The Radical Equation That Hides Everything

[00:00:00]Hello everyone. In this video, we're going to be solving a problem with radicals. We have a rationally radical or radically rational expression roo<unk> of x / x + roo<unk> of x + 1 is equal to a. And now we're going to be solving for the second expression in terms of a. So in other words, the first one is given as a. We need to now evaluate this expression in terms of a because they are related. Right? Great.

[00:00:29]So let's go ahead and take a look at this problem. I'm going to be presenting at least two methods. Maybe there there are three ways to solve it. Let's go ahead and see how we can do that. So I'm going to start with the first method.

[00:00:39]And usually my first method is kind of complicated. Okay, it's usually harder than the other methods. So we're going to start with the given expression. Of course, we have this radical thing and let's go ahead and rewrite it. square root of x / x + x + 1 is equal to a. So our answer should be in terms of a.

[00:01:03]Okay, a function of a. In other words, so to solve this equation, I'm going to cross multiply and I want to turn this into a radical equation. So square root of x, maybe I should multiply this first because that that looks more complicated. ax + a<unk>x + a is equal to square<unk> of x. And since I want to isolate the radical, I'm going to leave this as ax plus a and then put the square root of x on the other side.

[00:01:29]Square root of x - a * the square root of x. Now obviously I would like to square root of x, take it out and then square both sides. Now I can do it in two ways. I can square both sides as is or divide by 1 minus a first and then square. No big deal. It's going to turn out to be the same thing anyways. Okay.

[00:01:50]Now, let's go ahead and square both sides. Our goal is to get rid of the radicals. And of course, I'm trying to solve for x. Why? Because if you remember in the second expression, I have x and x squ. If I can get an expression for x, I can definitely plug it in, replace x with that, and hopefully when I simplify, if I can, it'll give me an expression in terms of a. Now, let's go ahead and square the left hand side. You could also do this.

[00:02:15]Maybe that'll make it a little easier. I don't know. But this will be a^2 * x^2 + 2x + 1. On the right hand side, you're going to have x * 1 - a^ 2, which is 1 + a 2 - 2 a. I write it in different ways.

[00:02:30]Don't worry about it. It's the same formula. Okay. Now, we're going to get a 2 x^2 + 2 a 2 x + a 2. And then from here, we're going to get x + a 2 x - 2 ax. Notice that I'm trying to write the x uh a first and then x because I want to turn this into a quadratic was a radical. Now I want to turn it into a quadratic equation in x. Okay, let's go ahead and put the x² then x together. So we have only one term with x^2. So it's good. We can keep that. Now I have 2 a 2 xus a 2 x. That's going to give me a 2 x. But I have to subtract this. It's going to be a 2 + 2 a because we're subtracting a negative. So this will be the coefficient of x. We've taken care of everything with x except for this one. And there's a minus one. So I'm going to have to include that in there so that we can package everything together. Make sense? And then our my constant is going to be of course a squared in this case. Beautiful, right?

[00:03:30]Don't you love that? This is a quadratic. A quadratic with a parameter.

[00:03:35]How nice is that? And definitely this will give you a family of quadratics.

[00:03:39]For every value of a except for a equals zero, you do get a parabola. So you have a family of parabas. How nice. And from here you can basically look at different values of a. What happens to the parabola? So on and so forth. But I'm not interested in any of that. I'm not doing a parametric analysis. I'm just interested in solving for x. Who cares about parameters, right? Sometimes you do. So let's go ahead and solve for x.

[00:04:03]And to be able to solve for x, I do need the quadratic formula because I don't think you're going to be able to factor this. You can give it a try. Maybe there's a way to do it. I can't see it right away. Now, the quadratic formula tells us negative b. This is b, the coefficient of x. So, I have to negate it. That's going to look like this.

[00:04:21]Plus minus uh oh, that looks like an h, which I don't like at all. definitely uh plus minus the square root of if you remember the formula b plus - of b square - 4 a c right b square so I'm going to square it so it doesn't matter whether it's the same or it's opposite squared it minus 4 a c minus 4 a ac is going to give me air conditioning but at the same time a to the fourth power and air conditioning is today is a warm day in southern California all over two a^2 awesome Now what am I going to do with this? Well, if you want to focus inside the radical because that is complicated.

[00:05:00]So let's call that delta Greek letter and delta is discriminant. Okay, let's go ahead and simplify this. So it's going to be again I squeeze these together so it doesn't doesn't look good. Oops. Okay, there we go. We should always have some space. Now delta is going to equal the following. I have to square everything. a 4th + 4 a 2 + 1 and then I will do the 2xy 2 y z so on and so forth. It's going to give me 4 a cubed - 2 a^ 2 and then finally - 4 a and then finally - 4 a to the 4th.

[00:05:35]Beautiful. How nice, right? Now, delta is going to be if you kind of, you know, simplify some terms. First of all, this is going to give you -3 a 4th + 4 a cubed. Okay. And then I have 4 a^ 2 - 2 a^ 2 which is a 2 a 2. And then you have the minus 4 a and then + one. Well, it didn't really turn out that nice, but again, we did get a solution, right?

[00:06:03]Didn't we? Now, if you go ahead and plug it in, x value is going to be something like this. And I probably made a mistake somewhere. I don't know. I'm too lazy.

[00:06:11]I'm not going to go back and try to fix it because maybe I didn't make a mistake. Who knows, right? Plus minus the square root of delta. Now, delta comes up again, but in a more simplified manner. 4 a cubed + 2 a 2 minus 4 a + 1.

[00:06:26]I don't think this is a perfect square, by the way, to be honest with you. So, I could be wrong, but this should. And of course at the end I have to divide by 2 a^ 2. Great. So this is your x value.

[00:06:36]Are you ready to plug it in? Just kidding. We're not going to do it because you know what? This is going to be super duper complicated. I mean who would who would want to substitute actually that I mean probably you can do it w from alpha but imagine plugging it into this and I think I should test it or if you someone can test it with alpha and see if you're going to get the simple answer that we're going to be get getting with the second method. Okay, here's the second method. A lot simpler, but let me know what you think. Okay, does the question was root of x / x + x + 1 is equal to a. And we're trying to solve for x / x^2 + x + 1 in terms of a.

[00:07:16]All right, so that's the problem. Let's see. We can solve it in a much much nicer way. So this what I'm going to do.

[00:07:23]Obviously square root of x cannot be zero, can it? If x is zero, then a is zero. Okay. And if x is zero, this is going to turn into a zero. Again, it's not really going to help us find our expression in terms of a. So, it's kind of useless, but you can see we're going to assume or we have to accept that x does not equal zero in this case. So, divide everything by square root of x.

[00:07:48]You're going to get something like this, something like this, and something like this. And that's still equal to a because we only only work with the fraction. This is square root of x. So from here I get 1 / roo<unk> of x + 1 + 1 / x = a. Let's go ahead and switch sides. x + 1 + 1 / x is equal to 1 / a.

[00:08:12]As you know, these are cross multiply.

[00:08:13]So we can switch them. Make sense? It's a very practical approach to solving equations. A lot of times people struggle with am I going to multiply?

[00:08:20]No, you can just switch them around and that'll give you the solution. Does that make sense? For example, let me give you an example. Let's say you're trying to solve this equation. Switch x and three you get x= 2/3. Easy. Now I want to put these two together.

[00:08:34]They're related. So and subtract one. So from here we get the following. If you isolate this expression, we get that.

[00:08:43]And of course I can make a common denominator and something like this.

[00:08:46]Great. Now this is helpful. And let's go ahead and erase this and write just replace it with 1 minus a over a because we know that it's going to equal that at the end. Great. Now, let's go ahead and save this because we're going to use it.

[00:08:57]How do we use it? Well, let's go ahead and work with our expression and then we'll come back to this. Okay, remember our expression that we're trying to evaluate was something like this. Well, let's call it equal to B. And our goal is going to be to solve for B. Okay, in terms of A. That's what we're looking for, right? To do that, I'm going to use the same strategy. Or can I can I divide the numerator and denominator by square root of X? No. You're going to divide by x this time. Why? Because we have an x here. So divide by x, you're going to get one. This is going to turn into an x. This is going to turn into a one. And this is going to turn into 1 /x. And b is unchanged. Beautiful. Now, how is that helpful? We can again do the same thing. Since we're looking for B, let's go ahead and save it, not change it. And let's go ahead and switch these around.

[00:09:45]And this is B. And since I'm looking for B, I need to know we're miss.

[00:09:52]Is that how you say it? If you're German, let us know in the comment section and correct my German. I learned I studied German for a while, but I forgot because I didn't speak it for too long. Now, this is what we're trying to find. So, it basically hinges upon this.

[00:10:05]I need to find x + 1/x. But can I find it? Absolutely. Look at that. You have root of x plus 1 over x. Let's go ahead and copy that. Square root of x + 1 over x is equal to, if you remember, it was 1 - a, right? Maybe I don't remember correctly. Okay, I remember correctly.

[00:10:21]Good. It's rare. Now what are we going to do? We need to find x plus 1 /x. To do that, tada, you need to square both sides. Easy, right? Math is easy if you know how to do it. Now when you square a plus b or x + y or m + n, you're going to square the terms and then you know do the 2 a thing. So it's going to be x + 1 /x + 2 * roo<unk> of x * 1 / of x. Of course, that cancels out. That's the beauty of this problem. And we did a similar problem before, by the way, a long time ago, but it wasn't a radical.

[00:10:54]Now, I got x + 1 /x, didn't I? If I subtract two from both sides, I actually got what I'm looking for almost. Okay, this is the answer. Now, you could definitely simplify this, square it, subtract two, whatever, so on and so forth. That doesn't matter that much. It's, you know, easy algebra. But here's what I need to worry about. replace x + 1 /x with that so that you can find b. What is that? It's 1 - a / aantity^ squar minus 2 and then of course there's a plus one which turns into a minus one.

[00:11:28]So that is the answer in terms of a.

[00:11:30]Uhoh. If we substituted this gigantic ginormous radical into this, would that work? I don't know. You need to test it out and let me know because I'm way too lazy to try it. But please let us know if you can do that. That would be awesome. I really appreciate it. But the answer is that. And if you wanted to simplify this more, you could square this, subtract one, make a common denominator, flip it, and get the answer. So let's do it. Why not, right?

[00:11:58]Since we already got the answer this far, let's go ahead and finish it up and, you know, uh, get the final answer.

[00:12:05]Now, here's what we're going to do.

[00:12:07]We're going to go ahead and square this expression. So b is what we're looking for. Remember, I'm going to square the numerator. 1 + a^ 2 minus uh 2 a and then it looks weird when you write it that way minus 1 and then of course I need to make a common denominator is going to be 1 + a^ 2 - 2 a - a 2 / a 2 a 2 cancels out how nice when you flip b is going to be a 2 / 1 - 2 a are you serious it can be that simple wild and surprising I didn't even know the answer when I wrote this problem. Anyways, this brings us to the end of this video.

[00:12:47]Thanks for watching. Hope you enjoyed it. Please let me know. Don't forget to comment, like, and subscribe. I'll see you next time in another video. Until then, be safe. Take care. Don't forget to check out A+BI. Until next time.

[00:12:58]Bye-bye.