The journey to the General Quadratic Formula is a long one. It should begin where there reader is, but I don’t know where that is. As a consequence, I will try to begin a little behind the reader so that I may after a short while catch up with him and then we may journey forward together.

Let us begin with the binomial. A binomial is an expression that has two terms connected by a plus sign, i.e.

Let it be understood now that and are stand ins. They might be representing complicated expressions. For example,

is a binomial with being replaced by $xy$ and being replaced by , and

is a binomial with being replaced by and being replaced by .

Let us now take two such binomials and and watch them interact. That is to say I will multiply by .

or, to summarize,

We will, for the most part, be concerned with a special case of this in which $A=C=x$. In this case,

or, to summarize,

That quadratic polynomial thing

Now recall the expression of the first chapter. Since , I can divide both sides of the equation by it and not change the value of required for the equation to hold true. That is to say, I can consider

If I am able to find values of and so that and , then

so that either and , that is to say or .

There are special cases in which we can, by trial and error, find and . Pursuing solutions in this way in an entertaining mental exercise similar to Sudoku, but, for practical purposes, a more general method is required and that general method is the General Quadratic Formula in whose derivation we are currently proceeding.