The standard way to derive “the quadratic formula” is by completing the square, as explained, for example, at cut-the-knot. As most standard things, it is a bit boring, maybe because everybody have seen it so many times and everyone knows it.
Once upon a time I thought about a more entertaining way to derive this well-known formula for the roots of Here is what I had come up with.
To find the roots and of our equation is the same as to factor its left-hand side like . By expanding the right-hand side of this identity, we can see that and .
Now, it would be nice to have a formula for in terms of and Then we could have found the roots. But the trouble is that the equation doesn’t know which one of its roots is and which one is , and don’t change when we interchange and , while changes sign. It looks like we are out of luck.
Still, getting up to a sign is good enough, and doesn’t change when we flip and ; finding a formula for that would be good enough. Let us work on this idea.
We already know that , but , whence , and now we can figure out the roots. The quadratic formula will appear, of course.
I have written a mathematica script that shows how the roots of depend on , you may find it amusing.