12. Where did the Quadratic Formula come from?
Show students the derivation of the quadratic formula. It is easy to understand, uses the techniques that they are just learning (completing the square, and factoring) and students seem to enjoy seeing where it comes from. Here it is below, for your reference.
Start with the general form of the second degree polynomial:
Move the 
term to the right side.
Divide both sides by
to get a coefficient of 1 in front of
Take 
, divide by 2, getting
then square top and bottom, getting
Add this to both sides, thus completing the square.
Since we completed the square, the left side factors exactly:
Put the right side over a common denominator:
Change 
to the more familiar form of
which we know as the “discriminant”.
Take the square root of both sides:
Simplify the radicals, and add the
because square roots can occur two ways:
Subtract from both sides:
Put both terms over a common denominator, and voila, its the quadratic formula!
