Towards the end of a GCSE paper, you’re quite frequently asked to simplify an algebraic fraction like:

$\frac{4x^2 + 12x - 7}{2x^2 + 5x - 3}$

Hold back the tears, dear students, hold back the tears. These are easier than they look. There’s one thing you need to know: algebraic fractions are happiest when they’re in brackets.

If you’re a regular reader, you’ll know how to put quadratic expressions in brackets - check out this article, or this one if you prefer.

To factorise the top, you convert it to $X^2 + 12X - 28$ (shuttling the 4) and factorise: $(X+14)(X-2)$. You then move a two from the 14 to the opposite $X$, and a two from the -2 to to opposite X to get $(2x - 7)(2x - 1)$. Lovely.

The bottom works much the same way: it becomes $X^2 + 5X - 6$, or $(X+6)(X-1)$. Shuttle a 2 from the -6 to the other X to get $(x-3)(2 x -1)$.

The fraction is now $\frac{(2 x -7)(2 x -1)}{(x-3)(2 x -1)}$. Aha! There’s a common factor of $(2 x -1)$ on the top and the bottom, so we can cancel that to get: $\frac{2x -7}{x-3}$ - which is our fully-simplified answer.

Once you’ve done a handful of these, you’ll start to get a Pavlovian response to this kind of algebraic fraction, and dive straight in!

One thing to look out for is difference of two squares, which comes up once in a while and catches some students out. But you’re smarter than that, right? Right.

* Edited 2014-09-11 for clarity and to fix LaTeX errors.