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 $5x^2 + 14x + 9$ – 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 $a x^2 + (a+c)x + c$, that its factorisation must be $(x+1)(ax+c)$.

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