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Which of the following are set of solutions for the following equation?t2x" + 5tx' + 4 = 0lt)2,t 21t)+2 lnt)lt)(t,lnt)...

Question

Which of the following are set of solutions for the following equation?t2x" + 5tx' + 4 = 0lt)2,t 21t)+2 lnt)lt)(t,lnt)

Which of the following are set of solutions for the following equation? t2x" + 5tx' + 4 = 0 lt) 2,t 21t) +2 lnt) lt) (t,lnt)



Answers

Find all real or imaginary solutions to each equation. Use the method of your choice. $$5 t^{2}+4 t-3=0$$

Given in question is the power one by two minus two people one by four plus one is equal Do zero. So hair Let us suppose be power one by four is equal X So here, people one by two we can write this deep Our one by four Foley square minus two people one by four Bless one equals zero. We can play this and x squared minus two x plus one is 4 to 0. So this isn't a fall off a minus beholds question but this X minus going hold square record of you. So that means we have actually doing so. If X is one witness we have access to people are owned by foreign quit which is given to use it One powerful digital So now we'll check this answer. So as we have these 4 to 1 So here we'll put one part on my do Mine is too one part one best wrestler which used one minus two plus months which is equal to zero. And since this is satisfying, so does solutions. It is left

Okay, let's go ahead and solve this equation. The first thing to kind of take a look at is the structure that we have. So we have something that's squared something that is to the first power and then just a number by itself. So even though it doesn't look like it, this is actually a quadratic type of equation as long as we do some U substitution. So let's do this. Let us let u equal t over t minus one. And if we do that, we end up with eight. You squared minus two U minus three is equal to zero. Now we can actually go ahead and factor that by using the A C method where A is equal to eight and C is equal to negative three. So we do a times C. We end up getting negative 24. So I need to think of two numbers that multiply to negative 24 will add to negative six. Thankfully, negative six and four. Well, multiply negative 24 they will also add to negative two. So that will look like this. Eight. You squared. Plus for you minus six. You minus three is equal to zero. And now let's go ahead and do some factor by grouping. So over here on the left portion, Um, I can take out a four you as the largest common term, which will leave to you plus one. And then, on the left side, the largest common term. Aiken, take out its three, and here's a pro tip. If you have a negative take out a negative, that's a negative three. And once we do that, we're left with two you plus one. So when it's all said and done, fully factored, it is to you, plus one times for U minus three equaling zero. Now let us use the zero product property to solve for you. So that way you is completely by itself. And I'll make it easier for when we have to sub back in our tea over t minus one. So solving this very quickly, we will see that we end up with U is equal to negative one half on the left. I got okay. And then on the right portion of our equation, once the end solving very quickly. Uh, we're just doing some simple algebra by adding and dividing. We end up with you is equal to 3/4. So now we can substitute back in what you was because we let u equal t over T minus one. So I have t over t minus one equals negative one half and t over T minus one equals 3/4. So now I really need to focus my attention on these equations. They're gonna be pretty similar, but they're going to be different, of course, because they have two different numbers. My recommendation is when you have a fraction, I'm not a big fan of fractions, so I would multiply by two on both sides first giving us to tea T minus one. This will be a negative one. Then we will multiply both sides by T minus one. Once again, the whole point is just to get rid of that denominator. I don't like dealing with fractions. So this is the easiest way that I have found. And make sure that you also distribute so negative one times to negative one times one the negative won t plus one and solving for this for tea, add one tea to both sides so three t equals one and then divide by three. So that T is equal to one third. Okay, the right. The equation on the right hand side, we very, very similar. But once again, just give me some different numbers. So here is the completed work for the equation on the right hand side. Okay. So very, very similar. Same process, literally multiplying by four over here on both sides instead of to to get rid of that fraction. Distributing and multiplying by T minus one. So you end up with a final answer of T is equal to one third and T is equal to negative one.

In this problem. We're gonna go ahead and solve for this linear solve this linear equation and find out if it has no solutions or all riel numbers as solutions. I'll start by distributing on the left side eight times three T, which is 2040 and eight times four, which is a plus 32 on the right side. Let's distribute seven is seven times to which is 14 and then ah, 14 t and then seven times the five, which is a plus 35. Let's combine some like terms on the right side. So 10 T plus 14 t is 2040 and then a Nega three and a plus 35 is a plus 32. Okay, when we go to subtract 2040 to get rid of one of the 2014 is we get rid of both of them. Actually, they both cancel. So we're left with 32 on the left side and 32 on the right side. This happens to be true. So we call this solution all riel numbers. They're all real numbers are solution

You've been in the COMESA pic about ticket about four minutes. When a vehicle could you the sort of look deep from the disagrees? No. Yeah. Do you square difficult to get material adding both side one not taking this wild boasts a We never will take us quite a boost said we have right here plus minus sign. So here you get these guys do less minus What now? We have been to build up this way. So you can I d s critical to one. Who are these guys? Will do minus one. Now take this crown again. Then you really get less minus suit under one. Here you get teachable toe last minus one under minus one. No! From here you get the below teach plus minus one and from here will get the developed last minus. I Rilya, I m squared ableto minus one in there. Giving in a complex number ticket in a complexion number. I is magical to minus one. So we were poor buildup. Be one minus one Jewish. Really, Bill. And do it in Arabic. I and minus I Thank you


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