Solving rational equations
Equation with two rational expressions (old example 3)
Solve and eliminate any extraneous solutions. And what they mean by extraneous solutions are, in the course of solving this rational equation right here, we might get some solutions that if we actually put it into the original problem would give us undefined expressions. And so those solutions are extraneous solutions. They actually don't apply. You actually want to throw them out. And so let's look at this equation. We have x squared over x plus 2 is equal to 4 over x plus 2. So right from the get-go, we don't know if this is going to necessarily be a solution to this equation. But we know, just looking at this, that if x is equal to negative 2, then this denominator and this denominator are going to be 0. And you're dividing by 0. It would be undefined. So we can, right from the get-go, exclude x is equal to negative 2. So x cannot be equal to negative 2. That would make either of these expressions undefined, on either side of the equation. So with that out of the way, let's try to solve it. So as a first step, we want to get the x plus 2 out of the denominator. So let's multiply both sides by x plus 2. X plus 2 divided by x plus 2 is just 1. And we can assume that x plus 2 isn't 0. So it's going to be defined. x plus 2 divided by x plus 2 is just 1. And so our equation has simplified to x squared is equal to 4. And you could probably do this in your head, but I want to do it properly. So you can write this. You could subtract 4 from both sides. Do it in kind of the proper quadratic equation form. So x squared minus 4 is equal to 0. I just subtracted 4 from both sides over here. And so you could factor this. This is a difference of squares. You get x plus 2 times x minus 2 is equal to 0. And then if this is equal to 0, if the product of two things are equal to 0, that means either one or both of them are equal to 0. So this tells us that x plus 2 is equal to 0 or x minus 2 is equal to 0. If you subtract 2 from both sides of this equation right here, you get x is equal to negative 2. If you add 2 to both sides of this equation right over here, you get x is equal to 2. And we're saying that either of these would make this last expression 0. Now, we know that we need to exclude one of them. We know that x cannot be equal to negative 2. So x equals negative 2 is an extraneous solution. It's not really a solution for-- it is a solution for this, once we got rid of the rational expressions. But it's not a solution for this original problem up here, because it would make the expressions undefined. It would cause you to divide by 0. So the only solution here is x is equal to 2. And you can check it yourself. If you do 2 squared, you get 4, over 2 plus 2 is 4. And that should be equal to 4 over 2 plus 2, over 4, which it definitely does. 1 is equal to 1.