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We're on problem 58. The graph of the equation y is equal to x squared minus 3x minus 4 is shown below. Fair enough. For what value or values of x is y equal to 0? So they're essentially saying is, when does this here equal 0? They want to know when does y equal 0? So what values of x does that happen? And we could factor this and solve for the roots, but they drew us the graph, so let's just inspect it. So when does y equal 0? So let me draw the line of y is equal to 0. So that's right here. Let me draw it as a line. y equals 0. That's y equals 0 right there. So what values of x makes y equal 0? If I can see this properly, it's when x is equal to negative 1 and when x is equal to 4. So x is equal to negative 1 or 4. And if we substitute either of these values into this right here, we should get y is equal to 0. And let's see. The choices, do they have negative 1 and 4? Yep, sure enough. Negative 1 and 4 right there. Next question, 59. Let me copy and paste it. OK, I've copied it. I'll paste it below. I'll do it right on top of this. There you go. 59. Let me erase this stuff right here. This is unrelated to this problem. So they are asking-- let me get the pen right-- which best represents the graph of y is equal to minus x squared plus 3? So here, just to get an intuition of what parabolas look like, because these are all parabolas, or the graph of a quadratic equation. So if I had the graph of y is equal to x squared, what does that look like? Let me just draw a quick and dirty x- and y-axis. And I think you're familiar with what that looks like. So if I were to just draw y is equal to x squared, that looks something like this. It looks something like this. It will look something like that. I think you're familiar with it. Because you're taking x squared, you always get positive values. Even if you have a negative number squared that still becomes a positive. And it's symmetric around the line x is equal to 0. That's the graph of y equals x squared. Now let me ask you a question. What is the graph of y is equal to minus x squared? Let me do that. y is equal to minus x squared. So it's essentially the same thing as this graph, but it's going to be the negative of whatever you get. So here, x squared is always going to be positive. You square any real number and you're going to get a positive number. Here, you square any real number, this part right here becomes positive. But then you take the negative of it. So this is always going to be a negative number. So x equals 0 is still going to be there, but regardless of whether you go in the positive x direction or the negative x direction, this is going to be positive. But then you put the negative sign, it's going to become negative. So the graph is going to look like this. The graph is going to look like that. I didn't draw that well, let me give that another shot. The graph is going to look like that. It's essentially like the mirror image of this one if you were to reflect it on the x-axis. This is y equals minus x squared. So now you're pointing it down. The u goes-- opens up downward. And hopefully that makes a little bit of sense. And now, what happens if you do plus or minus 3? So what is y is equal to x squared plus 3? Not minus x squared plus 3, but just x squared plus 3. So if you start with x squared, now every y value for every given x is just going to be 3 higher. So it's just going to shift the graph up by 3. It's going to look like that. So if you go from x squared to x squared plus 3, you're just shifting up by 3. Similar, if you go from minus x squared to minus x squared plus 3, which is what they gave us in the problem, you're just going to shift the graph up by 3. So I'll do that in this brown color. So it's just going to take this graph, which is minus x squared and you're going to shift it up by 3. So it's going to look something like this. It's going to look something like that. So let's see, out of all the choices they gave us, it should be opening downward and it should have its y-intercept at y is equal to 3. If you put x is equal to 0, y is equal to 3. So let's see. It's opening downwards. So these two are the only two that are opening downwards. And the y-intercept should be at 3 because we shifted it up by 3. So this is the choice B. Problem 60. Which quadratic funtion, when graphed, has x-intercepts of 4 and minus 3? So x-intercepts of 4 and minus 3 means that when you substitute x of either of these values into the equation, you get y is equal to 0. Because when y is equal to 0, you're at the x-intercepts. This is when y equals to 0. So that's what they mean by x-intercepts. So how do we set up an equation where if I put in one of these numbers I'm going to get 0? Well, if I make it the product of x minus the first root and x minus the second root. So x minus minus 3 is x plus 3. So think about it. If you put 4 here for x, you get 4 minus 4, which is 0. Times 4 plus 3 is 7. So 0 times 7 is 0. So that works. And then, for minus 3. Minus 3 minus 4 is minus 7. But then minus 3 plus 3 is a 0. So either of these, when you substitute it into this expression, you get 0. Let's see, which choice is that? x minus 4 times x plus 3. Have the x-intercepts of 4 and minus 3. Right. This should be right. They're being tricky right here. So x plus 3 is there. I see that in a couple of them, right? But I don't see the x minus 4 anywhere. But that's because we can multiply this by any constant. Because 0 times some number times some constant is still going to be 0. So if you look at this one right here, 2x minus 8. We could factor out a 2. That's the same thing as 2 times x minus 4. So choice B is x plus 3, times x minus 4, times some constant 2. So choice B is our answer. If this is equal to 0 when x is equal to 4 minus 3, this-- any constant, x minus 4 times x plus 3--I that's still going to be equal to 0. Because when x is equal to 4, this is going to be equal to 0. So 0 times anything times anything else is going to be 0. Same thing with x is equal to minus 3. So they just put a 2 here. That's a good problem. It made you realize that you could put a constant in there and it's a little tricky. Next problem. OK, so they want to know-- let me copy and paste it-- how many times does the graph of y equals 2x squared minus 2x plus 3 intersect the x-axis? So the easiest thing-- because maybe it doesn't intersect the x-axis at all. Maybe if you use a quadratic equation there are no real solutions. So let's just apply the quadratic. So the roots or the times that-- I guess the x values that solve this equation, 2x squared minus 2x plus 3 is equal to 0. And these are the x values where you intersect the x-axis. And why do I say that? Because the x-axis is the line y is equal to 0. So I set y equal to 0 and I get this. And we know from the quadratic equation the solution to this is negative b-- let me do this in another color. So negative b. So minus minus 2 is 2. Plus or minus the square root of b squared. Minus 2 squared is 4. Minus 4 times a, which is 2. Times c, times 3. All of that over 2a. 2 times 2, which is 4. Now they don't want us to figure out the roots or anything. They just want to know, how many times does it intersect the axis? So let's think about this. What happens under this radical sign? We have 4 times 2 times 3 is 24. So this becomes 2 plus or minus 4 minus 24 over 4. This is minus 20. So you end up with minus 20 under the radical sign. And we know if we're dealing with real numbers, if we want real solutions, you can't take the square root of minus 20. So this actually has no solutions or, another way to put it is, there is no x values where y is equal to 0. Or another way to put it is, this never does intersect the x-axis. So it's A. none. What gave that away was the fact that when you apply the quadratic equation, you get a negative number under the radical sign. So if we're dealing with real numbers, there's no answer there. Next question. 62. An object that is projected straight down-- oh, this is good, this is projectile motion-- is projected straight downward with initial velocity v feet per second, Travels a distance of s v times t plus 16t squared, where t equals time in seconds. If Ramon is standing on a balcony 84 feet above the ground and throws a penny straight down with an initial velocity of 10 feet per second, in how many seconds will it reach the ground? OK, so he's 84 feet above the ground. Let's draw a diagram. He's 84 feet. This is 84 feet above the ground. It says, how many seconds will it reach the ground? So we essentially want to know how many seconds will it take it to travel? s is distance, right? So s is equal to 84 feet. It has to go down 84 feet. Let's see if we can figure this out. Now this is something that might be a little bit-- so how long does it take it to go 84 feet, I guess is the best way to think about it. So we say 84 is equal to velocity times-- your initial velocity times time. And your initial velocity is 10 feet per second. So it's 10 feet per second. Everything is in feet I think. Right, everything is-- v feet per second. Initial velocity of v feet per second. So 10 feet per second times time. I just substituted what they gave us. 10 for v. Plus 16t squared. And now I just solve this quadratic. That is a t right there. So let me put everything on the same-- let me subtract 84 from both sides and I'll rearrange a little bit. So you get 16t squared plus 10t minus 84 is equal to 0. Well I swapped the sides. I put these on the left. Well, let me just show you what I did. I swapped the sides. So I made this 16t squared plus 10t equals 84. I just swapped them. And then I subtracted 84 from both sides to get this. And now we just have to solve when t equals 0. So I guess the first thing we could do is we could simplify this a little bit. Everything here is divisible by 2. So this is 8t squared plus-- I'm just dividing both sides of this equation by 2. Plus 5t minus what? Minus 42 is equal to 0. And then we can use the quadratic equation. So what are the solutions? t is equal to negatives b. So minus 5 plus or minus the square root of b squared, so 25, minus 4 times a. times 8, times c. c is minus 42. So instead of times minus 42, let's put a plus here and do plus 42. Just a negative times a negative is a positive. All of that, over 2 times a. 2a is 16. So let's see where that gets me. So I t is equal to minus 5 plus or minus the square root. What is this? 25 plus-- let's see. 4 times 8 times 42. That's 32 times 42. 32 times 42. 2 times 32 is 64. Put a 0. 4 times 2 is 8. 4 times 3 is 12. You end up with 4, 14, 3 and 1. So this is 1,344. And we're going to add this 25 here. So let me see. Plus 1,344. All of that over 16. Let's see. 1,344 plus 25. So it's minus 5 plus or minus the square root of-- what is this? 1,369 over 16. And actually, I don't know what the square root of 1369 is. Let me get the calculator. Let me open it up. Give me one second. Accessories, calculator. All right. So 1,369. 37. Look at that. OK, so it's minus 5 plus or minus 37. That's the square root of 1,369. So minus 5 plus or minus 37 over 16 is equal to the time. Now we don't have to worry about the minus because that's going to give us a negative number. Minus 5 minus 37 over 16. We don't want a negative time, we want a positive time. So let's just do the positive. So minus 5 plus 37. Let's see. Minus 5 plus 37 over 16. So that's 32/16, which equals to 2 seconds. And that's choice A. Anyway, see you in the next video.