Video: Finding the Solution to a Rational Equation Containing Three Algebraic Fractions

Solve (3/(𝑛² βˆ’ 4)) + (1/(𝑛 + 2)) = 2/(𝑛 βˆ’ 2).

02:21

Video Transcript

Solve three over 𝑛 squared minus four plus one over 𝑛 plus two equals two over 𝑛 minus two.

We have three algebraic fractions. And there are a number of ways we can solve this. The key to any of these methods, though, is spotting that 𝑛 squared minus four could be factored. We can write it using the difference of two squares. We write it as 𝑛 minus two times 𝑛 plus two. And then we notice that this is the product of our other two denominators. So we could create a common denominator of 𝑛 minus two times 𝑛 plus two and gather all our terms on the left-hand side. Alternatively, we could subtract one over 𝑛 plus two from both sides of our equation. Let’s see what that would look like.

Our equation becomes three over 𝑛 squared minus four equals two over 𝑛 minus two minus one over 𝑛 plus two. Next, we’ll multiply the numerator and denominator of our first fraction by 𝑛 plus two and of our second by 𝑛 minus two, creating a common denominator of 𝑛 minus two times 𝑛 plus two. The numerators on the right-hand side become, respectively, two times 𝑛 plus two and one times 𝑛 minus two. And in fact, we saw we could write the denominator on the left-hand side as 𝑛 minus two times 𝑛 plus two.

Now, the denominators are equal on the right-hand side. We’re going to subtract one times 𝑛 minus two from two times 𝑛 plus two. And the expression on the right-hand side becomes two times 𝑛 plus two minus one times 𝑛 minus two over 𝑛 minus two times 𝑛 plus two. Notice now that the denominators of these two fractions are equal. And we’re told the fractions themselves are equal. So this means our numerators must be equal also. So three must be equal to two times 𝑛 plus two minus one times 𝑛 minus two.

Let’s distribute these parentheses, remembering that, on our second set of parentheses, we’re multiplying everything by negative one and our equation becomes three equals two 𝑛 plus four minus 𝑛 plus two. We simplify the right-hand side to get 𝑛 plus six. And we now see we have quite a simple equation that we can solve for 𝑛. We subtract six from both sides, giving us 𝑛 equals negative three. So the solution to our equation is 𝑛 equals negative three.

Remember, we could check the solution by substituting it back into the original equation and making sure that both sides are then equal.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.