Solve each equation in Exercises 41–60 by making an appropriate s...

Question

Solve each equation in Exercises 41–60 by making an appropriate substitution.

x^(-2) - x^(-1) - 6 = 0

Accepted Answer

Welcome back. I'm so glad you're here. We've got this equation, we have six X to the negative second minus 17 X to the negative first plus five equals zero. And we are to solve using the substitution method. Well, what is the substitution method? Substitution method is used when you have something that looks like almost could be factored and you have some variables or factors that are really similar. Like you've got these two, X is here and you look at the first one and that is double the exponents of the second one. What you can do now is substitute in for the second one. So in this case X to the negative one and you can substitute whatever variable you want. In this case I'm going to use W. So it's now going to be negative 17 times W. And what do we do about that X to the negative two in the first term? Well, we recall that X to the negative one squared what we do to one side we do to the other negative one, raised to the second is going to be X raised to the negative two. So X to the negative two equals W squared. And that's exactly what we have for that first term. So it's going to be six times W squared minus 17. W plus five equals zero. And well, it's not intuitively like great, this is super easy to factor but we can factor this one. So we'll give it a go instead of looking at that six W squared. First. I want to look at this five because well five only has one in five as its only factors and they could be positive, they could be negative. We look at the middle and we see that they have to add up to be a negative 17. So both the one and the five are going to be negative in this case. So now we just have to think about that six W squared and how those factors can be used to get us to negative 17. We've got quite a ways to go down there. So let's multiply the negative five times three W to get negative 15 W. And then the other factor left is two W. So we'll be adding another negative to W. And yeah, that'll get us to negative 17 W. You can always go back and foil this really quick to make sure we did it correctly. Three W. Times two W. Six W squared. That's the first. Now the Outsides three W times negative five is negative 15. W. Our insides negative one times two W is negative two W. And our last negative one times negative five is plus five combined our middle like terms and we get six W squared minus 17. W plus five. And yes, this matches what we had before. So we chose our factors correctly. Now we can take our two factors three W minus one and two W minus five and we can set those both equal to zero to solve for W. We'll add one to both sides of our three, W minus one equals zero. And we have three. W equals one will divide both sides by three and we have W equals one third. For the other one will add five to both sides for two, W minus five equals zero. We'll have two W equals five and we'll divide both sides by two and we get W equals five halves. Well, before you rejoice that we have solved all of the problems, recall that we're not actually trying to solve for W we're trying to solve for X. So wherever we have a W we're going to substitute back in that X to the negative one. So instead of W equals one third, it's X to the negative one equals one third. And instead of W equals five halves. It's X. To the negative one equals five halves. And how do we get to that? Get rid of that to the -1. Well let's raise it to a higher power. Let's raise it to a negative one because then we're multiplying negative one times negative one. And that's just going to be positive one. But what we do to one side, we have to do the other. So we get X equals well one third to the negative one. That means we have to flip the numerator and the denominator. So X equals three as one of our results. And we'll do the same thing to this equation X to the negative one equals five. House will raise both sides to the negative one, and so we'll get X equals five halves to the negative one, which is the same as 2/5. Gotta flip the numerator and the denominator. So we look at our answer choices 2/5 and three and that matches answer choice B. Well done, we'll catch you on the next one.

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