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## Solving rational equations

# Equation with two rational expressions (old example 3)

## Video transcript

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.