0 = x−x

Extract

Everyone knows that x−x = 0, but how many people realise the power of reversing this equality and writing 0 = x−x? What I mean is that we can always add and subtract the same quantity to and from any algebraic expression without altering its value. In general, we may choose infinitely many values of x, and the skill lies in choosing the most suitable x so that progress can be made in a specific problem. Beginners in mathematics are often mystified when a particular x is chosen and almost invariably respond with something like “I agree that what you are doing is correct, but why on earth did you pick that particular x?” There is no answer to this except perhaps “because it works!”.