Fancy Maths Stuff!

[Xei]

I just discovered quite a neat little thing which has made me happy. I thought it'd be cool to set it as a puzzle; you only need elementary mathematics. Also it'll be interesting to see if people have ever been taught this (I've never seen it). When you get it, it's very satisfying to watch how all the algebra comes together. Puzzle: find the expression in terms of x for the gradient of the general quadratic curve, y = ax^2 + bx + c, for every arbitrarily chosen value of x, without the use of any calculus. Spoiler for Big clue: Draw a tangent line to a quadratic curve an arbitrary value of x, and think about how it's different from all the other lines which touch the curve at this value of x (but don't have the right gradient).