Solve each equation. See Example 2. 2x/(x-2) = 5 + 4x^2/(x-2)

Question

Accepted Answer

Hey, everyone in this problem for the following equation we're asked to solve for X and we're given four X over X minus four is equal to eight plus six X squared over X minus four. So rewriting what we're given for X over X minus four is equal to eight plus six X squared over X minus four. All right. So this looks a little messy right now for an equation that we're solving for X four. But what we noticed is that we have this denominator of X minus four K two spots. If we can get this equation to have one main term, can one rational expression on one side with zero on the other side, then it's gonna be easier to solve for X. OK. So let's go ahead and find a common denominator um for all three terms so that we can simplify. OK. So again, we have X minus four on the left and the right. So let's get a denominator of X minus four on this term for eight as well. So how do we do that? Well, we have four X over X minus four. OK. And this is just like finding a common denominator when you're working with numbers, OK? We want to multiply eight by X minus four so that we can divide it by X minus four as well. OK. Now we're multiplying by X minus four, divided by X minus four. So just multiplying by one. So it doesn't change the equation. And then we have six X squared over X minus four. OK. So now we have a common denominator for all three terms and we can combine them. So let's move everything to the left hand side where we have our leading term. OK? Or sorry to the right hand side where we have our leading term, the left hand side, we're gonna leave a zero. So we're gonna move this over. We have eight X minus 32. OK. If we expand in the numerator, we have plus six X squared and then we have minus four X for when we move this term from the left hand side. And all of this is divided by our common denominator X minus four. All right. So let's just simplify the numerator. We get six X squared plus four X 32 divided by X minus four. Now you'll notice that everything in the numerator is divisible by two. OK. So let's go ahead divide both sides by two. And that's just gonna make our numbers a little bit smaller and simpler to work with. So divided by two. On the left hand side, we have zero divided by two, which just gives us zero. And then on the right hand side, we're gonna get three X squared plus two X minus 16. These are added and subtracted together. So we have to apply the division to each term and we're still dividing by X minus four. All right. So first thing we wanna do, we have this quadratic in the numerator. Let's go ahead and factor it because we know that those factors are gonna relate to when this is equal to zero. OK? We have everything factored, everything is multiplied together and it's easier to break it down into pieces. So you can factor this any way you'd like. OK. And remember if you get stuck factoring, you can always use a quadratic formula to help you find those roots. OK? And work backwards from there if you're looking for the factors. All right. So let me factor this. We're gonna have three X plus eight times X minus two and we're still dividing everything by X minus four. All right. So when is this equation going to equal zero? Right? Well, this equation is gonna equal zero when this numerator is equal to zero, right? Because zero divided by something is just gonna be zero. So what we're looking for is one is the numerator equal to zero. OK? So we can actually write this just as three X plus eight times X minus two. OK? We can ignore that denominator because we just want to find when the numerator is there, the only thing we have to be careful of is that we can't have a solution of X equals four. If we have a solution of X equals four, then in this original expression, we're gonna be dividing by zero, which we know we can't do because we have this restriction on our equation where X cannot be equal to four. OK? And it's important to write that down and to carry that through your work. And then when you get your final answer at the end, you can double check that you have um a solution that isn't a restriction. All right. So we have this equation on the right hand side is equal to zero. OK? These two things are multiplied. So if either one of them is zero, then the expression on the right, it's going to be zero and the equation is gonna be satisfied. So we are going to get from this first term, OK? If this is zero, then X is going to be negative 8/3. And from the second term, if X minus two is zero, then X is going to be equal to two. OK? And so we have two solutions that we found X equals negative 8/3 and X equals two. And we see that neither of them are this restriction of X equals or X not equal to four. OK. So we don't have any issues there. We can go back up to our answer choices. And we see that we found that our solution is going to be X equals negative 8/3 or X equals two. And that's gonna be answer D Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each equation. See Example 2. 2x/(x-2) = 5 + 4x^2/(x-2)

Key Concepts

Rational Equations

Cross Multiplication

Polynomial Functions

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