When I was in my first year at secondary school I used to walk down the road to the bus stop with a fellow pupil two years older than me. He was a rounded lad with, I recall, red hair and his name was Phil. I was learning to solve linear and simultaneous equations at the time and was becoming aware of quadratic equations, a topic which fascinated me. I also knew he had already studied this in his maths class ...
Suppose we have x2 - 5x + 6 = 0, or, less mysteriously to a 12 year old,
5x = x2 + 6.
Look! x occurs on both sides of the equation but not in a way which can be eliminated. There seems no easy way to proceed, but I had heard that there was a mysterious technique using "factorisation". I badgered Phil to tell me what the secret was.
As we sat on the 25 minute ride to the centre of Bristol, Phil wearily explained that you write:
x2 - 5x + 6 = (x-3)(x-2) = 0.
"So x must be either 3 or 2, see?" he concluded brusquely.
I did not see. Surely x couldn't be two things at once. I was happy that the values of 2 and 3 both 'worked' in the sense that each satisfied the equation - and this itself was a source of wonderment, two different values and they both work! The teacher eventually explained that if two things, when multiplied together, equal zero, then one or the other must itself be zero because otherwise their product would be something non-zero. So take your pick of the two factors.
This is more sophisticated reasoning than it appears. The equation doesn't tell you a fact about a pre-existing x (as I thought, hence my confusion as to how it could be two different things at once). Instead, the equation is a kind of sieve, a constraint which "checks" all possible values for x and only allows through those for which it is true. I confess I did not have a mental model of the entire number line flowing through the equation, with just 2 and 3 being sieved out!
Quadratic equations are not trivial. Consider that we could rearrange x2 - 5x + 6 = 0 as an iterative equation
xn+1 = (xn2 + 6)/5.
A solution emerges when xn+1 and xn are equal, but if you start with a blind guess, perhaps the sequence you get by plugging each xn+1 back into the equation's right-hand-side will 'converge' to the right answer?
Is it obvious if, or when, this happens? Look at the spreadsheet below: starting values run along the top row in yellow; below we have twenty iterations. When stuff starts getting big, it really gets big and blows Excel apart, so not all the cells have numbers in them.
It's a bit more advanced to work out the domains of convergence, and to realise that 2 is a local attractor and 3 isn't, so you'd get just one of the two solutions if you relied on naive iteration.
I wonder how many kids in their school maths classes today experience the wow factor of getting to really know quadratic equations?