[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.58,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.58,0:00:02.58,Default,,0000,0000,0000,,We're on problem 58.
Dialogue: 0,0:00:02.58,0:00:06.04,Default,,0000,0000,0000,,The graph of the equation y is\Nequal to x squared minus 3x
Dialogue: 0,0:00:06.04,0:00:07.30,Default,,0000,0000,0000,,minus 4 is shown below.
Dialogue: 0,0:00:07.30,0:00:09.27,Default,,0000,0000,0000,,Fair enough.
Dialogue: 0,0:00:09.27,0:00:14.38,Default,,0000,0000,0000,,For what value or values\Nof x is y equal to 0?
Dialogue: 0,0:00:14.38,0:00:16.35,Default,,0000,0000,0000,,So they're essentially\Nsaying is, when does
Dialogue: 0,0:00:16.35,0:00:18.47,Default,,0000,0000,0000,,this here equal 0?
Dialogue: 0,0:00:18.47,0:00:20.03,Default,,0000,0000,0000,,They want to know when\Ndoes y equal 0?
Dialogue: 0,0:00:20.03,0:00:22.59,Default,,0000,0000,0000,,So what values of x\Ndoes that happen?
Dialogue: 0,0:00:22.59,0:00:25.18,Default,,0000,0000,0000,,And we could factor this and\Nsolve for the roots, but they
Dialogue: 0,0:00:25.18,0:00:26.97,Default,,0000,0000,0000,,drew us the graph, so let's\Njust inspect it.
Dialogue: 0,0:00:26.97,0:00:28.27,Default,,0000,0000,0000,,So when does y equal 0?
Dialogue: 0,0:00:28.27,0:00:31.18,Default,,0000,0000,0000,,So let me draw the line\Nof y is equal to 0.
Dialogue: 0,0:00:31.18,0:00:32.59,Default,,0000,0000,0000,,So that's right here.
Dialogue: 0,0:00:32.59,0:00:34.13,Default,,0000,0000,0000,,Let me draw it as a line.
Dialogue: 0,0:00:34.13,0:00:35.16,Default,,0000,0000,0000,,y equals 0.
Dialogue: 0,0:00:35.16,0:00:36.97,Default,,0000,0000,0000,,That's y equals 0 right there.
Dialogue: 0,0:00:36.97,0:00:41.24,Default,,0000,0000,0000,,So what values of x\Nmakes y equal 0?
Dialogue: 0,0:00:41.24,0:00:43.97,Default,,0000,0000,0000,,If I can see this properly,\Nit's when x is equal to
Dialogue: 0,0:00:43.97,0:00:47.33,Default,,0000,0000,0000,,negative 1 and when\Nx is equal to 4.
Dialogue: 0,0:00:47.33,0:00:50.72,Default,,0000,0000,0000,,So x is equal to negative\N1 or 4.
Dialogue: 0,0:00:50.72,0:00:53.48,Default,,0000,0000,0000,,And if we substitute either of\Nthese values into this right
Dialogue: 0,0:00:53.48,0:00:57.24,Default,,0000,0000,0000,,here, we should get\Ny is equal to 0.
Dialogue: 0,0:00:57.24,0:00:57.85,Default,,0000,0000,0000,,And let's see.
Dialogue: 0,0:00:57.85,0:01:00.85,Default,,0000,0000,0000,,The choices, do they have\Nnegative 1 and 4?
Dialogue: 0,0:01:00.85,0:01:02.18,Default,,0000,0000,0000,,Yep, sure enough.
Dialogue: 0,0:01:02.18,0:01:05.15,Default,,0000,0000,0000,,Negative 1 and 4 right there.
Dialogue: 0,0:01:05.15,0:01:12.45,Default,,0000,0000,0000,,Next question, 59.
Dialogue: 0,0:01:12.45,0:01:16.72,Default,,0000,0000,0000,,Let me copy and paste it.
Dialogue: 0,0:01:16.72,0:01:20.19,Default,,0000,0000,0000,,OK, I've copied it.
Dialogue: 0,0:01:20.19,0:01:22.45,Default,,0000,0000,0000,,I'll paste it below.
Dialogue: 0,0:01:22.45,0:01:24.30,Default,,0000,0000,0000,,I'll do it right\Non top of this.
Dialogue: 0,0:01:24.30,0:01:24.73,Default,,0000,0000,0000,,There you go.
Dialogue: 0,0:01:24.73,0:01:25.28,Default,,0000,0000,0000,,59.
Dialogue: 0,0:01:25.28,0:01:28.56,Default,,0000,0000,0000,,Let me erase this stuff\Nright here.
Dialogue: 0,0:01:28.56,0:01:31.96,Default,,0000,0000,0000,,This is unrelated\Nto this problem.
Dialogue: 0,0:01:31.96,0:01:35.97,Default,,0000,0000,0000,,So they are asking-- let me get\Nthe pen right-- which best
Dialogue: 0,0:01:35.97,0:01:39.06,Default,,0000,0000,0000,,represents the graph of\Ny is equal to minus x
Dialogue: 0,0:01:39.06,0:01:41.47,Default,,0000,0000,0000,,squared plus 3?
Dialogue: 0,0:01:41.47,0:01:44.02,Default,,0000,0000,0000,,So here, just to get an\Nintuition of what parabolas
Dialogue: 0,0:01:44.02,0:01:46.20,Default,,0000,0000,0000,,look like, because these are all\Nparabolas, or the graph of
Dialogue: 0,0:01:46.20,0:01:48.01,Default,,0000,0000,0000,,a quadratic equation.
Dialogue: 0,0:01:48.01,0:01:52.23,Default,,0000,0000,0000,,So if I had the graph of y is\Nequal to x squared, what does
Dialogue: 0,0:01:52.23,0:01:54.01,Default,,0000,0000,0000,,that look like?
Dialogue: 0,0:01:54.01,0:01:57.38,Default,,0000,0000,0000,,Let me just draw a quick and\Ndirty x- and y-axis.
Dialogue: 0,0:01:57.38,0:02:00.51,Default,,0000,0000,0000,,And I think you're familiar\Nwith what that looks like.
Dialogue: 0,0:02:00.51,0:02:02.75,Default,,0000,0000,0000,,So if I were to just draw y is\Nequal to x squared, that looks
Dialogue: 0,0:02:02.75,0:02:03.86,Default,,0000,0000,0000,,something like this.
Dialogue: 0,0:02:03.86,0:02:07.75,Default,,0000,0000,0000,,It looks something like this.
Dialogue: 0,0:02:07.75,0:02:09.21,Default,,0000,0000,0000,,It will look something\Nlike that.
Dialogue: 0,0:02:09.21,0:02:10.93,Default,,0000,0000,0000,,I think you're familiar\Nwith it.
Dialogue: 0,0:02:10.93,0:02:12.27,Default,,0000,0000,0000,,Because you're taking x squared,\Nyou always get
Dialogue: 0,0:02:12.27,0:02:13.16,Default,,0000,0000,0000,,positive values.
Dialogue: 0,0:02:13.16,0:02:14.93,Default,,0000,0000,0000,,Even if you have a negative\Nnumber squared that still
Dialogue: 0,0:02:14.93,0:02:16.87,Default,,0000,0000,0000,,becomes a positive.
Dialogue: 0,0:02:16.87,0:02:22.74,Default,,0000,0000,0000,,And it's symmetric around the\Nline x is equal to 0.
Dialogue: 0,0:02:22.74,0:02:24.18,Default,,0000,0000,0000,,That's the graph of y\Nequals x squared.
Dialogue: 0,0:02:24.18,0:02:24.91,Default,,0000,0000,0000,,Now let me ask you a question.
Dialogue: 0,0:02:24.91,0:02:28.60,Default,,0000,0000,0000,,What is the graph of y is equal\Nto minus x squared?
Dialogue: 0,0:02:28.60,0:02:29.37,Default,,0000,0000,0000,,Let me do that.
Dialogue: 0,0:02:29.37,0:02:32.68,Default,,0000,0000,0000,,y is equal to minus x squared.
Dialogue: 0,0:02:32.68,0:02:35.43,Default,,0000,0000,0000,,So it's essentially the same\Nthing as this graph, but it's
Dialogue: 0,0:02:35.43,0:02:39.15,Default,,0000,0000,0000,,going to be the negative\Nof whatever you get.
Dialogue: 0,0:02:39.15,0:02:41.86,Default,,0000,0000,0000,,So here, x squared is always\Ngoing to be positive.
Dialogue: 0,0:02:41.86,0:02:43.51,Default,,0000,0000,0000,,You square any real number\Nand you're going to
Dialogue: 0,0:02:43.51,0:02:44.67,Default,,0000,0000,0000,,get a positive number.
Dialogue: 0,0:02:44.67,0:02:47.37,Default,,0000,0000,0000,,Here, you square any real\Nnumber, this part right here
Dialogue: 0,0:02:47.37,0:02:49.07,Default,,0000,0000,0000,,becomes positive.
Dialogue: 0,0:02:49.07,0:02:50.43,Default,,0000,0000,0000,,But then you take the\Nnegative of it.
Dialogue: 0,0:02:50.43,0:02:52.47,Default,,0000,0000,0000,,So this is always going to\Nbe a negative number.
Dialogue: 0,0:02:52.47,0:02:57.52,Default,,0000,0000,0000,,So x equals 0 is still going to\Nbe there, but regardless of
Dialogue: 0,0:02:57.52,0:02:59.67,Default,,0000,0000,0000,,whether you go in the positive\Nx direction or the negative x
Dialogue: 0,0:02:59.67,0:03:01.55,Default,,0000,0000,0000,,direction, this is going\Nto be positive.
Dialogue: 0,0:03:01.55,0:03:02.46,Default,,0000,0000,0000,,But then you put the\Nnegative sign, it's
Dialogue: 0,0:03:02.46,0:03:03.33,Default,,0000,0000,0000,,going to become negative.
Dialogue: 0,0:03:03.33,0:03:05.81,Default,,0000,0000,0000,,So the graph is going\Nto look like this.
Dialogue: 0,0:03:05.81,0:03:08.58,Default,,0000,0000,0000,,The graph is going to\Nlook like that.
Dialogue: 0,0:03:08.58,0:03:11.65,Default,,0000,0000,0000,,I didn't draw that well, let\Nme give that another shot.
Dialogue: 0,0:03:11.65,0:03:15.10,Default,,0000,0000,0000,,The graph is going to\Nlook like that.
Dialogue: 0,0:03:15.10,0:03:18.27,Default,,0000,0000,0000,,It's essentially like the mirror\Nimage of this one if
Dialogue: 0,0:03:18.27,0:03:21.21,Default,,0000,0000,0000,,you were to reflect\Nit on the x-axis.
Dialogue: 0,0:03:21.21,0:03:22.82,Default,,0000,0000,0000,,This is y equals minus\Nx squared.
Dialogue: 0,0:03:22.82,0:03:24.15,Default,,0000,0000,0000,,So now you're pointing\Nit down.
Dialogue: 0,0:03:24.15,0:03:26.44,Default,,0000,0000,0000,,The u goes-- opens\Nup downward.
Dialogue: 0,0:03:26.44,0:03:28.99,Default,,0000,0000,0000,,And hopefully that makes\Na little bit of sense.
Dialogue: 0,0:03:28.99,0:03:31.20,Default,,0000,0000,0000,,And now, what happens if\Nyou do plus or minus 3?
Dialogue: 0,0:03:31.20,0:03:33.65,Default,,0000,0000,0000,,So what is y is equal\Nto x squared plus 3?
Dialogue: 0,0:03:33.65,0:03:37.51,Default,,0000,0000,0000,,
Dialogue: 0,0:03:37.51,0:03:39.82,Default,,0000,0000,0000,,Not minus x squared plus 3,\Nbut just x squared plus 3.
Dialogue: 0,0:03:39.82,0:03:43.41,Default,,0000,0000,0000,,So if you start with x squared,\Nnow every y value for
Dialogue: 0,0:03:43.41,0:03:45.45,Default,,0000,0000,0000,,every given x is just going\Nto be 3 higher.
Dialogue: 0,0:03:45.45,0:03:48.39,Default,,0000,0000,0000,,So it's just going to shift\Nthe graph up by 3.
Dialogue: 0,0:03:48.39,0:03:50.91,Default,,0000,0000,0000,,It's going to look like that.
Dialogue: 0,0:03:50.91,0:03:54.19,Default,,0000,0000,0000,,So if you go from x squared to\Nx squared plus 3, you're just
Dialogue: 0,0:03:54.19,0:03:55.69,Default,,0000,0000,0000,,shifting up by 3.
Dialogue: 0,0:03:55.69,0:03:58.89,Default,,0000,0000,0000,,Similar, if you go from minus\Nx squared to minus x squared
Dialogue: 0,0:03:58.89,0:04:01.20,Default,,0000,0000,0000,,plus 3, which is what they gave\Nus in the problem, you're
Dialogue: 0,0:04:01.20,0:04:03.38,Default,,0000,0000,0000,,just going to shift\Nthe graph up by 3.
Dialogue: 0,0:04:03.38,0:04:05.55,Default,,0000,0000,0000,,So I'll do that in\Nthis brown color.
Dialogue: 0,0:04:05.55,0:04:07.50,Default,,0000,0000,0000,,So it's just going to take this\Ngraph, which is minus x
Dialogue: 0,0:04:07.50,0:04:09.59,Default,,0000,0000,0000,,squared and you're going\Nto shift it up by 3.
Dialogue: 0,0:04:09.59,0:04:11.84,Default,,0000,0000,0000,,So it's going to look\Nsomething like this.
Dialogue: 0,0:04:11.84,0:04:14.20,Default,,0000,0000,0000,,It's going to look something\Nlike that.
Dialogue: 0,0:04:14.20,0:04:16.61,Default,,0000,0000,0000,,So let's see, out of all the\Nchoices they gave us, it
Dialogue: 0,0:04:16.61,0:04:19.74,Default,,0000,0000,0000,,should be opening downward and\Nit should have its y-intercept
Dialogue: 0,0:04:19.74,0:04:23.52,Default,,0000,0000,0000,,at y is equal to 3.
Dialogue: 0,0:04:23.52,0:04:24.95,Default,,0000,0000,0000,,If you put x is equal to\N0, y is equal to 3.
Dialogue: 0,0:04:24.95,0:04:25.75,Default,,0000,0000,0000,,So let's see.
Dialogue: 0,0:04:25.75,0:04:28.29,Default,,0000,0000,0000,,It's opening downwards.
Dialogue: 0,0:04:28.29,0:04:30.65,Default,,0000,0000,0000,,So these two are the only two\Nthat are opening downwards.
Dialogue: 0,0:04:30.65,0:04:32.62,Default,,0000,0000,0000,,And the y-intercept should\Nbe at 3 because we
Dialogue: 0,0:04:32.62,0:04:33.78,Default,,0000,0000,0000,,shifted it up by 3.
Dialogue: 0,0:04:33.78,0:04:36.90,Default,,0000,0000,0000,,So this is the choice B.
Dialogue: 0,0:04:36.90,0:04:39.47,Default,,0000,0000,0000,,Problem 60.
Dialogue: 0,0:04:39.47,0:04:44.40,Default,,0000,0000,0000,,Which quadratic funtion, when\Ngraphed, has x-intercepts of 4
Dialogue: 0,0:04:44.40,0:04:46.18,Default,,0000,0000,0000,,and minus 3?
Dialogue: 0,0:04:46.18,0:04:48.86,Default,,0000,0000,0000,,So x-intercepts of 4 and minus\N3 means that when you
Dialogue: 0,0:04:48.86,0:04:50.88,Default,,0000,0000,0000,,substitute x of either of\Nthese values into the
Dialogue: 0,0:04:50.88,0:04:54.45,Default,,0000,0000,0000,,equation, you get\Ny is equal to 0.
Dialogue: 0,0:04:54.45,0:04:56.63,Default,,0000,0000,0000,,Because when y is equal to 0,\Nyou're at the x-intercepts.
Dialogue: 0,0:04:56.63,0:04:58.37,Default,,0000,0000,0000,,This is when y equals to 0.
Dialogue: 0,0:04:58.37,0:04:59.62,Default,,0000,0000,0000,,So that's what they mean\Nby x-intercepts.
Dialogue: 0,0:04:59.62,0:05:02.55,Default,,0000,0000,0000,,
Dialogue: 0,0:05:02.55,0:05:05.24,Default,,0000,0000,0000,,So how do we set up an equation\Nwhere if I put in one
Dialogue: 0,0:05:05.24,0:05:08.17,Default,,0000,0000,0000,,of these numbers I'm\Ngoing to get 0?
Dialogue: 0,0:05:08.17,0:05:16.09,Default,,0000,0000,0000,,Well, if I make it the product\Nof x minus the first root and
Dialogue: 0,0:05:16.09,0:05:17.79,Default,,0000,0000,0000,,x minus the second root.
Dialogue: 0,0:05:17.79,0:05:22.04,Default,,0000,0000,0000,,So x minus minus\N3 is x plus 3.
Dialogue: 0,0:05:22.04,0:05:22.72,Default,,0000,0000,0000,,So think about it.
Dialogue: 0,0:05:22.72,0:05:26.79,Default,,0000,0000,0000,,If you put 4 here for x, you\Nget 4 minus 4, which is 0.
Dialogue: 0,0:05:26.79,0:05:27.94,Default,,0000,0000,0000,,Times 4 plus 3 is 7.
Dialogue: 0,0:05:27.94,0:05:30.08,Default,,0000,0000,0000,,So 0 times 7 is 0.
Dialogue: 0,0:05:30.08,0:05:31.35,Default,,0000,0000,0000,,So that works.
Dialogue: 0,0:05:31.35,0:05:32.88,Default,,0000,0000,0000,,And then, for minus 3.
Dialogue: 0,0:05:32.88,0:05:34.78,Default,,0000,0000,0000,,Minus 3 minus 4 is minus 7.
Dialogue: 0,0:05:34.78,0:05:36.96,Default,,0000,0000,0000,,But then minus 3\Nplus 3 is a 0.
Dialogue: 0,0:05:36.96,0:05:38.88,Default,,0000,0000,0000,,So either of these, when you\Nsubstitute it into this
Dialogue: 0,0:05:38.88,0:05:40.82,Default,,0000,0000,0000,,expression, you get 0.
Dialogue: 0,0:05:40.82,0:05:42.18,Default,,0000,0000,0000,,Let's see, which\Nchoice is that?
Dialogue: 0,0:05:42.18,0:05:49.21,Default,,0000,0000,0000,,x minus 4 times x plus 3.
Dialogue: 0,0:05:49.21,0:05:57.59,Default,,0000,0000,0000,,Have the x-intercepts\Nof 4 and minus 3.
Dialogue: 0,0:05:57.59,0:05:58.35,Default,,0000,0000,0000,,Right.
Dialogue: 0,0:05:58.35,0:06:00.61,Default,,0000,0000,0000,,This should be right.
Dialogue: 0,0:06:00.61,0:06:01.92,Default,,0000,0000,0000,,They're being tricky\Nright here.
Dialogue: 0,0:06:01.92,0:06:03.62,Default,,0000,0000,0000,,So x plus 3 is there.
Dialogue: 0,0:06:03.62,0:06:05.33,Default,,0000,0000,0000,,I see that in a couple\Nof them, right?
Dialogue: 0,0:06:05.33,0:06:07.99,Default,,0000,0000,0000,,But I don't see the x\Nminus 4 anywhere.
Dialogue: 0,0:06:07.99,0:06:11.91,Default,,0000,0000,0000,,But that's because we can\Nmultiply this by any constant.
Dialogue: 0,0:06:11.91,0:06:14.35,Default,,0000,0000,0000,,Because 0 times some number\Ntimes some constant is still
Dialogue: 0,0:06:14.35,0:06:15.44,Default,,0000,0000,0000,,going to be 0.
Dialogue: 0,0:06:15.44,0:06:20.27,Default,,0000,0000,0000,,So if you look at this one\Nright here, 2x minus 8.
Dialogue: 0,0:06:20.27,0:06:21.03,Default,,0000,0000,0000,,We could factor out a 2.
Dialogue: 0,0:06:21.03,0:06:26.46,Default,,0000,0000,0000,,That's the same thing as\N2 times x minus 4.
Dialogue: 0,0:06:26.46,0:06:30.72,Default,,0000,0000,0000,,So choice B is x plus 3,\Ntimes x minus 4, times
Dialogue: 0,0:06:30.72,0:06:32.10,Default,,0000,0000,0000,,some constant 2.
Dialogue: 0,0:06:32.10,0:06:34.74,Default,,0000,0000,0000,,So choice B is our answer.
Dialogue: 0,0:06:34.74,0:06:39.56,Default,,0000,0000,0000,,If this is equal to 0 when x is\Nequal to 4 minus 3, this--
Dialogue: 0,0:06:39.56,0:06:45.22,Default,,0000,0000,0000,,any constant, x minus 4 times x\Nplus 3--I that's still going
Dialogue: 0,0:06:45.22,0:06:46.26,Default,,0000,0000,0000,,to be equal to 0.
Dialogue: 0,0:06:46.26,0:06:47.92,Default,,0000,0000,0000,,Because when x is equal\Nto 4, this is going
Dialogue: 0,0:06:47.92,0:06:48.74,Default,,0000,0000,0000,,to be equal to 0.
Dialogue: 0,0:06:48.74,0:06:50.45,Default,,0000,0000,0000,,So 0 times anything times\Nanything else
Dialogue: 0,0:06:50.45,0:06:51.19,Default,,0000,0000,0000,,is going to be 0.
Dialogue: 0,0:06:51.19,0:06:53.07,Default,,0000,0000,0000,,Same thing with x is\Nequal to minus 3.
Dialogue: 0,0:06:53.07,0:06:54.71,Default,,0000,0000,0000,,So they just put a 2 here.
Dialogue: 0,0:06:54.71,0:06:55.60,Default,,0000,0000,0000,,That's a good problem.
Dialogue: 0,0:06:55.60,0:06:57.91,Default,,0000,0000,0000,,It made you realize that you\Ncould put a constant in there
Dialogue: 0,0:06:57.91,0:06:59.11,Default,,0000,0000,0000,,and it's a little tricky.
Dialogue: 0,0:06:59.11,0:07:00.36,Default,,0000,0000,0000,,Next problem.
Dialogue: 0,0:07:00.36,0:07:05.79,Default,,0000,0000,0000,,
Dialogue: 0,0:07:05.79,0:07:16.13,Default,,0000,0000,0000,,OK, so they want to know-- let\Nme copy and paste it-- how
Dialogue: 0,0:07:16.13,0:07:19.65,Default,,0000,0000,0000,,many times does the graph of y\Nequals 2x squared minus 2x
Dialogue: 0,0:07:19.65,0:07:23.17,Default,,0000,0000,0000,,plus 3 intersect the x-axis?
Dialogue: 0,0:07:23.17,0:07:25.45,Default,,0000,0000,0000,,So the easiest thing-- because\Nmaybe it doesn't intersect the
Dialogue: 0,0:07:25.45,0:07:26.42,Default,,0000,0000,0000,,x-axis at all.
Dialogue: 0,0:07:26.42,0:07:28.45,Default,,0000,0000,0000,,Maybe if you use a quadratic\Nequation
Dialogue: 0,0:07:28.45,0:07:29.65,Default,,0000,0000,0000,,there are no real solutions.
Dialogue: 0,0:07:29.65,0:07:31.28,Default,,0000,0000,0000,,So let's just apply\Nthe quadratic.
Dialogue: 0,0:07:31.28,0:07:36.67,Default,,0000,0000,0000,,So the roots or the times that--\NI guess the x values
Dialogue: 0,0:07:36.67,0:07:42.65,Default,,0000,0000,0000,,that solve this equation, 2x\Nsquared minus 2x plus 3 is
Dialogue: 0,0:07:42.65,0:07:44.03,Default,,0000,0000,0000,,equal to 0.
Dialogue: 0,0:07:44.03,0:07:45.54,Default,,0000,0000,0000,,And these are the x\Nvalues where you
Dialogue: 0,0:07:45.54,0:07:46.73,Default,,0000,0000,0000,,intersect the x-axis.
Dialogue: 0,0:07:46.73,0:07:47.50,Default,,0000,0000,0000,,And why do I say that?
Dialogue: 0,0:07:47.50,0:07:50.82,Default,,0000,0000,0000,,Because the x-axis is the\Nline y is equal to 0.
Dialogue: 0,0:07:50.82,0:07:53.20,Default,,0000,0000,0000,,So I set y equal to\N0 and I get this.
Dialogue: 0,0:07:53.20,0:07:55.53,Default,,0000,0000,0000,,And we know from the quadratic\Nequation the solution to this
Dialogue: 0,0:07:55.53,0:07:58.68,Default,,0000,0000,0000,,is negative b-- let me do\Nthis in another color.
Dialogue: 0,0:07:58.68,0:08:00.21,Default,,0000,0000,0000,,So negative b.
Dialogue: 0,0:08:00.21,0:08:02.50,Default,,0000,0000,0000,,So minus minus 2 is 2.
Dialogue: 0,0:08:02.50,0:08:06.37,Default,,0000,0000,0000,,Plus or minus the square\Nroot of b squared.
Dialogue: 0,0:08:06.37,0:08:09.10,Default,,0000,0000,0000,,Minus 2 squared is 4.
Dialogue: 0,0:08:09.10,0:08:16.32,Default,,0000,0000,0000,,Minus 4 times a, which is 2.
Dialogue: 0,0:08:16.32,0:08:18.22,Default,,0000,0000,0000,,Times c, times 3.
Dialogue: 0,0:08:18.22,0:08:19.86,Default,,0000,0000,0000,,All of that over 2a.
Dialogue: 0,0:08:19.86,0:08:22.67,Default,,0000,0000,0000,,2 times 2, which is 4.
Dialogue: 0,0:08:22.67,0:08:23.84,Default,,0000,0000,0000,,Now they don't want\Nus to figure out
Dialogue: 0,0:08:23.84,0:08:24.50,Default,,0000,0000,0000,,the roots or anything.
Dialogue: 0,0:08:24.50,0:08:25.42,Default,,0000,0000,0000,,They just want to know,\Nhow many times does it
Dialogue: 0,0:08:25.42,0:08:26.57,Default,,0000,0000,0000,,intersect the axis?
Dialogue: 0,0:08:26.57,0:08:27.75,Default,,0000,0000,0000,,So let's think about this.
Dialogue: 0,0:08:27.75,0:08:29.14,Default,,0000,0000,0000,,What happens under this\Nradical sign?
Dialogue: 0,0:08:29.14,0:08:31.35,Default,,0000,0000,0000,,We have 4 times 2\Ntimes 3 is 24.
Dialogue: 0,0:08:31.35,0:08:37.97,Default,,0000,0000,0000,,So this becomes 2 plus or\Nminus 4 minus 24 over 4.
Dialogue: 0,0:08:37.97,0:08:40.96,Default,,0000,0000,0000,,This is minus 20.
Dialogue: 0,0:08:40.96,0:08:44.01,Default,,0000,0000,0000,,So you end up with minus 20\Nunder the radical sign.
Dialogue: 0,0:08:44.01,0:08:46.22,Default,,0000,0000,0000,,And we know if we're dealing\Nwith real numbers, if we want
Dialogue: 0,0:08:46.22,0:08:50.43,Default,,0000,0000,0000,,real solutions, you can't take\Nthe square root of minus 20.
Dialogue: 0,0:08:50.43,0:08:53.04,Default,,0000,0000,0000,,So this actually has no\Nsolutions or, another way to
Dialogue: 0,0:08:53.04,0:08:56.70,Default,,0000,0000,0000,,put it is, there is no x values\Nwhere y is equal to 0.
Dialogue: 0,0:08:56.70,0:08:58.72,Default,,0000,0000,0000,,Or another way to put it\Nis, this never does
Dialogue: 0,0:08:58.72,0:09:00.82,Default,,0000,0000,0000,,intersect the x-axis.
Dialogue: 0,0:09:00.82,0:09:01.68,Default,,0000,0000,0000,,So it's A.
Dialogue: 0,0:09:01.68,0:09:04.27,Default,,0000,0000,0000,,none.
Dialogue: 0,0:09:04.27,0:09:06.97,Default,,0000,0000,0000,,What gave that away was the fact\Nthat when you apply the
Dialogue: 0,0:09:06.97,0:09:09.75,Default,,0000,0000,0000,,quadratic equation, you get a\Nnegative number under the
Dialogue: 0,0:09:09.75,0:09:10.41,Default,,0000,0000,0000,,radical sign.
Dialogue: 0,0:09:10.41,0:09:11.57,Default,,0000,0000,0000,,So if we're dealing\Nwith real numbers,
Dialogue: 0,0:09:11.57,0:09:14.34,Default,,0000,0000,0000,,there's no answer there.
Dialogue: 0,0:09:14.34,0:09:15.59,Default,,0000,0000,0000,,Next question.
Dialogue: 0,0:09:15.59,0:09:19.74,Default,,0000,0000,0000,,
Dialogue: 0,0:09:19.74,0:09:27.70,Default,,0000,0000,0000,,62.
Dialogue: 0,0:09:27.70,0:09:30.26,Default,,0000,0000,0000,,An object that is projected\Nstraight down-- oh, this is
Dialogue: 0,0:09:30.26,0:09:32.07,Default,,0000,0000,0000,,good, this is projectile\Nmotion-- is projected straight
Dialogue: 0,0:09:32.07,0:09:35.65,Default,,0000,0000,0000,,downward with initial velocity\Nv feet per second, Travels a
Dialogue: 0,0:09:35.65,0:09:40.40,Default,,0000,0000,0000,,distance of s v times t plus\N16t squared, where t equals
Dialogue: 0,0:09:40.40,0:09:41.58,Default,,0000,0000,0000,,time in seconds.
Dialogue: 0,0:09:41.58,0:09:45.52,Default,,0000,0000,0000,,If Ramon is standing on a\Nbalcony 84 feet above the
Dialogue: 0,0:09:45.52,0:09:48.17,Default,,0000,0000,0000,,ground and throws a penny\Nstraight down with an initial
Dialogue: 0,0:09:48.17,0:09:51.76,Default,,0000,0000,0000,,velocity of 10 feet per second,\Nin how many seconds
Dialogue: 0,0:09:51.76,0:09:54.33,Default,,0000,0000,0000,,will it reach the ground?
Dialogue: 0,0:09:54.33,0:09:58.84,Default,,0000,0000,0000,,OK, so he's 84 feet\Nabove the ground.
Dialogue: 0,0:09:58.84,0:10:00.45,Default,,0000,0000,0000,,Let's draw a diagram.
Dialogue: 0,0:10:00.45,0:10:01.80,Default,,0000,0000,0000,,He's 84 feet.
Dialogue: 0,0:10:01.80,0:10:05.18,Default,,0000,0000,0000,,This is 84 feet above\Nthe ground.
Dialogue: 0,0:10:05.18,0:10:07.58,Default,,0000,0000,0000,,It says, how many seconds will\Nit reach the ground?
Dialogue: 0,0:10:07.58,0:10:10.50,Default,,0000,0000,0000,,So we essentially want to know\Nhow many seconds will it take
Dialogue: 0,0:10:10.50,0:10:13.08,Default,,0000,0000,0000,,it to travel? s is\Ndistance, right?
Dialogue: 0,0:10:13.08,0:10:14.56,Default,,0000,0000,0000,,So s is equal to 84 feet.
Dialogue: 0,0:10:14.56,0:10:17.69,Default,,0000,0000,0000,,It has to go down 84 feet.
Dialogue: 0,0:10:17.69,0:10:19.49,Default,,0000,0000,0000,,Let's see if we can\Nfigure this out.
Dialogue: 0,0:10:19.49,0:10:22.24,Default,,0000,0000,0000,,Now this is something that might\Nbe a little bit-- so how
Dialogue: 0,0:10:22.24,0:10:24.62,Default,,0000,0000,0000,,long does it take it to go 84\Nfeet, I guess is the best way
Dialogue: 0,0:10:24.62,0:10:25.54,Default,,0000,0000,0000,,to think about it.
Dialogue: 0,0:10:25.54,0:10:31.49,Default,,0000,0000,0000,,So we say 84 is equal to\Nvelocity times-- your initial
Dialogue: 0,0:10:31.49,0:10:32.87,Default,,0000,0000,0000,,velocity times time.
Dialogue: 0,0:10:32.87,0:10:37.09,Default,,0000,0000,0000,,And your initial velocity\Nis 10 feet per second.
Dialogue: 0,0:10:37.09,0:10:38.71,Default,,0000,0000,0000,,So it's 10 feet per second.
Dialogue: 0,0:10:38.71,0:10:40.15,Default,,0000,0000,0000,,Everything is in feet I think.
Dialogue: 0,0:10:40.15,0:10:42.01,Default,,0000,0000,0000,,Right, everything is--\Nv feet per second.
Dialogue: 0,0:10:42.01,0:10:45.11,Default,,0000,0000,0000,,Initial velocity of\Nv feet per second.
Dialogue: 0,0:10:45.11,0:10:47.10,Default,,0000,0000,0000,,So 10 feet per second\Ntimes time.
Dialogue: 0,0:10:47.10,0:10:48.87,Default,,0000,0000,0000,,I just substituted what\Nthey gave us.
Dialogue: 0,0:10:48.87,0:10:50.67,Default,,0000,0000,0000,,10 for v.
Dialogue: 0,0:10:50.67,0:10:53.92,Default,,0000,0000,0000,,Plus 16t squared.
Dialogue: 0,0:10:53.92,0:10:56.59,Default,,0000,0000,0000,,And now I just solve\Nthis quadratic.
Dialogue: 0,0:10:56.59,0:10:58.07,Default,,0000,0000,0000,,That is a t right there.
Dialogue: 0,0:10:58.07,0:11:00.45,Default,,0000,0000,0000,,So let me put everything on the\Nsame-- let me subtract 84
Dialogue: 0,0:11:00.45,0:11:02.08,Default,,0000,0000,0000,,from both sides and I'll\Nrearrange a little bit.
Dialogue: 0,0:11:02.08,0:11:11.70,Default,,0000,0000,0000,,So you get 16t squared plus 10t\Nminus 84 is equal to 0.
Dialogue: 0,0:11:11.70,0:11:13.72,Default,,0000,0000,0000,,Well I swapped the sides.
Dialogue: 0,0:11:13.72,0:11:14.94,Default,,0000,0000,0000,,I put these on the left.
Dialogue: 0,0:11:14.94,0:11:16.73,Default,,0000,0000,0000,,Well, let me just show\Nyou what I did.
Dialogue: 0,0:11:16.73,0:11:17.71,Default,,0000,0000,0000,,I swapped the sides.
Dialogue: 0,0:11:17.71,0:11:23.18,Default,,0000,0000,0000,,So I made this 16t squared\Nplus 10t equals 84.
Dialogue: 0,0:11:23.18,0:11:24.11,Default,,0000,0000,0000,,I just swapped them.
Dialogue: 0,0:11:24.11,0:11:26.91,Default,,0000,0000,0000,,And then I subtracted 84 from\Nboth sides to get this.
Dialogue: 0,0:11:26.91,0:11:29.65,Default,,0000,0000,0000,,And now we just have to\Nsolve when t equals 0.
Dialogue: 0,0:11:29.65,0:11:32.19,Default,,0000,0000,0000,,So I guess the first thing we\Ncould do is we could simplify
Dialogue: 0,0:11:32.19,0:11:32.84,Default,,0000,0000,0000,,this a little bit.
Dialogue: 0,0:11:32.84,0:11:35.08,Default,,0000,0000,0000,,Everything here is\Ndivisible by 2.
Dialogue: 0,0:11:35.08,0:11:39.89,Default,,0000,0000,0000,,So this is 8t squared plus-- I'm\Njust dividing both sides
Dialogue: 0,0:11:39.89,0:11:41.15,Default,,0000,0000,0000,,of this equation by 2.
Dialogue: 0,0:11:41.15,0:11:43.93,Default,,0000,0000,0000,,Plus 5t minus what?
Dialogue: 0,0:11:43.93,0:11:48.67,Default,,0000,0000,0000,,Minus 42 is equal to 0.
Dialogue: 0,0:11:48.67,0:11:51.20,Default,,0000,0000,0000,,And then we can use the\Nquadratic equation.
Dialogue: 0,0:11:51.20,0:11:55.20,Default,,0000,0000,0000,,So what are the solutions? t\Nis equal to negatives b.
Dialogue: 0,0:11:55.20,0:12:02.98,Default,,0000,0000,0000,,So minus 5 plus or minus the\Nsquare root of b squared, so
Dialogue: 0,0:12:02.98,0:12:07.88,Default,,0000,0000,0000,,25, minus 4 times a.
Dialogue: 0,0:12:07.88,0:12:11.36,Default,,0000,0000,0000,,times 8, times c.\Nc is minus 42.
Dialogue: 0,0:12:11.36,0:12:13.72,Default,,0000,0000,0000,,So instead of times minus\N42, let's put a plus
Dialogue: 0,0:12:13.72,0:12:16.54,Default,,0000,0000,0000,,here and do plus 42.
Dialogue: 0,0:12:16.54,0:12:18.61,Default,,0000,0000,0000,,Just a negative times a negative\Nis a positive.
Dialogue: 0,0:12:18.61,0:12:22.54,Default,,0000,0000,0000,,All of that, over 2 times a.
Dialogue: 0,0:12:22.54,0:12:24.65,Default,,0000,0000,0000,,2a is 16.
Dialogue: 0,0:12:24.65,0:12:26.15,Default,,0000,0000,0000,,So let's see where\Nthat gets me.
Dialogue: 0,0:12:26.15,0:12:32.03,Default,,0000,0000,0000,,So I t is equal to minus 5 plus\Nor minus the square root.
Dialogue: 0,0:12:32.03,0:12:32.55,Default,,0000,0000,0000,,What is this?
Dialogue: 0,0:12:32.55,0:12:35.33,Default,,0000,0000,0000,,25 plus-- let's see.
Dialogue: 0,0:12:35.33,0:12:39.44,Default,,0000,0000,0000,,4 times 8 times 42.
Dialogue: 0,0:12:39.44,0:12:41.58,Default,,0000,0000,0000,,That's 32 times 42.
Dialogue: 0,0:12:41.58,0:12:46.84,Default,,0000,0000,0000,,32 times 42.
Dialogue: 0,0:12:46.84,0:12:49.59,Default,,0000,0000,0000,,2 times 32 is 64.
Dialogue: 0,0:12:49.59,0:12:50.34,Default,,0000,0000,0000,,Put a 0.
Dialogue: 0,0:12:50.34,0:12:52.25,Default,,0000,0000,0000,,4 times 2 is 8.
Dialogue: 0,0:12:52.25,0:12:56.97,Default,,0000,0000,0000,,4 times 3 is 12.
Dialogue: 0,0:12:56.97,0:13:05.81,Default,,0000,0000,0000,,You end up with 4,\N14, 3 and 1.
Dialogue: 0,0:13:05.81,0:13:06.88,Default,,0000,0000,0000,,So this is 1,344.
Dialogue: 0,0:13:06.88,0:13:08.53,Default,,0000,0000,0000,,And we're going to\Nadd this 25 here.
Dialogue: 0,0:13:08.53,0:13:09.20,Default,,0000,0000,0000,,So let me see.
Dialogue: 0,0:13:09.20,0:13:11.62,Default,,0000,0000,0000,,Plus 1,344.
Dialogue: 0,0:13:11.62,0:13:13.80,Default,,0000,0000,0000,,All of that over 16.
Dialogue: 0,0:13:13.80,0:13:13.97,Default,,0000,0000,0000,,Let's see.
Dialogue: 0,0:13:13.97,0:13:15.64,Default,,0000,0000,0000,,1,344 plus 25.
Dialogue: 0,0:13:15.64,0:13:20.28,Default,,0000,0000,0000,,So it's minus 5 plus or\Nminus the square root
Dialogue: 0,0:13:20.28,0:13:24.37,Default,,0000,0000,0000,,of-- what is this?
Dialogue: 0,0:13:24.37,0:13:29.15,Default,,0000,0000,0000,,1,369 over 16.
Dialogue: 0,0:13:29.15,0:13:30.23,Default,,0000,0000,0000,,And actually, I don't\Nknow what the square
Dialogue: 0,0:13:30.23,0:13:31.43,Default,,0000,0000,0000,,root of 1369 is.
Dialogue: 0,0:13:31.43,0:13:34.04,Default,,0000,0000,0000,,Let me get the calculator.
Dialogue: 0,0:13:34.04,0:13:37.04,Default,,0000,0000,0000,,Let me open it up.
Dialogue: 0,0:13:37.04,0:13:38.06,Default,,0000,0000,0000,,Give me one second.
Dialogue: 0,0:13:38.06,0:13:42.54,Default,,0000,0000,0000,,Accessories, calculator.
Dialogue: 0,0:13:42.54,0:13:43.51,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:13:43.51,0:13:48.63,Default,,0000,0000,0000,,So 1,369.
Dialogue: 0,0:13:48.63,0:13:49.41,Default,,0000,0000,0000,,37.
Dialogue: 0,0:13:49.41,0:13:50.39,Default,,0000,0000,0000,,Look at that.
Dialogue: 0,0:13:50.39,0:13:53.44,Default,,0000,0000,0000,,OK, so it's minus 5\Nplus or minus 37.
Dialogue: 0,0:13:53.44,0:13:55.44,Default,,0000,0000,0000,,That's the square\Nroot of 1,369.
Dialogue: 0,0:13:55.44,0:14:01.60,Default,,0000,0000,0000,,So minus 5 plus or minus 37 over\N16 is equal to the time.
Dialogue: 0,0:14:01.60,0:14:03.02,Default,,0000,0000,0000,,Now we don't have to worry about\Nthe minus because that's
Dialogue: 0,0:14:03.02,0:14:03.95,Default,,0000,0000,0000,,going to give us a\Nnegative number.
Dialogue: 0,0:14:03.95,0:14:06.11,Default,,0000,0000,0000,,Minus 5 minus 37 over 16.
Dialogue: 0,0:14:06.11,0:14:08.78,Default,,0000,0000,0000,,We don't want a negative time,\Nwe want a positive time.
Dialogue: 0,0:14:08.78,0:14:10.08,Default,,0000,0000,0000,,So let's just do the positive.
Dialogue: 0,0:14:10.08,0:14:13.24,Default,,0000,0000,0000,,So minus 5 plus 37.
Dialogue: 0,0:14:13.24,0:14:13.45,Default,,0000,0000,0000,,Let's see.
Dialogue: 0,0:14:13.45,0:14:16.83,Default,,0000,0000,0000,,Minus 5 plus 37 over 16.
Dialogue: 0,0:14:16.83,0:14:21.34,Default,,0000,0000,0000,,So that's 32/16, which\Nequals to 2 seconds.
Dialogue: 0,0:14:21.34,0:14:23.29,Default,,0000,0000,0000,,And that's choice A.
Dialogue: 0,0:14:23.29,0:14:25.59,Default,,0000,0000,0000,,Anyway, see you in\Nthe next video.