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We're on problem 32. What are the solutions to the equation 1 plus 1 over x squared is equal to 3 over x? So at first. This looks like a pretty daunting equation. You have these x's in the denominator and x squared in the denominator. But I think we can simplify it if we can just get rid of these x squares in the denominator. The easiest way to do that is to multiply everything by x squared. So let's multiply both sides of this equation by x squared. Then we'll get x squared times 1 is x squared, x squared times 1 over x squared, that's just 1. Then x squared times 3 over x, that's 3x squared over x. x squared divided by x is just x, so that is equal to 3x. We can subtract 3x from both sides and you get x squared minus 3x plus 1 is equal to 0. This is a simple quadratic. It's not obvious that you can factor it. In fact, two numbers when you multiply them equal 1, and then when you add them equal minus 3. I'm guessing it might even be imaginary. It will probably not be imaginary, but it's just a strange number. So let's use the quadratic equation. When in doubt, use the quadratic equation. So minus B, this is B right? B is this 3 right there, negative 3. B is negative 3. So minus B is going to be plus 3, plus or minus the square root of B squared. Minus 3 squared is 9, minus 4 times A, which is 1, times C which is 1. So it's minus 4, all of that over 2A. A is 1 so it's just over 2. That is equal to 3/2 plus or minus the square root of 5 over 2. I just separated these out because I'm looking at the choices and it seems like they did that. So you could've said that's 3/2 plus square root of 5 over 2 or 3/2 minus the square root of 5 over 2. I just did that because it seems like that's how they write it. That is choice A. Next problem, 33. I think this one actually might be good to copy and paste the problem. Let me see if I can do this. OK, there are two numbers with the following properties. Let me write down the properties and let me copy and paste it for you. OK, I've copied it. Let me go here. Then I've pasted it for you. All right. So there's two numbers with the following properties. The second number is 3 more than the first number. So let's say S for second number and F for first number. So the second number is 3 more than the first number. So the second number is equal to the first number plus 3. That's from statement 1. The product of the two numbers is 9 more than the sum. So the product of the two numbers, that's S times F is 9 more, 9 plus their sum, plus S plus F. So let's see, we have two equations and two unknowns. This is nonlinear because I'm multiplying these two variables. But I think we should be able to solve them one way or the other. So let's see, we have what S is equal to. So let's just substitute that back into this equation. So let's say that S is equal to F plus 3. So if we substitute for these S's, we get F plus 3 times F is equal to 9 plus F plus 3, right, instead of an S, F plus 3 and then plus F. Let's see if we can simplify this. F times F is F squared plus 3F is equal to 9 plus 3 is 12 plus 2F. Subtract 2F from both sides, you get F squared plus F is equal to 12. Subtract 12, you get F squared plus F minus 12 is equal to 0. This one looks factorable. I don't have to take out the quadratic equation. Let's see, this is F plus 4 times F minus 3, right? Because when you multiply those, you get negative 12. When you add those, you get plus 1, so that is equal to 0. So in order for this to be true, one or both of these have to be equal to 0. So if F plus 4 is equal to 0, that means F could be equal to minus 4. If F minus 3 is equal to 0, then that says that F could be 3. So F could be minus 4 or 3. Now, S is F plus 3, so if we're dealing with the minus 4 scenario, if F is equal to minus 4, then what is S? Then S is going to be minus 4 plus 3, and S is going to be equal to minus 1. Then if F is equal to 3, than S is equal to 6. So let's see if we see either of these combinations. Minus 4, minus 1, that's choice B. Excellent. All right, problem 34. Let me see, maybe I should copy and paste these word problems so we can see how we parse the problems. So I've copied it, then I'll go here, and then pasted it. Jenny is solving the equation x squared minus 8x equals 9 by completing the square. What number should be added to both sides of the equation to complete the square? So x squared minus 8x is equal to 9. I wrote it with space for a reason. When you're completing the square, you're trying to turn the left-hand side of this equation into some type of a perfect square. So if it's a perfect square, I have two numbers. It's the same number that when you add them together, you get minus 8, and when you square them, you should get something else, right? So what's half of minus 8? Half of minus 8 is minus 4. So minus 4 squared is 16. So if I add 16 to both sides, I'm all set. Why did that work? Well, now it's a perfect square. This is now x minus 4 squared is equal to 9 plus 16 is 25. They're not even asking us to solve it. They just want to know what we had to add to both sides. So it's 16, D. Remember the whole logic here, and I've done a few videos on completing the squares, is what number do I add here to make this a perfect square? You say, OK, I have a minus 8x, so I take half of this number, because the same number added to itself twice is going to become minus 8. I take half of that number, then I squared it. So half of minus 8 is minus 4. If you square it, you get the 16. So add 16 to both sides, you get this. You can actually solve for this. x minus 4 is plus or minus 5. You keep going. That's actually where the quadratic equation comes from. Anyway, next problem. 16 was choice number D. I'm going to copy and paste this entire problem here. Let's go here and paste it here. OK, which of the following most accurately describes the translation of the graph y is equal to x plus 3 squared minus 2 to the graph y equals x minus 2 squared plus 2? So the y translation tends to be pretty easy to figure out. Let me just draw some example graph. So if I had the graph x squared, the graph x squared looks something like this. Let's see if I can draw it. The graph x squared looks something like this, right? And it intersects. When x is equal to 0, we're at our minimum point. And any other value increases in both directions. The graph of x squared plus 2, you're shifting up. This is the graph of x squared plus 2. You would shift it up by 2. The graph of x squared minus 2, you would shift down by 2. This would be x squared plus 2. This would be x squared minus 2. So the shift in the y direction is very easy to see. So if we're going from something minus 2 to plus 2 we're going to be shifting it up 4, right? So that's always the easy one to just eyeball and figure out. So we're definitely going to be shifting from minus 2 to 2. So it's up 4. It's either going to be choice A or choice D. The left/right shift is often a little bit more hard for people to visualize or to at least internalize, but I'll give you an attempt. Let's just go back to this. This is the graph of x squared, this yellow line right there. That's the graph of x squared. Let me ask you a question. What is the graph of x minus 3 squared? So does this shift it down 3 to the negative direction or 3 to the positive direction? Your intuition might say, oh, I'm subtracting 3. When I did minus 2, I shifted down. But it's actually the opposite here. Because you have to think about for what value of x am I going to have a 0 squared here? That happens with x is equal to 3. So you can think of it this way. Now, when we're at this point, when x is equal to 3, it's the same thing as this point when we have just x squared. When you put 3 in here, this whole expression becomes zero. As you get above 3, that's like going above zero. As you go below 3, that's like going below zero. So this graph will just get shifted to the right by 3. That's x minus 3 shifts to the right by 3. x plus three would go in the other direction, because when x is minus 3, that's when it would equal to zero. I haven't written that down. So let's think about this. We're going from x plus 3, so if this is x squared, x plus 3 would look something-- let me do it in a different color. x plus 3 is actually shifted to the left. The way I always think about it, there's two ways to think about it. The y shift is intuitive and the x shift might not be. If you have a plus 3 here, you're actually shifting in the downward direction. The way to actually think about the intuition is when will this whole expression equal 0? This whole expression equals 0 when x is equal to minus 3. So that's the point at which you're getting 0 squared. When I'm drawing these graphs, I'm not doing the y shift here. So this is going to be shifted to the left 3. This is going to be shifted to the right by 2. So if this is shifted to the left 3 and this is shifted to the right by 2, to go from this to this, you're shifting to the right by 5. So the actual graph x plus 3 squared minus 2 is going to be here. Then to go here, you have a plus 2, so you're shifting the graph up by 4, and then you're going to x minus 2. So this graph right here is going to be up here. So you're shifting up by 4 and then you're shifting to the right by 5. Actually, even if you're confused with your shifting left or right, you can just say the difference between plus 3 and minus 2 is 5, and 5 is only there. But you should hopefully understand the problems a little bit deeper than that. Anyway, I'll see you in the next video.