Question Video: Adding Algebraic Fractions with 𝑥 in the Denomerator Mathematics

Simplify (4/(𝑥 + 3)) + (2/(𝑥 − 2)).


Video Transcript

Simplify four over 𝑥 plus three plus two over 𝑥 minus two. Well it looks a little bit more complicated, but the process we’re gonna follow is exactly the same as what we’ve just been doing. Now the one thing that I would recommend is putting brackets around your denominators, cause that’s gonna make life a little bit easier as we go through here and make sure we don’t make any mistakes. So we’re gonna take our first denominator and multiply the top and the bottom of the other fraction by that value. And we’ll take the second denominator and multiply the top and the bottom of the first fraction by that value. So the principle still sticks, 𝑥 minus two divided by 𝑥 minus two is just one. So we’ve still got one times the first fraction, so it’s still the first fraction. And 𝑥 plus three divided by 𝑥 plus three is still one. So we’ve got one times the second fraction. So it’s still just two over 𝑥 minus two. Okay let’s multiply those out. Well we combine them into one fraction here. We’ve got four lots of 𝑥 minus two plus two lots of 𝑥 plus three all over the common denominator of 𝑥 minus two times 𝑥 plus three. So now we’re going to do four lots of 𝑥 and four lots of negative two and we’re gonna add two lots of 𝑥 and two lots of positive three. So four lots of 𝑥 is four 𝑥 and four lots of negative two is negative eight. Two lots of 𝑥 is positive two 𝑥 and two lots of three is positive six. Now I’m not multiplying out the denominator at this stage, because we’re not finished tidying up the numerator. Who knows? Things might factorise and something might cancel out. If I multiply the denominator out, I wouldn’t be able to spot that.

Well four 𝑥 plus two 𝑥 is six 𝑥 and negative eight plus six is negative two. So we’ve got six 𝑥 minus two over 𝑥 minus two times 𝑥 plus three. But in fact, if we look carefully at the numerator, we can see that that will factorise. Six and two got common factor of two. So what do I have to multiply two by to get six 𝑥? Well that would be three 𝑥. And what do I have to multiply two by to get negative two? Well that would be minus one. So there we go. Now what we’ve ended up with inside the bracket isn’t the same as any of the brackets of the bottom. So nothing is gonna cancel, but this is a nice factored format of our answer. Now if I had multiplied out the denominator and I hadn’t factored the numerator, that’s the solution I would have got and still be a perfectly correct answer. But people tend to leave it in this factored format rather than this multiplied out format.

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