> 00:05:54.910
This is equal to-- negative 3
times 1/9 is negative 1/3.

00:05:54.910 --> 00:05:59.540
This is plus 4/3, and then
minus 1 or negative 1,

00:05:59.540 --> 00:06:01.780
depending on how you view it,
it's the same thing as

00:06:01.780 --> 00:06:04.420
minus 3 over 3.

00:06:04.420 --> 00:06:08.180
So this is going to be negative
1 plus 4, minus 3,

00:06:08.180 --> 00:06:11.480
all of that over 3,
which is indeed 0.

00:06:11.480 --> 00:06:13.690
So we also have the
x-intercept.

00:06:13.690 --> 00:06:19.310
x is equal to 1/3,
y is equal to 0.

00:06:19.310 --> 00:06:20.620
Let's do another one.

00:06:20.620 --> 00:06:32.270
Let's say we have y is
equal to 1/2x squared

00:06:32.270 --> 00:06:36.890
minus 2x, plus 3.

00:06:36.890 --> 00:06:40.470
Let's see if we can get its
roots, or the 0's, of this

00:06:40.470 --> 00:06:42.870
equation, where it intersects
the x-axis.

00:06:42.870 --> 00:06:44.390
So let's get our calculator
back.

00:06:44.390 --> 00:06:46.900
And we're going to have to
zoom out of this thing.

00:06:46.900 --> 00:06:49.140
So let me do zoom.

00:06:49.140 --> 00:06:51.770
I think I can just use
the previous zoom.

00:06:51.770 --> 00:06:53.900
Well, the previous zoom
is still a little bit

00:06:53.900 --> 00:06:54.890
not too zoomed in.

00:06:54.890 --> 00:06:57.095
Let me just do the previous
zoom before that.

00:07:04.580 --> 00:07:07.800
And then, I'll do another
zoom-- oh, it just keeps

00:07:07.800 --> 00:07:13.480
alternating between the zooms,
so maybe I have to clear out.

00:07:13.480 --> 00:07:15.490
I am not an expert at this.

00:07:15.490 --> 00:07:16.510
All right.

00:07:16.510 --> 00:07:18.700
So let me just input
my new graph.

00:07:18.700 --> 00:07:19.900
So it's 1/2x squared.

00:07:19.900 --> 00:07:29.470
I could write that as 0.5x
squared minus 2x, plus 3.

00:07:29.470 --> 00:07:31.740
And then I want to graph this.

00:07:34.690 --> 00:07:37.100
Oh, so it's on that super-zoomed
in version, so

00:07:37.100 --> 00:07:38.350
let me zoom out.

00:07:41.690 --> 00:07:43.390
There you go, zoom standard,
that resets it.

00:07:43.390 --> 00:07:43.960
There you go.

00:07:43.960 --> 00:07:46.380
That's all I had to do,
the standard zoom.

00:07:46.380 --> 00:07:47.420
So you see.

00:07:47.420 --> 00:07:49.770
Let's see, where does it
intersect the x-axis?

00:07:49.770 --> 00:07:51.490
Well, it doesn't intersect
the x-axis.

00:07:51.490 --> 00:07:54.160
It just goes down here,
then keeps going up.

00:07:54.160 --> 00:07:59.720
So this quadratic function
right here will have no

00:07:59.720 --> 00:08:00.160
x-intercepts.

00:08:00.160 --> 00:08:01.900
I mean, you can see it
graphically here.

00:08:01.900 --> 00:08:05.020
It does not intersect
the x-axis.

00:08:05.020 --> 00:08:16.240
So, if you were trying to
solve the equation 1/2x

00:08:16.240 --> 00:08:20.250
squared minus 2x, plus 3 is
equal to 0, what you're

00:08:20.250 --> 00:08:22.880
essentially trying to do is say,
what x-values give me y

00:08:22.880 --> 00:08:24.160
is equal to 0 here?

00:08:24.160 --> 00:08:27.950
We just graphed it, it never
intersected the x-axis.

00:08:27.950 --> 00:08:29.410
It just did something
like this.

00:08:29.410 --> 00:08:32.750
It never equaled y
is equal to 0.

00:08:32.750 --> 00:08:38.100
So this right here will have
no real solutions.

00:08:38.100 --> 00:08:41.419
And I use the word real in
the mathematical sense.

00:08:41.419 --> 00:08:43.929
There no real number
solutions.

00:08:43.929 --> 00:08:46.970
We'll see, not too far off in
the future, that there are

00:08:46.970 --> 00:08:52.860
things called complex numbers,
and those can be solutions to

00:08:52.860 --> 00:08:56.170
quadratics that don't intersect
the x-axis, at least

00:08:56.170 --> 00:08:58.900
in kind of the real domain.

00:08:58.900 --> 00:09:00.300
Now let's do one
more of these.

00:09:00.300 --> 00:09:09.480
Let's say we have x squared plus
6x, plus 9 is equal to 0.

00:09:09.480 --> 00:09:12.520
And we should actually be able
to factor this in our head.

00:09:12.520 --> 00:09:15.280
What two numbers add up to
6 and when you multiply

00:09:15.280 --> 00:09:16.025
them you get 9?

00:09:16.025 --> 00:09:17.470
It probably jumps out
of your head.

00:09:17.470 --> 00:09:18.610
It's 3 and 3.

00:09:18.610 --> 00:09:24.960
So this is x plus 3, times
x plus 3 is equal to 0.

00:09:24.960 --> 00:09:28.770
Or-- and we just have to use
one of these-- x plus 3 is

00:09:28.770 --> 00:09:30.270
equal to 0.

00:09:30.270 --> 00:09:34.550
Subtract 3 from both sides.
x is equal to negative 3.

00:09:34.550 --> 00:09:36.310
So this has one 0.

00:09:36.310 --> 00:09:38.930
So how do you think this is
going to look if it only

00:09:38.930 --> 00:09:42.950
intersects the x-axis at
exactly one point?

00:09:42.950 --> 00:09:45.710
Well, the best guess is is that
that one point is going

00:09:45.710 --> 00:09:47.260
to be the vertex.

00:09:47.260 --> 00:09:52.010
So let's graph it just to
verify for ourselves.

00:09:52.010 --> 00:09:56.750
So let me to clear this one
out, and we have x squared

00:09:56.750 --> 00:10:04.590
plus 6x, plus 9.

00:10:04.590 --> 00:10:08.580
And we want to graph it.

00:10:08.580 --> 00:10:10.630
And there you go.

00:10:10.630 --> 00:10:13.840
The vertex is right at x is
equal to negative 3, y is

00:10:13.840 --> 00:10:14.260
equal to 0.

00:10:14.260 --> 00:10:15.990
It just hits it right
over there.

00:10:15.990 --> 00:10:18.150
So our intuition was correct.

00:10:18.150 --> 00:10:20.610
And just as a bit of review--
and I should have pointed this

00:10:20.610 --> 00:10:23.910
out when I was showing you the
other graphs-- notice, this

00:10:23.910 --> 00:10:26.190
has a positive coefficient
on x-squared.

00:10:26.190 --> 00:10:27.820
It is upward-opening.

00:10:27.820 --> 00:10:30.360
The second problem we did also
had a positive coefficient.

00:10:30.360 --> 00:10:32.830
It didn't intersect the x-axis,
but it was also

00:10:32.830 --> 00:10:33.800
upward-opening.

00:10:33.800 --> 00:10:36.960
And if you rewind this video to
the very first problem we

00:10:36.960 --> 00:10:40.400
did, you saw it had a negative
coefficient, and it also had a

00:10:40.400 --> 00:10:45.310
downward-opening parabola, or
downward-opening u-shape.

00:10:45.310 --> 00:10:48.440
Well, anyway, hopefully you
found this little practice

00:10:48.440 --> 00:10:50.800
with the graphing calculator
helpful.


Multiple examples graphing parabolas using roots and vertices

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.800
In this video, I want to show
some examples of graphing

00:00:03.800 --> 00:00:05.620
quadratic functions.

00:00:05.620 --> 00:00:08.790
But before I do that, I want
to make one fairly minor

00:00:08.790 --> 00:00:11.260
correction to the end of the
last video where we did some

00:00:11.260 --> 00:00:12.880
factoring with grouping.

00:00:12.880 --> 00:00:17.350
When I showed you the proof of
why that works, I multiplied

00:00:17.350 --> 00:00:24.180
fx plus g times hx plus j.

00:00:24.180 --> 00:00:27.020
When I multiplied them, I
did the first term, I

00:00:27.020 --> 00:00:29.280
said fx times hx.

00:00:29.280 --> 00:00:32.530
When I did it in the last
video, I just wrote fhx.

00:00:32.530 --> 00:00:34.980
And we know that that's not
true, we have f times h

00:00:34.980 --> 00:00:37.050
times x times x.

00:00:37.050 --> 00:00:39.950
It's fhx squared.

00:00:39.950 --> 00:00:43.690
That times that is fhx squared
and then the rest of it I had

00:00:43.690 --> 00:00:51.200
gotten right, it's fx times j,
so plus we get fjx and then we

00:00:51.200 --> 00:00:56.110
had g times hx so we could
write that as plus ghx.

00:00:56.110 --> 00:01:00.300
And then finally g times
j, which gives us gj.

00:01:00.300 --> 00:01:03.330
So I apologize for that error,
I had forgotten to write the

00:01:03.330 --> 00:01:05.660
exponent there and I think I'd
forgotten to write it in the

00:01:05.660 --> 00:01:08.580
step after that, but it didn't
change the argument for the

00:01:08.580 --> 00:01:10.370
proof, I at least didn't
mess up there.

00:01:10.370 --> 00:01:12.810
So I just want to make that
quick correction in case that

00:01:12.810 --> 00:01:13.635
confused anyone.

00:01:13.635 --> 00:01:16.340
And I also made a small
annotation in that video if

00:01:16.340 --> 00:01:17.890
you're watching it on YouTube.

00:01:17.890 --> 00:01:21.130
With that out of the way, let's
learn to graph some of

00:01:21.130 --> 00:01:23.260
these quadratic functions.

00:01:23.260 --> 00:01:28.020
So let's say I have the function
y is equal to x

00:01:28.020 --> 00:01:32.820
squared minus 2x minus 8.

00:01:32.820 --> 00:01:34.910
And what we're going to do is
we're going to rewrite this

00:01:34.910 --> 00:01:36.860
quadratic expression on
the right-hand side.

00:01:36.860 --> 00:01:39.410
We're going to write it
in intercept form.

00:01:39.410 --> 00:01:40.860
That essentially means
just writing it

00:01:40.860 --> 00:01:42.470
in it factored form.

00:01:42.470 --> 00:01:45.390
Then doing that, we're able to
figure out the x-intercept,

00:01:45.390 --> 00:01:47.950
where we're intersecting the
x-access, and then using that

00:01:47.950 --> 00:01:49.370
we'll be able to figure
out the vertex.

00:01:49.370 --> 00:01:51.110
And I'll show you you what the
vertex is in a second.

00:01:51.110 --> 00:01:53.800
So let's just factor
this quadratic on

00:01:53.800 --> 00:01:54.680
the right-hand side.

00:01:54.680 --> 00:01:58.720
So we get y is equal to-- so
what are two numbers where

00:01:58.720 --> 00:02:01.090
their product is a negative 8,
so they're going to have

00:02:01.090 --> 00:02:05.310
different signs and then their
difference is negative 2?

00:02:05.310 --> 00:02:09.889
Let's see, if we have a negative
4, and plus 2.

00:02:09.889 --> 00:02:12.915
When you take the product, you
get negative 8, when you add

00:02:12.915 --> 00:02:16.160
them, you get negative 2.

00:02:16.160 --> 00:02:21.010
So we could rewrite y as being
equal to x minus 4

00:02:21.010 --> 00:02:25.090
times x plus 2.

00:02:25.090 --> 00:02:27.910
So this is essentially writing
this quadratic function in

00:02:27.910 --> 00:02:29.490
intercept form.

00:02:29.490 --> 00:02:32.870
And now let's figure out where
this intersects the x-axis So

00:02:32.870 --> 00:02:34.580
let me label my axis.

00:02:34.580 --> 00:02:38.210
This is, of course, the x-axis,
that is the y-axis.

00:02:38.210 --> 00:02:40.580
So when will something
intersect the x-axis?

00:02:40.580 --> 00:02:44.090
The x-axis is when y is
equal to 0, right?

00:02:44.090 --> 00:02:46.270
So let's set this equal to 0.

00:02:46.270 --> 00:02:52.390
0 is equal to x minus
4 times x plus 2.

00:02:52.390 --> 00:02:55.690
We've seen this multiple times
before, this means that either

00:02:55.690 --> 00:03:00.500
x minus 4 or x plus 2 or both of
them have to be equal to 0.

00:03:00.500 --> 00:03:05.490
So x minus four is equal to 0
or x plus two is equal to 0.

00:03:05.490 --> 00:03:08.670
So that means x could be equal
to 4-- we're just adding 4 to

00:03:08.670 --> 00:03:12.790
both sides of that equation--
or subtracting 2 from both

00:03:12.790 --> 00:03:17.230
sides of this equation, x could
be equal to negative 2.

00:03:17.230 --> 00:03:21.350
So these points where we
intersect, when x is 4, this

00:03:21.350 --> 00:03:23.840
term is 0 so the whole
thing is 0.

00:03:23.840 --> 00:03:25.390
That's how we solved for this.

00:03:25.390 --> 00:03:31.850
So the points 4, 0, and the
point 2, 0 are going to both

00:03:31.850 --> 00:03:33.030
be on this parabola.

00:03:33.030 --> 00:03:34.510
So let's graph them.

00:03:34.510 --> 00:03:41.280
So 4, 0-- let me do this in a
darker color-- 1, 2 3, 4,

00:03:41.280 --> 00:03:44.210
that's 4, 0.

00:03:44.210 --> 00:03:46.810
And then the other point
is negative 2, 0.

00:03:46.810 --> 00:03:51.200
1, 2 negative 2, 0
is right there.

00:03:51.200 --> 00:03:55.960
So that's the two points where
we intersect the x-axis.

00:03:55.960 --> 00:03:57.900
And now we're going to determine
something called the

00:03:57.900 --> 00:03:59.980
vertex of our parabola.

00:03:59.980 --> 00:04:05.400
And the vertex, just to give you
a sneak preview, parabolas

00:04:05.400 --> 00:04:08.930
are the graphs of quadratic
functions and so the graphic

00:04:08.930 --> 00:04:11.740
is going to look either like an
upward u or a downward u,

00:04:11.740 --> 00:04:13.760
and it's going to be shifted
around on this.

00:04:13.760 --> 00:04:16.839
But the vertex is either
this minimum point or

00:04:16.839 --> 00:04:18.110
this maximum point.

00:04:18.110 --> 00:04:21.750
And the word parabola just
describes this function shape.

00:04:21.750 --> 00:04:25.090
It's the shape of the graph
of a quadratic.

00:04:25.090 --> 00:04:27.560
Let me write that word down,
these are parabolas.

00:04:30.630 --> 00:04:34.045
So to figure out the vertex,
you actually just take the

00:04:34.045 --> 00:04:37.520
x-value that's halfway in
between the two intercepts.

00:04:37.520 --> 00:04:40.700
No matter what, the intercepts
are going to be equidistant

00:04:40.700 --> 00:04:41.950
from the vertex.

00:04:43.980 --> 00:04:47.980
So if that is the graph, those
would be our intercepts and

00:04:47.980 --> 00:04:52.300
then our vertex is going to be
an x-value exactly halfway

00:04:52.300 --> 00:04:53.520
between them.

00:04:53.520 --> 00:04:57.620
So to figure out the x-value of
our vertex, we just average

00:04:57.620 --> 00:05:01.330
these two x-values of
our x-intercepts.

00:05:01.330 --> 00:05:02.770
So let's do that.

00:05:02.770 --> 00:05:08.250
We could say the x for the
vertex, let's call that x for

00:05:08.250 --> 00:05:10.910
the vertex-- that's why I wrote
a v there-- is going to

00:05:10.910 --> 00:05:12.840
be equal to the average
of these.

00:05:12.840 --> 00:05:16.290
So 4 plus negative 2 over 2.

00:05:16.290 --> 00:05:20.810
Which is equal to this 4 minus
2, which is 2 over

00:05:20.810 --> 00:05:22.590
2 is equal to 1.

00:05:22.590 --> 00:05:25.540
So x is going to be equal
to 1 and what is y when

00:05:25.540 --> 00:05:26.970
x is equal to 1?

00:05:26.970 --> 00:05:31.480
So the y for the vertex is
going to be equal to 1

00:05:31.480 --> 00:05:36.500
squared-- I'll just write that
as 1-- minus 2 times 1, so

00:05:36.500 --> 00:05:39.800
minus 2, minus 8.

00:05:39.800 --> 00:05:40.510
So what is this?

00:05:40.510 --> 00:05:45.230
This is 1 minus 10, which
is equal to negative 9.

00:05:45.230 --> 00:05:50.090
So the vertex is going to be
at the point 1, negative 9.

00:05:50.090 --> 00:05:52.830
So we go one and then
we go down nine.

00:05:52.830 --> 00:05:56.010
So x is 1 and then
y is negative 9.

00:05:56.010 --> 00:06:00.860
1, 2, 3, 4, 5, 6, 7, 8, 9.

00:06:00.860 --> 00:06:03.010
So it's right there.

00:06:03.010 --> 00:06:06.010
And so if we were to graph this
parabola, the line of

00:06:06.010 --> 00:06:08.330
symmetry is on the x-value
of the vertex.

00:06:08.330 --> 00:06:11.680
So, I want to be doing it in
a darker color than that.

00:06:11.680 --> 00:06:17.280
So the graph should be symmetric
around x is equal to

00:06:17.280 --> 00:06:20.000
1, and the graph will look
something like this.

00:06:20.000 --> 00:06:22.300
I'll try my best to draw it.

00:06:22.300 --> 00:06:22.555
Bam!

00:06:22.555 --> 00:06:24.990
It'll intersect the x-axis
there, it'll be symmetric,

00:06:24.990 --> 00:06:27.940
it'll look just like that
on the other side.

00:06:27.940 --> 00:06:30.950
And it'll go bam,
just like that.

00:06:30.950 --> 00:06:32.660
I think you get the
general idea.

00:06:32.660 --> 00:06:34.620
And then the other point that
you may or may not be

00:06:34.620 --> 00:06:36.760
interested in is the
actual y-intercept.

00:06:36.760 --> 00:06:39.250
If x is equal to 0, you
immediately see that y is

00:06:39.250 --> 00:06:40.280
equal to negative 8.

00:06:40.280 --> 00:06:44.270
So the point 0, negative 8
should also be on this graph.

00:06:44.270 --> 00:06:45.510
Let's see if I drew it right.

00:06:45.510 --> 00:06:50.520
0, negative eight right there
should also be on that graph.

00:06:50.520 --> 00:06:51.770
Let's do another one of these.

00:06:55.030 --> 00:06:57.810
I'm kind of taking up some
of the space for the next

00:06:57.810 --> 00:07:01.110
problem, so let me clear this.

00:07:01.110 --> 00:07:03.200
Let's do another one.

00:07:03.200 --> 00:07:04.550
All right.

00:07:04.550 --> 00:07:14.170
So let's say we have y is
equal to 2x squared

00:07:14.170 --> 00:07:20.510
plus 6x plus 4.

00:07:20.510 --> 00:07:23.150
So here, the thing that
immediately jumps out at me is

00:07:23.150 --> 00:07:24.860
that I can factor a 2
out of everything.

00:07:24.860 --> 00:07:29.210
So I could rewrite this as y is
equal to 2 times x squared

00:07:29.210 --> 00:07:32.080
plus 3x plus 2.

00:07:32.080 --> 00:07:35.170
That saves me the pain of having
to do the factoring by

00:07:35.170 --> 00:07:37.500
grouping that we talked about
in the last video.

00:07:37.500 --> 00:07:39.830
And this we can factor in a
pretty straightforward way.

00:07:39.830 --> 00:07:47.330
This is 2 times 1 is 2, 2 plus
1 is 3, so this is x plus 2

00:07:47.330 --> 00:07:50.220
times x plus 1, right?

00:07:50.220 --> 00:07:53.590
2 plus 1 is 3, 2 times 1 is 2.

00:07:53.590 --> 00:07:55.710
So that's what y is equal to.

00:07:55.710 --> 00:08:00.500
So if we wanted to know the
x-intercepts, we figure out

00:08:00.500 --> 00:08:03.880
where this expression is equal
to 0, so we said 0 is equal to

00:08:03.880 --> 00:08:08.180
2 times x plus 2
times x plus 1.

00:08:08.180 --> 00:08:10.150
Remember, we want to figure
out where we intercept the

00:08:10.150 --> 00:08:12.020
x-axis, that's the
x-axis there.

00:08:12.020 --> 00:08:14.250
Y is 0 on the x-axis.

00:08:18.710 --> 00:08:20.050
So let's get back
to this problem.

00:08:20.050 --> 00:08:25.220
So either of these can
be equal to 0.

00:08:25.220 --> 00:08:28.170
Obviously the 2 can't be equal
to 0, so we get the situation

00:08:28.170 --> 00:08:34.510
where x minus x plus 2 is equal
to 0 or x plus 1 is

00:08:34.510 --> 00:08:35.190
equal to 0.

00:08:35.190 --> 00:08:37.110
I didn't have to write the
parentheses there.

00:08:37.110 --> 00:08:39.890
Subtract 2 from both sides of
this equation, we get x is

00:08:39.890 --> 00:08:41.340
equal to negative 2.

00:08:41.340 --> 00:08:45.250
Subtract 1 from both sides of
this question or x could be

00:08:45.250 --> 00:08:47.650
equal to negative 1.

00:08:47.650 --> 00:08:50.740
So the point negative 2, 0.

00:08:50.740 --> 00:08:52.930
You put negative 2 here, this
is obviously going to make y

00:08:52.930 --> 00:08:59.300
equal to 0 and the point
negative 1, 0, same argument.

00:08:59.300 --> 00:09:01.050
Those are our x-intercepts.

00:09:01.050 --> 00:09:02.230
Let me graph it.

00:09:02.230 --> 00:09:07.340
We have negative 2,
0 right there.

00:09:07.340 --> 00:09:10.120
Actually let me do this graph
a little bit bigger.

00:09:10.120 --> 00:09:13.900
Let's say that this is right
here is-- let me label it-- in

00:09:13.900 --> 00:09:15.120
double steps.

00:09:15.120 --> 00:09:18.050
So let's say this is negative 1,
let's say this is negative

00:09:18.050 --> 00:09:28.200
2, that is negative 3,
1, 2, this is our 3.

00:09:28.200 --> 00:09:32.640
It's a 1, 2, 3 just like that,
negative 1, negative 2,

00:09:32.640 --> 00:09:33.800
negative 3.

00:09:33.800 --> 00:09:36.390
So, I'm using two blocks
to represent one.

00:09:36.390 --> 00:09:39.900
So the point negative 2, 0
is on our graph, with our

00:09:39.900 --> 00:09:42.570
x-intercept, so negative
2, 0 is right there.

00:09:42.570 --> 00:09:46.200
And then we have negative 1,
0, which is right there.

00:09:46.200 --> 00:09:49.330
And then we want to figure
out the vertex.

00:09:49.330 --> 00:09:51.900
Well, the vertex is going to be
the average of those two,

00:09:51.900 --> 00:09:55.330
so the x for the vertex is going
to be negative 2 plus

00:09:55.330 --> 00:09:59.940
negative 1 over 2, that's
negative 3/2 which is the same

00:09:59.940 --> 00:10:02.590
thing is negative 1.5.

00:10:02.590 --> 00:10:05.720
Now what is y equal to when
x is negative 1.5?

00:10:05.720 --> 00:10:07.270
This is going to be a
little bit involved.

00:10:07.270 --> 00:10:09.000
I'll use actual 3/2.

00:10:09.000 --> 00:10:18.390
So so we get y is equal to 2
times negative 3/2 squared

00:10:18.390 --> 00:10:23.720
plus 6 times negative
3/2 plus 4.

00:10:23.720 --> 00:10:26.310
And let us figure this out.

00:10:26.310 --> 00:10:32.090
So this is equal to 2 times 9
over 4 minus-- you have that

00:10:32.090 --> 00:10:34.950
negative sign there-- and then
we could make this 6 divided

00:10:34.950 --> 00:10:37.720
by 2 is just the same
thing is 3/1.

00:10:37.720 --> 00:10:42.890
So this is minus 9,
3 times 3 plus 4.

00:10:42.890 --> 00:10:49.930
This 2/4 the same thing as
1/2, so this becomes 9/2

00:10:49.930 --> 00:10:52.820
minus 9 plus 4.

00:10:52.820 --> 00:10:54.240
So let's simplify this.

00:10:54.240 --> 00:10:59.290
Minus 9 plus 4 is the same
thing as minus 5, right?

00:10:59.290 --> 00:11:00.990
Minus 9 plus 4.

00:11:00.990 --> 00:11:06.010
Minus 5 we can rewrite is
equal to minus 10/2.

00:11:06.010 --> 00:11:11.520
9 minus 10, all of that
over 2, is minus 1/2.

00:11:11.520 --> 00:11:17.390
So, our vertex of this parabola
is the point minus

00:11:17.390 --> 00:11:21.250
1.5 or negative 1.5,
negative 1/2.

00:11:21.250 --> 00:11:24.980
This is our vertex, so we
can graph it right here.

00:11:24.980 --> 00:11:28.240
Negative 1.5, negative
1/2, our vertex

00:11:28.240 --> 00:11:30.480
is right over there.

00:11:30.480 --> 00:11:32.100
And then we can a graph
this parabola.

00:11:32.100 --> 00:11:33.850
If we want to do the
y-intercept, we can.

00:11:33.850 --> 00:11:37.120
The y-intercept is going
to be here when x is 0,

00:11:37.120 --> 00:11:38.190
y is equal to 4.

00:11:38.190 --> 00:11:39.720
These terms cancel out.

00:11:39.720 --> 00:11:45.120
X is 0, y is 1, 2, 3,
4 right over there.

00:11:45.120 --> 00:11:47.100
And then we can graph
this parabola.

00:11:47.100 --> 00:11:53.030
It will look like that, that's
my best, it should be

00:11:53.030 --> 00:11:58.720
symmetric around x-coordinate of
the vertex just like that.

00:11:58.720 --> 00:12:01.480
Let's do one more of these.

00:12:01.480 --> 00:12:03.920
And, once again, I've used all
my real estate up, let me

00:12:03.920 --> 00:12:06.310
clear this out of the way.

00:12:06.310 --> 00:12:07.760
Let's do one more.

00:12:07.760 --> 00:12:14.130
Let's say that I have y is equal
to negative x squared

00:12:14.130 --> 00:12:16.640
plus 10x minus 21.

00:12:16.640 --> 00:12:19.290
Once again, let's write this
in intercept form, we just

00:12:19.290 --> 00:12:20.940
factor the right-hand side.

00:12:20.940 --> 00:12:23.276
We get y is equal to-- well the
first thing I want to do

00:12:23.276 --> 00:12:26.580
is it is factor this negative
1 out-- negative 1 times x

00:12:26.580 --> 00:12:30.770
squared minus 10x plus 21.

00:12:30.770 --> 00:12:33.400
Now what two numbers, when
I multiply them, they're

00:12:33.400 --> 00:12:34.550
positive 21?

00:12:34.550 --> 00:12:36.860
So they have to be the same
side, and when I add them I

00:12:36.860 --> 00:12:38.080
get negative 10.

00:12:38.080 --> 00:12:41.130
At least in my brain, negative
7 and negative 3 jump out.

00:12:41.130 --> 00:12:44.390
I mean, there's not that many
factors of 21, so 7 and 3 are

00:12:44.390 --> 00:12:47.630
probably two that usually jump
out because they add to 10.

00:12:47.630 --> 00:12:52.590
So you have y is equal to
negative 1 times x minus 3

00:12:52.590 --> 00:12:54.920
times x minus 7.

00:12:54.920 --> 00:12:58.590
Negative 3 plus negative 7 is
negative 10, negative 3 times

00:12:58.590 --> 00:13:01.500
negative 7 is positive 21.

00:13:01.500 --> 00:13:04.280
So if you want to find the
x-intecepts, the x-intercepts

00:13:04.280 --> 00:13:07.390
are going to be where-- so we
could set this whole equation

00:13:07.390 --> 00:13:12.400
equal to 0, negative 1 times x
minus 3 times x minus 7 are

00:13:12.400 --> 00:13:15.900
x-intercepts or the x-values
that make y equal to 0 are

00:13:15.900 --> 00:13:20.090
either the ones that make x
minus 3 equal to 0 or the

00:13:20.090 --> 00:13:23.100
x-values that make x
minus 7 equal to 0.

00:13:23.100 --> 00:13:25.660
Add 3 to both sides of this
equation, you got x is equal

00:13:25.660 --> 00:13:29.940
to 3, add 7 to both sides of
this equation, you get x is

00:13:29.940 --> 00:13:30.830
equal to 7.

00:13:30.830 --> 00:13:35.150
So so we get our two
x-intercepts, the point 3, 0--

00:13:35.150 --> 00:13:44.440
1, 2, 3, 0-- and the point 7,
0-- 1, 2, 3, 4, 5, 6, 7-- 0.

00:13:44.440 --> 00:13:46.790
Now, we know that our vertex
is going to be right in

00:13:46.790 --> 00:13:47.830
between them.

00:13:47.830 --> 00:13:53.000
So the x for the vertex is going
to be 3 plus 7 over 2,

00:13:53.000 --> 00:13:56.130
which is equal to 10/2,
which is equal to 5.

00:13:56.130 --> 00:14:00.370
And to the y-value, let's
figure it out, y for our

00:14:00.370 --> 00:14:07.540
vertex going to be negative 5
squared, so negative 25 plus

00:14:07.540 --> 00:14:14.080
10 times 5 plus 50 minus 21.

00:14:14.080 --> 00:14:14.850
So what is this?

00:14:14.850 --> 00:14:20.380
This is equal to-- we have a
negative 25 and negative 21,

00:14:20.380 --> 00:14:23.680
you add those together you
get a negative 46.

00:14:23.680 --> 00:14:27.780
Add that to 50, you get
y is equal to 4.

00:14:27.780 --> 00:14:32.450
So the vertex is at 5, 4.

00:14:32.450 --> 00:14:37.070
Let me graph that, the
vertex is 5, 4.

00:14:37.070 --> 00:14:40.160
So 1, 2, 3, 4, 5, right
in between.

00:14:40.160 --> 00:14:43.610
And then up four, 1, 2, 3, 4.

00:14:43.610 --> 00:14:46.940
So notice something, in this
graph over here, the last two

00:14:46.940 --> 00:14:49.240
graphs, actually I think I just
cleared them, you saw

00:14:49.240 --> 00:14:53.810
that the vertex was
below the x-axis.

00:14:53.810 --> 00:14:57.450
So, when we plotted the
x-intercept and then the

00:14:57.450 --> 00:15:01.590
vertex, the upward-forming
u-shape was the only option.

00:15:01.590 --> 00:15:05.810
But here, the only option
is a downward u-shape.

00:15:05.810 --> 00:15:08.460
This being either the minimum or
maximum point and going for

00:15:08.460 --> 00:15:11.200
both of these points and having
a u-shape, our u-shape

00:15:11.200 --> 00:15:15.450
is going to have to look
something like this.

00:15:15.450 --> 00:15:20.180
Now,-- let me draw that little
bit better-- the u is going to

00:15:20.180 --> 00:15:23.530
have to look something
like that.

00:15:23.530 --> 00:15:26.070
That's a better shot at it.

00:15:26.070 --> 00:15:29.100
Now notice what was the
difference between this

00:15:29.100 --> 00:15:31.240
equation and the
previous ones?

00:15:31.240 --> 00:15:34.580
The difference was is that this
one had a negative in

00:15:34.580 --> 00:15:36.100
front of the x squared.

00:15:36.100 --> 00:15:37.850
The coefficient in front
of the x squared

00:15:37.850 --> 00:15:39.970
was a negative number.

00:15:39.970 --> 00:15:43.190
And that's why we have kind of
a downward opening parabola.

00:15:43.190 --> 00:15:47.130
If this was a positive number,
we would have an

00:15:47.130 --> 00:15:48.380
upward-opening parabola.

00:15:48.380 --> 00:15:50.760
So that's why this one is kind
of a downward shape and its

00:15:50.760 --> 00:15:53.140
vertex represents
a maximum point.

00:15:53.140 --> 00:15:55.310
It's the highest point
on the parabola.

00:15:55.310 --> 00:15:57.840
In the previous videos, we had
an upward opening, or in the

00:15:57.840 --> 00:16:01.330
previous problems, with an
upward-opening parabola and

00:16:01.330 --> 00:16:03.930
our vertex was the
minimum point.