The quadratic formula
Is there one formula that solves every quadratic, and where does it come from?
Where you are 4 inside
What this area is about
Every equation ax^2 + bx + c = 0 with a \neq 0 is solved by
It is not a new idea. It is completing the square done once, on letters instead of numbers, so you never have to do it again.
How these fit together
Where it comes from. Deriving the quadratic formula completes the square on the general equation. Read it and the formula stops being something to memorize.
The procedure, and the judgment. Using the quadratic formula is reading off a, b and c, substituting with care over signs, and simplifying the radical. It also covers choosing how to solve: the formula always works, but factoring, square roots or completing the square is often faster.
The part under the root. The discriminant is b^2 - 4ac. For real a, b and c, its sign says how many real roots there are and how the parabola meets the x-axis, and a negative value, where the formula gives complex roots, means the parabola never touches it.
A path through it
Derivation, then use and choosing a method, then the discriminant. The derivation assumes completing the square with any leading coefficient.
The formula does the solving in word problems, and it turns up well outside quadratics: in the golden ratio, and in finding the eigenvalues of a 2 \times 2 matrix.