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We're on problem 58.

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The graph of the equation y is
equal to x squared minus 3x

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minus 4 is shown below.

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Fair enough.

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For what value or values
of x is y equal to 0?

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So they're essentially
saying is, when does

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this here equal 0?

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They want to know when
does y equal 0?

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So what values of x
does that happen?

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And we could factor this and
solve for the roots, but they

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drew us the graph, so let's
just inspect it.

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So when does y equal 0?

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So let me draw the line
of y is equal to 0.

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So that's right here.

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Let me draw it as a line.

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y equals 0.

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That's y equals 0 right there.

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So what values of x
makes y equal 0?

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If I can see this properly,
it's when x is equal to

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negative 1 and when
x is equal to 4.

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So x is equal to negative
1 or 4.

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And if we substitute either of
these values into this right

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here, we should get
y is equal to 0.

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And let's see.

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The choices, do they have
negative 1 and 4?

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Yep, sure enough.

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Negative 1 and 4 right there.

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Next question, 59.

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Let me copy and paste it.

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OK, I've copied it.

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I'll paste it below.

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I'll do it right
on top of this.

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There you go.

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59.

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Let me erase this stuff
right here.

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This is unrelated
to this problem.

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So they are asking-- let me get
the pen right-- which best

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represents the graph of
y is equal to minus x

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squared plus 3?

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So here, just to get an
intuition of what parabolas

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look like, because these are all
parabolas, or the graph of

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a quadratic equation.

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So if I had the graph of y is
equal to x squared, what does

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that look like?

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Let me just draw a quick and
dirty x- and y-axis.

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And I think you're familiar
with what that looks like.

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So if I were to just draw y is
equal to x squared, that looks

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something like this.

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It looks something like this.

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It will look something
like that.

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I think you're familiar
with it.

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Because you're taking x squared,
you always get

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positive values.

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Even if you have a negative
number squared that still

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becomes a positive.

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And it's symmetric around the
line x is equal to 0.

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That's the graph of y
equals x squared.

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Now let me ask you a question.

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What is the graph of y is equal
to minus x squared?

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Let me do that.

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y is equal to minus x squared.

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So it's essentially the same
thing as this graph, but it's

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going to be the negative
of whatever you get.

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So here, x squared is always
going to be positive.

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You square any real number
and you're going to

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get a positive number.

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Here, you square any real
number, this part right here

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becomes positive.

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But then you take the
negative of it.

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So this is always going to
be a negative number.

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So x equals 0 is still going to
be there, but regardless of

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whether you go in the positive
x direction or the negative x

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direction, this is going
to be positive.

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But then you put the
negative sign, it's

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going to become negative.

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So the graph is going
to look like this.

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The graph is going to
look like that.

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I didn't draw that well, let
me give that another shot.

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The graph is going to
look like that.

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It's essentially like the mirror
image of this one if

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you were to reflect
it on the x-axis.

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This is y equals minus
x squared.

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So now you're pointing
it down.

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The u goes-- opens
up downward.

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And hopefully that makes
a little bit of sense.

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And now, what happens if
you do plus or minus 3?

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So what is y is equal
to x squared plus 3?

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Not minus x squared plus 3,
but just x squared plus 3.

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So if you start with x squared,
now every y value for

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every given x is just going
to be 3 higher.

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So it's just going to shift
the graph up by 3.

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It's going to look like that.

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So if you go from x squared to
x squared plus 3, you're just

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shifting up by 3.

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Similar, if you go from minus
x squared to minus x squared

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plus 3, which is what they gave
us in the problem, you're

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just going to shift
the graph up by 3.

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So I'll do that in
this brown color.

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So it's just going to take this
graph, which is minus x

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squared and you're going
to shift it up by 3.

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So it's going to look
something like this.

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It's going to look something
like that.

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So let's see, out of all the
choices they gave us, it

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should be opening downward and
it should have its y-intercept

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at y is equal to 3.

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If you put x is equal to
0, y is equal to 3.

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So let's see.

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It's opening downwards.

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So these two are the only two
that are opening downwards.

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And the y-intercept should
be at 3 because we

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shifted it up by 3.

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So this is the choice B.

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Problem 60.

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Which quadratic funtion, when
graphed, has x-intercepts of 4

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and minus 3?

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So x-intercepts of 4 and minus
3 means that when you

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substitute x of either of
these values into the

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equation, you get
y is equal to 0.

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Because when y is equal to 0,
you're at the x-intercepts.

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This is when y equals to 0.

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So that's what they mean
by x-intercepts.

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So how do we set up an equation
where if I put in one

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of these numbers I'm
going to get 0?

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Well, if I make it the product
of x minus the first root and

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x minus the second root.

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So x minus minus
3 is x plus 3.

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So think about it.

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If you put 4 here for x, you
get 4 minus 4, which is 0.

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Times 4 plus 3 is 7.

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So 0 times 7 is 0.

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So that works.

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And then, for minus 3.

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Minus 3 minus 4 is minus 7.

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But then minus 3
plus 3 is a 0.

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So either of these, when you
substitute it into this

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expression, you get 0.

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Let's see, which
choice is that?

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x minus 4 times x plus 3.

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Have the x-intercepts
of 4 and minus 3.

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Right.

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This should be right.

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They're being tricky
right here.

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So x plus 3 is there.

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I see that in a couple
of them, right?

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But I don't see the x
minus 4 anywhere.

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But that's because we can
multiply this by any constant.

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Because 0 times some number
times some constant is still

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going to be 0.

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So if you look at this one
right here, 2x minus 8.

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We could factor out a 2.

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That's the same thing as
2 times x minus 4.

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So choice B is x plus 3,
times x minus 4, times

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some constant 2.

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So choice B is our answer.

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If this is equal to 0 when x is
equal to 4 minus 3, this--

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any constant, x minus 4 times x
plus 3--I that's still going

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to be equal to 0.

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Because when x is equal
to 4, this is going

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to be equal to 0.

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So 0 times anything times
anything else

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is going to be 0.

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Same thing with x is
equal to minus 3.

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So they just put a 2 here.

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That's a good problem.

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It made you realize that you
could put a constant in there

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and it's a little tricky.

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Next problem.

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OK, so they want to know-- let
me copy and paste it-- how

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many times does the graph of y
equals 2x squared minus 2x

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plus 3 intersect the x-axis?

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So the easiest thing-- because
maybe it doesn't intersect the

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x-axis at all.

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Maybe if you use a quadratic
equation

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there are no real solutions.

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So let's just apply
the quadratic.

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So the roots or the times that--
I guess the x values

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that solve this equation, 2x
squared minus 2x plus 3 is

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equal to 0.

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And these are the x
values where you

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intersect the x-axis.

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And why do I say that?

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Because the x-axis is the
line y is equal to 0.

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So I set y equal to
0 and I get this.

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And we know from the quadratic
equation the solution to this

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is negative b-- let me do
this in another color.

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So negative b.

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So minus minus 2 is 2.

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Plus or minus the square
root of b squared.

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Minus 2 squared is 4.

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Minus 4 times a, which is 2.

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Times c, times 3.

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All of that over 2a.

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2 times 2, which is 4.

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Now they don't want
us to figure out

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the roots or anything.

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They just want to know,
how many times does it

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intersect the axis?

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So let's think about this.

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What happens under this
radical sign?

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We have 4 times 2
times 3 is 24.

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So this becomes 2 plus or
minus 4 minus 24 over 4.

206
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This is minus 20.

207
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So you end up with minus 20
under the radical sign.

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And we know if we're dealing
with real numbers, if we want

209
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real solutions, you can't take
the square root of minus 20.

210
00:08:50,430 --> 00:08:53,040
So this actually has no
solutions or, another way to

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put it is, there is no x values
where y is equal to 0.

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00:08:56,700 --> 00:08:58,720
Or another way to put it
is, this never does

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00:08:58,720 --> 00:09:00,820
intersect the x-axis.

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So it's A.

215
00:09:01,680 --> 00:09:04,270
none.

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00:09:04,270 --> 00:09:06,970
What gave that away was the fact
that when you apply the

217
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quadratic equation, you get a
negative number under the

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00:09:09,750 --> 00:09:10,410
radical sign.

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So if we're dealing
with real numbers,

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there's no answer there.

221
00:09:14,340 --> 00:09:15,590
Next question.

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00:09:15,590 --> 00:09:19,740


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00:09:19,740 --> 00:09:27,700
62.

224
00:09:27,700 --> 00:09:30,260
An object that is projected
straight down-- oh, this is

225
00:09:30,260 --> 00:09:32,070
good, this is projectile
motion-- is projected straight

226
00:09:32,070 --> 00:09:35,650
downward with initial velocity
v feet per second, Travels a

227
00:09:35,650 --> 00:09:40,400
distance of s v times t plus
16t squared, where t equals

228
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time in seconds.

229
00:09:41,580 --> 00:09:45,520
If Ramon is standing on a
balcony 84 feet above the

230
00:09:45,520 --> 00:09:48,170
ground and throws a penny
straight down with an initial

231
00:09:48,170 --> 00:09:51,760
velocity of 10 feet per second,
in how many seconds

232
00:09:51,760 --> 00:09:54,330
will it reach the ground?

233
00:09:54,330 --> 00:09:58,840
OK, so he's 84 feet
above the ground.

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Let's draw a diagram.

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He's 84 feet.

236
00:10:01,800 --> 00:10:05,180
This is 84 feet above
the ground.

237
00:10:05,180 --> 00:10:07,580
It says, how many seconds will
it reach the ground?

238
00:10:07,580 --> 00:10:10,500
So we essentially want to know
how many seconds will it take

239
00:10:10,500 --> 00:10:13,080
it to travel? s is
distance, right?

240
00:10:13,080 --> 00:10:14,560
So s is equal to 84 feet.

241
00:10:14,560 --> 00:10:17,690
It has to go down 84 feet.

242
00:10:17,690 --> 00:10:19,490
Let's see if we can
figure this out.

243
00:10:19,490 --> 00:10:22,240
Now this is something that might
be a little bit-- so how

244
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long does it take it to go 84
feet, I guess is the best way

245
00:10:24,620 --> 00:10:25,540
to think about it.

246
00:10:25,540 --> 00:10:31,490
So we say 84 is equal to
velocity times-- your initial

247
00:10:31,490 --> 00:10:32,870
velocity times time.

248
00:10:32,870 --> 00:10:37,090
And your initial velocity
is 10 feet per second.

249
00:10:37,090 --> 00:10:38,710
So it's 10 feet per second.

250
00:10:38,710 --> 00:10:40,150
Everything is in feet I think.

251
00:10:40,150 --> 00:10:42,010
Right, everything is--
v feet per second.

252
00:10:42,010 --> 00:10:45,110
Initial velocity of
v feet per second.

253
00:10:45,110 --> 00:10:47,100
So 10 feet per second
times time.

254
00:10:47,100 --> 00:10:48,870
I just substituted what
they gave us.

255
00:10:48,870 --> 00:10:50,670
10 for v.

256
00:10:50,670 --> 00:10:53,920
Plus 16t squared.

257
00:10:53,920 --> 00:10:56,590
And now I just solve
this quadratic.

258
00:10:56,590 --> 00:10:58,070
That is a t right there.

259
00:10:58,070 --> 00:11:00,450
So let me put everything on the
same-- let me subtract 84

260
00:11:00,450 --> 00:11:02,080
from both sides and I'll
rearrange a little bit.

261
00:11:02,080 --> 00:11:11,700
So you get 16t squared plus 10t
minus 84 is equal to 0.

262
00:11:11,700 --> 00:11:13,720
Well I swapped the sides.

263
00:11:13,720 --> 00:11:14,940
I put these on the left.

264
00:11:14,940 --> 00:11:16,730
Well, let me just show
you what I did.

265
00:11:16,730 --> 00:11:17,710
I swapped the sides.

266
00:11:17,710 --> 00:11:23,180
So I made this 16t squared
plus 10t equals 84.

267
00:11:23,180 --> 00:11:24,110
I just swapped them.

268
00:11:24,110 --> 00:11:26,910
And then I subtracted 84 from
both sides to get this.

269
00:11:26,910 --> 00:11:29,650
And now we just have to
solve when t equals 0.

270
00:11:29,650 --> 00:11:32,190
So I guess the first thing we
could do is we could simplify

271
00:11:32,190 --> 00:11:32,840
this a little bit.

272
00:11:32,840 --> 00:11:35,080
Everything here is
divisible by 2.

273
00:11:35,080 --> 00:11:39,886
So this is 8t squared plus-- I'm
just dividing both sides

274
00:11:39,886 --> 00:11:41,150
of this equation by 2.

275
00:11:41,150 --> 00:11:43,930
Plus 5t minus what?

276
00:11:43,930 --> 00:11:48,670
Minus 42 is equal to 0.

277
00:11:48,670 --> 00:11:51,200
And then we can use the
quadratic equation.

278
00:11:51,200 --> 00:11:55,200
So what are the solutions? t
is equal to negatives b.

279
00:11:55,200 --> 00:12:02,980
So minus 5 plus or minus the
square root of b squared, so

280
00:12:02,980 --> 00:12:07,880
25, minus 4 times a.

281
00:12:07,880 --> 00:12:11,360
times 8, times c.
c is minus 42.

282
00:12:11,360 --> 00:12:13,720
So instead of times minus
42, let's put a plus

283
00:12:13,720 --> 00:12:16,540
here and do plus 42.

284
00:12:16,540 --> 00:12:18,610
Just a negative times a negative
is a positive.

285
00:12:18,610 --> 00:12:22,540
All of that, over 2 times a.

286
00:12:22,540 --> 00:12:24,652
2a is 16.

287
00:12:24,652 --> 00:12:26,150
So let's see where
that gets me.

288
00:12:26,150 --> 00:12:32,030
So I t is equal to minus 5 plus
or minus the square root.

289
00:12:32,030 --> 00:12:32,550
What is this?

290
00:12:32,550 --> 00:12:35,330
25 plus-- let's see.

291
00:12:35,330 --> 00:12:39,440
4 times 8 times 42.

292
00:12:39,440 --> 00:12:41,580
That's 32 times 42.

293
00:12:41,580 --> 00:12:46,840
32 times 42.

294
00:12:46,840 --> 00:12:49,590
2 times 32 is 64.

295
00:12:49,590 --> 00:12:50,340
Put a 0.

296
00:12:50,340 --> 00:12:52,250
4 times 2 is 8.

297
00:12:52,250 --> 00:12:56,970
4 times 3 is 12.

298
00:12:56,970 --> 00:13:05,810
You end up with 4,
14, 3 and 1.

299
00:13:05,810 --> 00:13:06,880
So this is 1,344.

300
00:13:06,880 --> 00:13:08,530
And we're going to
add this 25 here.

301
00:13:08,530 --> 00:13:09,200
So let me see.

302
00:13:09,200 --> 00:13:11,620
Plus 1,344.

303
00:13:11,620 --> 00:13:13,800
All of that over 16.

304
00:13:13,800 --> 00:13:13,970
Let's see.

305
00:13:13,970 --> 00:13:15,640
1,344 plus 25.

306
00:13:15,640 --> 00:13:20,280
So it's minus 5 plus or
minus the square root

307
00:13:20,280 --> 00:13:24,370
of-- what is this?

308
00:13:24,370 --> 00:13:29,150
1,369 over 16.

309
00:13:29,150 --> 00:13:30,230
And actually, I don't
know what the square

310
00:13:30,230 --> 00:13:31,430
root of 1369 is.

311
00:13:31,430 --> 00:13:34,040
Let me get the calculator.

312
00:13:34,040 --> 00:13:37,040
Let me open it up.

313
00:13:37,040 --> 00:13:38,060
Give me one second.

314
00:13:38,060 --> 00:13:42,540
Accessories, calculator.

315
00:13:42,540 --> 00:13:43,510
All right.

316
00:13:43,510 --> 00:13:48,630
So 1,369.

317
00:13:48,630 --> 00:13:49,410
37.

318
00:13:49,410 --> 00:13:50,390
Look at that.

319
00:13:50,390 --> 00:13:53,440
OK, so it's minus 5
plus or minus 37.

320
00:13:53,440 --> 00:13:55,440
That's the square
root of 1,369.

321
00:13:55,440 --> 00:14:01,600
So minus 5 plus or minus 37 over
16 is equal to the time.

322
00:14:01,600 --> 00:14:03,020
Now we don't have to worry about
the minus because that's

323
00:14:03,020 --> 00:14:03,950
going to give us a
negative number.

324
00:14:03,950 --> 00:14:06,110
Minus 5 minus 37 over 16.

325
00:14:06,110 --> 00:14:08,780
We don't want a negative time,
we want a positive time.

326
00:14:08,780 --> 00:14:10,080
So let's just do the positive.

327
00:14:10,080 --> 00:14:13,240
So minus 5 plus 37.

328
00:14:13,240 --> 00:14:13,450
Let's see.

329
00:14:13,450 --> 00:14:16,830
Minus 5 plus 37 over 16.

330
00:14:16,830 --> 00:14:21,340
So that's 32/16, which
equals to 2 seconds.

331
00:14:21,340 --> 00:14:23,290
And that's choice A.

332
00:14:23,290 --> 00:14:25,590
Anyway, see you in
the next video.