I thought of this new method of deriving the quadratic formula using the relationship between the zeroes and the coefficients of a quadratic equation, namely, the fact that if we have a quadratic equation ax^2 + bx + c, then the sum of its zeroes is (-b/a) and their product is (c/a). Using this, we can find what the difference will be and we will have a system of linear equations in two variables which we can easily solve thereafter.

Here's the link to a question I posted on Mathematics Stack Exchange : https://math.stackexchange.com/questions/3632052/a-new-method-to-derive-the-quadratic-formula

Let me know aout your views on it, thank you for your time.

Good stuff. It's related to the fact that the zeros are symmetrical about the vertical line through the vertex. Some people use the formula to find the zeros and then average them to find the x-coordinate of the vertex. Others, sadly, just give them x=-b/2a and away we go.

Sorry, that's a bit of a rant, but your proof is good stuff - Ian.