Deriving the Quadratic Formula
April 25, 2021
In any high school pre-calculus course, students are taught various strategies for finding the zeros of quadratic functions, one of those being the use of the quadratic-formula.
For reference, here is the formula where and are the zeros of a quadratic function and , and are its coefficients:
Students are often told to memorize this formula without any further explaination, though I believe that if some extra insight as to how the formula can be found was provided along with its introduction, that task of memorization may feel a little less daunting for some.
The common methods of deriving the quadratic-formula can feel just as mysterious to students as the formula itself, however,
in this post I'd like to walk through and explain the steps for a method of deriving the formula which I find to be fairly intuitive.
A general problem
Imagine a scenerio where we have no prior knowledge of the quadratic-formula and we are told to find the zeros of a quadratic function whose coefficients are arbitrary constants, namely and .
Such a function would have the following form:
Defining the zeros
One's first instinct may be to factor the function, but that is going to be tough to do since we don't know anything about the coefficients other than the fact that they are numbers.
However, what we can do is introduce a few new variables in order to write in its factored form.
Let's see what that would look like:
While is in this form, we can see that if either or .
If we isolate both terms and solve for we will have found the zeros of , those being and .
So we've found our zeros, however, their definitions are in terms of variables we don't yet know the values of.
The only values which are known to us are the constant coefficients of , so we need to find a way to redefine both and in terms of and .
Redefining the coefficients
Even though we have defined the factored form of in terms of a few arbitrary variables, it is still representative of our original function, meaning that if we were to expand the factored form of it would provide us with new definitions for and in terms of our new variables.
This expansion goes as follows:
Here is a clearer view of our new definitions:
We now have formal definitions for each of our coefficients, but in a similar fashion to and , they are defined in terms of the variables we recently introduced and know nothing about, so these definitions aren't going to help us much right now.
Finding a middle-ground
At this point it's not all too obvious what should be done next, however, I'm going to take a moment to bring up a property which will help us manipulate our definitions in order to eliminate some of the unknown variables.
The property goes as follows:
The middle value between any two numbers and is equal to
It may not be clear as to how this property could help us, but try and stick with me as we go through these next couple of steps, as things should soon start making sense.
Let's recall our definiton of .
Since is the sum of and , by the previous property we can see that the value which lies in between and should be .
Let's call this new value for middle.
Going the distance
There is a small detail in the property I previously mentioned which I ommitted for clarity purposes, but we now need it in order to move forward.
The following is a version of the previous property which includes this detail:
The middle value between any two numbers and is equal to and resides at a distance from both and
This additional detail should make sense, as a middle object is defined by the fact that the distance between it and all of it's surrounding objects is equal
This version of the property denotes this distance by , and it can be used for traveling to the left and right of in order to reach both and respectively.
For our particular instance, this means we can travel a distance to the left and right of in order to reach both and respectively.
This travel can be represented in both and by rewriting them as the following:
With these two new equations we can now define in terms of and .
Now we can solve for .
We've diverged pretty far from our original problem, but we now have all of the nessesary tools in order to solve for the zeros of as we initially set out to do.
Let's recall the definition of these zeros
As I had mentioned before, this definition isn't very useful since it is not in terms that we can work with, but we can now fix that using the new definitions we've worked to find.
We can now write our zeros out as the following:
Now we can make a few substitutions and fully simplify.
And there we have it, the quadratic-formula.