0:00:00.000,0:00:00.580


0:00:00.580,0:00:02.580
We're on problem 58.

0:00:02.580,0:00:06.040
The graph of the equation y is[br]equal to x squared minus 3x

0:00:06.040,0:00:07.300
minus 4 is shown below.

0:00:07.300,0:00:09.270
Fair enough.

0:00:09.270,0:00:14.380
For what value or values[br]of x is y equal to 0?

0:00:14.380,0:00:16.350
So they're essentially[br]saying is, when does

0:00:16.350,0:00:18.470
this here equal 0?

0:00:18.470,0:00:20.030
They want to know when[br]does y equal 0?

0:00:20.030,0:00:22.590
So what values of x[br]does that happen?

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

0:00:25.180,0:00:26.970
drew us the graph, so let's[br]just inspect it.

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So when does y equal 0?

0:00:28.270,0:00:31.180
So let me draw the line[br]of y is equal to 0.

0:00:31.180,0:00:32.590
So that's right here.

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Let me draw it as a line.

0:00:34.130,0:00:35.160
y equals 0.

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That's y equals 0 right there.

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So what values of x[br]makes y equal 0?

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

0:00:43.970,0:00:47.330
negative 1 and when[br]x is equal to 4.

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So x is equal to negative[br]1 or 4.

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And if we substitute either of[br]these values into this right

0:00:53.480,0:00:57.240
here, we should get[br]y is equal to 0.

0:00:57.240,0:00:57.850
And let's see.

0:00:57.850,0:01:00.850
The choices, do they have[br]negative 1 and 4?

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Yep, sure enough.

0:01:02.180,0:01:05.150
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[br]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[br]right here.

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This is unrelated[br]to this problem.

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

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represents the graph of[br]y is equal to minus x

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squared plus 3?

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So here, just to get an[br]intuition of what parabolas

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look like, because these are all[br]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[br]equal to x squared, what does

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that look like?

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

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

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So if I were to just draw y is[br]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[br]like that.

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I think you're familiar[br]with it.

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Because you're taking x squared,[br]you always get

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positive values.

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Even if you have a negative[br]number squared that still

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becomes a positive.

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

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That's the graph of y[br]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[br]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[br]thing as this graph, but it's

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going to be the negative[br]of whatever you get.

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

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You square any real number[br]and you're going to

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get a positive number.

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Here, you square any real[br]number, this part right here

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becomes positive.

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But then you take the[br]negative of it.

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So this is always going to[br]be a negative number.

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

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whether you go in the positive[br]x direction or the negative x

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direction, this is going[br]to be positive.

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But then you put the[br]negative sign, it's

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going to become negative.

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So the graph is going[br]to look like this.

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The graph is going to[br]look like that.

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

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The graph is going to[br]look like that.

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

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you were to reflect[br]it on the x-axis.

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This is y equals minus[br]x squared.

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So now you're pointing[br]it down.

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The u goes-- opens[br]up downward.

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And hopefully that makes[br]a little bit of sense.

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

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So what is y is equal[br]to x squared plus 3?

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Not minus x squared plus 3,[br]but just x squared plus 3.

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

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every given x is just going[br]to be 3 higher.

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So it's just going to shift[br]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[br]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[br]x squared to minus x squared

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

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just going to shift[br]the graph up by 3.

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So I'll do that in[br]this brown color.

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

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

0:04:09.590,0:04:11.840
So it's going to look[br]something like this.

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It's going to look something[br]like that.

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

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should be opening downward and[br]it should have its y-intercept

0:04:19.740,0:04:23.520
at y is equal to 3.

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If you put x is equal to[br]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[br]that are opening downwards.

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And the y-intercept should[br]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[br]graphed, has x-intercepts of 4

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and minus 3?

0:04:46.180,0:04:48.860
So x-intercepts of 4 and minus[br]3 means that when you

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substitute x of either of[br]these values into the

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equation, you get[br]y is equal to 0.

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Because when y is equal to 0,[br]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[br]by x-intercepts.

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0:05:02.550,0:05:05.240
So how do we set up an equation[br]where if I put in one

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of these numbers I'm[br]going to get 0?

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

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x minus the second root.

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So x minus minus[br]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[br]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[br]plus 3 is a 0.

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So either of these, when you[br]substitute it into this

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expression, you get 0.

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Let's see, which[br]choice is that?

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x minus 4 times x plus 3.

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Have the x-intercepts[br]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[br]right here.

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So x plus 3 is there.

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I see that in a couple[br]of them, right?

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

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

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Because 0 times some number[br]times some constant is still

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going to be 0.

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So if you look at this one[br]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[br]2 times x minus 4.

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So choice B is x plus 3,[br]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[br]equal to 4 minus 3, this--

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

0:06:45.220,0:06:46.260
to be equal to 0.

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Because when x is equal[br]to 4, this is going

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to be equal to 0.

0:06:48.740,0:06:50.450
So 0 times anything times[br]anything else

0:06:50.450,0:06:51.190
is going to be 0.

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Same thing with x is[br]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[br]could put a constant in there

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and it's a little tricky.

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Next problem.

0:07:00.360,0:07:05.790


0:07:05.790,0:07:16.130
OK, so they want to know-- let[br]me copy and paste it-- how

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

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plus 3 intersect the x-axis?

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

0:07:25.450,0:07:26.420
x-axis at all.

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Maybe if you use a quadratic[br]equation

0:07:28.450,0:07:29.650
there are no real solutions.

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So let's just apply[br]the quadratic.

0:07:31.280,0:07:36.670
So the roots or the times that--[br]I guess the x values

0:07:36.670,0:07:42.650
that solve this equation, 2x[br]squared minus 2x plus 3 is

0:07:42.650,0:07:44.030
equal to 0.

0:07:44.030,0:07:45.540
And these are the x[br]values where you

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intersect the x-axis.

0:07:46.730,0:07:47.500
And why do I say that?

0:07:47.500,0:07:50.820
Because the x-axis is the[br]line y is equal to 0.

0:07:50.820,0:07:53.200
So I set y equal to[br]0 and I get this.

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And we know from the quadratic[br]equation the solution to this

0:07:55.530,0:07:58.680
is negative b-- let me do[br]this in another color.

0:07:58.680,0:08:00.210
So negative b.

0:08:00.210,0:08:02.500
So minus minus 2 is 2.

0:08:02.500,0:08:06.370
Plus or minus the square[br]root of b squared.

0:08:06.370,0:08:09.100
Minus 2 squared is 4.

0:08:09.100,0:08:16.320
Minus 4 times a, which is 2.

0:08:16.320,0:08:18.220
Times c, times 3.

0:08:18.220,0:08:19.860
All of that over 2a.

0:08:19.860,0:08:22.670
2 times 2, which is 4.

0:08:22.670,0:08:23.840
Now they don't want[br]us to figure out

0:08:23.840,0:08:24.500
the roots or anything.

0:08:24.500,0:08:25.420
They just want to know,[br]how many times does it

0:08:25.420,0:08:26.570
intersect the axis?

0:08:26.570,0:08:27.750
So let's think about this.

0:08:27.750,0:08:29.140
What happens under this[br]radical sign?

0:08:29.140,0:08:31.350
We have 4 times 2[br]times 3 is 24.

0:08:31.350,0:08:37.970
So this becomes 2 plus or[br]minus 4 minus 24 over 4.

0:08:37.970,0:08:40.960
This is minus 20.

0:08:40.960,0:08:44.010
So you end up with minus 20[br]under the radical sign.

0:08:44.010,0:08:46.220
And we know if we're dealing[br]with real numbers, if we want

0:08:46.220,0:08:50.430
real solutions, you can't take[br]the square root of minus 20.

0:08:50.430,0:08:53.040
So this actually has no[br]solutions or, another way to

0:08:53.040,0:08:56.700
put it is, there is no x values[br]where y is equal to 0.

0:08:56.700,0:08:58.720
Or another way to put it[br]is, this never does

0:08:58.720,0:09:00.820
intersect the x-axis.

0:09:00.820,0:09:01.680
So it's A.

0:09:01.680,0:09:04.270
none.

0:09:04.270,0:09:06.970
What gave that away was the fact[br]that when you apply the

0:09:06.970,0:09:09.750
quadratic equation, you get a[br]negative number under the

0:09:09.750,0:09:10.410
radical sign.

0:09:10.410,0:09:11.570
So if we're dealing[br]with real numbers,

0:09:11.570,0:09:14.340
there's no answer there.

0:09:14.340,0:09:15.590
Next question.

0:09:15.590,0:09:19.740


0:09:19.740,0:09:27.700
62.

0:09:27.700,0:09:30.260
An object that is projected[br]straight down-- oh, this is

0:09:30.260,0:09:32.070
good, this is projectile[br]motion-- is projected straight

0:09:32.070,0:09:35.650
downward with initial velocity[br]v feet per second, Travels a

0:09:35.650,0:09:40.400
distance of s v times t plus[br]16t squared, where t equals

0:09:40.400,0:09:41.580
time in seconds.

0:09:41.580,0:09:45.520
If Ramon is standing on a[br]balcony 84 feet above the

0:09:45.520,0:09:48.170
ground and throws a penny[br]straight down with an initial

0:09:48.170,0:09:51.760
velocity of 10 feet per second,[br]in how many seconds

0:09:51.760,0:09:54.330
will it reach the ground?

0:09:54.330,0:09:58.840
OK, so he's 84 feet[br]above the ground.

0:09:58.840,0:10:00.450
Let's draw a diagram.

0:10:00.450,0:10:01.800
He's 84 feet.

0:10:01.800,0:10:05.180
This is 84 feet above[br]the ground.

0:10:05.180,0:10:07.580
It says, how many seconds will[br]it reach the ground?

0:10:07.580,0:10:10.500
So we essentially want to know[br]how many seconds will it take

0:10:10.500,0:10:13.080
it to travel? s is[br]distance, right?

0:10:13.080,0:10:14.560
So s is equal to 84 feet.

0:10:14.560,0:10:17.690
It has to go down 84 feet.

0:10:17.690,0:10:19.490
Let's see if we can[br]figure this out.

0:10:19.490,0:10:22.240
Now this is something that might[br]be a little bit-- so how

0:10:22.240,0:10:24.620
long does it take it to go 84[br]feet, I guess is the best way

0:10:24.620,0:10:25.540
to think about it.

0:10:25.540,0:10:31.490
So we say 84 is equal to[br]velocity times-- your initial

0:10:31.490,0:10:32.870
velocity times time.

0:10:32.870,0:10:37.090
And your initial velocity[br]is 10 feet per second.

0:10:37.090,0:10:38.710
So it's 10 feet per second.

0:10:38.710,0:10:40.150
Everything is in feet I think.

0:10:40.150,0:10:42.010
Right, everything is--[br]v feet per second.

0:10:42.010,0:10:45.110
Initial velocity of[br]v feet per second.

0:10:45.110,0:10:47.100
So 10 feet per second[br]times time.

0:10:47.100,0:10:48.870
I just substituted what[br]they gave us.

0:10:48.870,0:10:50.670
10 for v.

0:10:50.670,0:10:53.920
Plus 16t squared.

0:10:53.920,0:10:56.590
And now I just solve[br]this quadratic.

0:10:56.590,0:10:58.070
That is a t right there.

0:10:58.070,0:11:00.450
So let me put everything on the[br]same-- let me subtract 84

0:11:00.450,0:11:02.080
from both sides and I'll[br]rearrange a little bit.

0:11:02.080,0:11:11.700
So you get 16t squared plus 10t[br]minus 84 is equal to 0.

0:11:11.700,0:11:13.720
Well I swapped the sides.

0:11:13.720,0:11:14.940
I put these on the left.

0:11:14.940,0:11:16.730
Well, let me just show[br]you what I did.

0:11:16.730,0:11:17.710
I swapped the sides.

0:11:17.710,0:11:23.180
So I made this 16t squared[br]plus 10t equals 84.

0:11:23.180,0:11:24.110
I just swapped them.

0:11:24.110,0:11:26.910
And then I subtracted 84 from[br]both sides to get this.

0:11:26.910,0:11:29.650
And now we just have to[br]solve when t equals 0.

0:11:29.650,0:11:32.190
So I guess the first thing we[br]could do is we could simplify

0:11:32.190,0:11:32.840
this a little bit.

0:11:32.840,0:11:35.080
Everything here is[br]divisible by 2.

0:11:35.080,0:11:39.886
So this is 8t squared plus-- I'm[br]just dividing both sides

0:11:39.886,0:11:41.150
of this equation by 2.

0:11:41.150,0:11:43.930
Plus 5t minus what?

0:11:43.930,0:11:48.670
Minus 42 is equal to 0.

0:11:48.670,0:11:51.200
And then we can use the[br]quadratic equation.

0:11:51.200,0:11:55.200
So what are the solutions? t[br]is equal to negatives b.

0:11:55.200,0:12:02.980
So minus 5 plus or minus the[br]square root of b squared, so

0:12:02.980,0:12:07.880
25, minus 4 times a.

0:12:07.880,0:12:11.360
times 8, times c.[br]c is minus 42.

0:12:11.360,0:12:13.720
So instead of times minus[br]42, let's put a plus

0:12:13.720,0:12:16.540
here and do plus 42.

0:12:16.540,0:12:18.610
Just a negative times a negative[br]is a positive.

0:12:18.610,0:12:22.540
All of that, over 2 times a.

0:12:22.540,0:12:24.652
2a is 16.

0:12:24.652,0:12:26.150
So let's see where[br]that gets me.

0:12:26.150,0:12:32.030
So I t is equal to minus 5 plus[br]or minus the square root.

0:12:32.030,0:12:32.550
What is this?

0:12:32.550,0:12:35.330
25 plus-- let's see.

0:12:35.330,0:12:39.440
4 times 8 times 42.

0:12:39.440,0:12:41.580
That's 32 times 42.

0:12:41.580,0:12:46.840
32 times 42.

0:12:46.840,0:12:49.590
2 times 32 is 64.

0:12:49.590,0:12:50.340
Put a 0.

0:12:50.340,0:12:52.250
4 times 2 is 8.

0:12:52.250,0:12:56.970
4 times 3 is 12.

0:12:56.970,0:13:05.810
You end up with 4,[br]14, 3 and 1.

0:13:05.810,0:13:06.880
So this is 1,344.

0:13:06.880,0:13:08.530
And we're going to[br]add this 25 here.

0:13:08.530,0:13:09.200
So let me see.

0:13:09.200,0:13:11.620
Plus 1,344.

0:13:11.620,0:13:13.800
All of that over 16.

0:13:13.800,0:13:13.970
Let's see.

0:13:13.970,0:13:15.640
1,344 plus 25.

0:13:15.640,0:13:20.280
So it's minus 5 plus or[br]minus the square root

0:13:20.280,0:13:24.370
of-- what is this?

0:13:24.370,0:13:29.150
1,369 over 16.

0:13:29.150,0:13:30.230
And actually, I don't[br]know what the square

0:13:30.230,0:13:31.430
root of 1369 is.

0:13:31.430,0:13:34.040
Let me get the calculator.

0:13:34.040,0:13:37.040
Let me open it up.

0:13:37.040,0:13:38.060
Give me one second.

0:13:38.060,0:13:42.540
Accessories, calculator.

0:13:42.540,0:13:43.510
All right.

0:13:43.510,0:13:48.630
So 1,369.

0:13:48.630,0:13:49.410
37.

0:13:49.410,0:13:50.390
Look at that.

0:13:50.390,0:13:53.440
OK, so it's minus 5[br]plus or minus 37.

0:13:53.440,0:13:55.440
That's the square[br]root of 1,369.

0:13:55.440,0:14:01.600
So minus 5 plus or minus 37 over[br]16 is equal to the time.

0:14:01.600,0:14:03.020
Now we don't have to worry about[br]the minus because that's

0:14:03.020,0:14:03.950
going to give us a[br]negative number.

0:14:03.950,0:14:06.110
Minus 5 minus 37 over 16.

0:14:06.110,0:14:08.780
We don't want a negative time,[br]we want a positive time.

0:14:08.780,0:14:10.080
So let's just do the positive.

0:14:10.080,0:14:13.240
So minus 5 plus 37.

0:14:13.240,0:14:13.450
Let's see.

0:14:13.450,0:14:16.830
Minus 5 plus 37 over 16.

0:14:16.830,0:14:21.340
So that's 32/16, which[br]equals to 2 seconds.

0:14:21.340,0:14:23.290
And that's choice A.

0:14:23.290,0:14:25.590
Anyway, see you in[br]the next video.