Look, they can’t all be huge posts. Sometimes you just spot something and it’s worth noting down.

For example, someone asked how to factorise 5x2+14x+9 – the kind of quadratic I could never factorise at school.

These days, I can immediately spot that it’s (x+1)(5x+9), barely having to think.

What’s the secret? It’s just that the x-coefficient in the middle (14) is the sum of the other two (5+9). And – having recently taught the 11yo all of my multiplying-by-11 tricks as a birthday present1, I had the idea in my head – when you multiply, say, 59 by 11, you write down the 5 in the hundreds place, the 9 in the ones, and add them up to get the tens (maybe cursing slightly about having to do a carry).

Multiplying by x+1 is the same thing – only you don’t need to fuss about the carry (or, as Bill would call it, the exchange. I have to concede that’s a better name).

It’s a small step from there to noticing that whenever your quadratic is of the form ax2+(a+c)x+c, that its factorisation must be (x+1)(ax+c).

  1. It’s a toss-up whether he preferred that or the lego. ↩