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41 polnts SNlglrg' ` 1C30Find all solutions ct thc glven cquatlon_ (Entcr your answccomma-scparatcd Ilst. LctIntcger Round tcrms twc dccima Places wherc approp...

Question

41 polnts SNlglrg' ` 1C30Find all solutions ct thc glven cquatlon_ (Entcr your answccomma-scparatcd Ilst. LctIntcger Round tcrms twc dccima Places wherc appropriatc )cos(0)

41 polnts SNlglrg' ` 1C30 Find all solutions ct thc glven cquatlon_ (Entcr your answc comma-scparatcd Ilst. Lct Intcger Round tcrms twc dccima Places wherc appropriatc ) cos(0)



Answers

In Problems $41-72$, find the real solutions of each equation. $$x-4 x \sqrt{x}=0$$

Okay. Here. We're using an additional subtraction formula to solve the following, which is co sign they, uh, co Same three theta minus Scient. Ada signed three data equals zero. OK, so we're gonna use the identity which says co sign a co sign B minus. Sign A signed B is equal to co sign of a plus B. So the A for us a state A and B is three data so we can collapse this whole left hand side into CO ST Data plus three data go zero. So we get co sign of fourth data. Is he quit his era? This means that fourth data is he quit a pi over two. It's also equal to three pi over two. Okay, When we divide these two numbers by four, we're gonna get pi over eight and three pi over a okay being the two solutions in the interval from 0 to 2 pi. But there could be more when we divide by four. It can bring solutions into our interval. Okay, so we're gonna keep these going. We could have five pi over 27 pi over two nine pi over to essentially any odd number of pi over two. Okay, I'm just gonna throw a bunch of these down, maybe stop there for now and we'll make sure when we divide these by full if the resulting number is less than two pi. OK, so we'll have five pi over eight five divided by a is less than two seven pi over eight. That's less than two nine. Pie of eight That's less than two 11 over. A is bigger than one, but less than two. The team pie of a great maybe anymore 15 pi over two, then 17 by over to stop there because I run out of space. So 15 by over eight. That's just less than two right now. 17 pi over a 17 divided by eight bigger than two. So this one is not in our interval. So a solution air is going to be these. How many we got 12345678 Those eight solutions will solve this problem in the given interval

Okay, here we are. Solving Cho Sang Data Hussein. Three data by the scientific data signed three data because there are We're on the interval from 0 to 2 pi. So we're going to use the additions. Attraction Identity co sign a coastline B minus. Sign a signed. The is equal to co sign of a minus bi. Okay, so the A for us commuter be three data with data. I'm gonna make it. The three data means that might be is the data matter. We trade around, they go. So we condense this left hand side into coastline three data minus data. Okay, that dropped down nicely. So we'll get co same tooth data you call zero. We know that this means that to data can equal pi over two. Okay? Can also equal to reap i over to. Could also be five pi over two. Could also be seven pi over two Could also be nine by over two. We'll go ahead and divide all these by two. And to give his pi over full three pi over full five pi over four Again. We're checking all these a lesson to pie, which they are so far, seven Pi over four nine pile before is not less than two, so we'll stop with that one and we'll have just these four ounces right here to solve this problem.

When we're solving an equation that have the highest expo native to those are called quadratic six. And we have four different techniques at least to solve quadratic equations in this particular problem. This is a quadratic, the highest exponent would be too. And what we would normally do is to foil this piece, distribute this negative six, collect are like terms and have a quadratic without all of these parentheses and then choose one of our techniques to solve it and that will work. However, there's a lot of steps and more steps means more errors sometimes. So there's a little bit shorter way to do this. And what it is is we call this use substitution where we substitute you in different variables or different values. In the original problem we choose the you by being the sort of the middle term without the coefficient. So I'm gonna let the U. B. This three X plus four. And if I substitute the you in to this problem you see that I get you squared -6. You plus nine equals zero. And this is also in quadratic form but a lot easier quadratic than what's the one we had before. So this we can solve by factoring this would be U -3 times U -3 equals zero. And setting each factor equal to zero. Well I guess I only need to do one since it's the same factor When we get you equals three and we're pretty happy. But we have to remember that our original problem was in terms of X is not used so we don't care what you is. So what we need to do is to put this back into X. Is so we're gonna plug in you being three back here and then solve for X. So if you equals three you'd have three equals three. X plus four. And now solving for the X. You can see that we end up with X equals negative one third. And so we solved it, we got the right answer, but we just did this in a lot fewer steps. And so our solution is just negative one third.

It was solving following trick equation that sign three data coastline data I minus co sign three data scientists Torrico zero. So we're gonna use an identity here. Which is that sine a coastline Be minus co sign a signed B is equal to sign of a minus bi. So that a for us, This is three data and the B he's just stated. Hey, so this whole left hand side up here will collapse. It's a sign. Three data minus data. This is equal to zero. So we're going to get signed to data equals 20 So we can say that to theatre is equal to We're looking for places on the unit circle where the y coordinate zero. So over here and over here, this would be zero by two pi. I'm gonna write some more out. Three by full pyre. Five pi. So on. And so Okay, the reason I wrote a couple more because they now have to divide this to off. So I divide all of these by two. I'm just gonna keep the ones that are less than pi over two. So zero divided by two is 20 Ex solution is pi over two to fire over to is just fine. I've got three pi over two four. Pi over two is just two pi was lost. Solution five pi over two. I don't need that one because that's greater than two pi. OK, I'm actually not going to use the to fight either, because my interval was zero the two pi and not including two pi right with the RNC. Then it doesn't. It's not included, So actually go with just thieves. Florence is right here.


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