Quadratic Formula

The derivation of the quadratic formula

This is the most popular method for solving quadratic equations since it is easy to remember and applies to many quadratic equations that arise in science, engineering and even finances.

Many science and engineering problems produce quadratic equations that must be solved to obtain the desired answers. Most scientists and engineers immediately use the quadratic formula to quickly solve such equations. Generally they have this formula memorized and write it down without much thought. It is a great tool and can be applied to so many situations that it is worth memorizing.

In this article, we will show how to derive the quadratic equation and then show a couple of examples of how to use it.

A quadratic equation is an equation that can be put in the form:

Whereis the unknown variable and , and are the known coefficients. It is called “quadratic” because it is a second order polynomial equation in a single variable. In other words, there is an unknown variable of second degree or raised to the power 2. The word, “quad…” actually refers to squares which have 4 sides, but you “square” the length of one side (meaning raising it to the power 2) to get the area of a square.

Scientists, engineers and a whole bunch of high school students are very familiar with the quick solution to a quadratic equation called the quadratic formula. The formula looks like this:

This is actually two solutions for in terms of the known coefficients, which is exactly how many solutions you expect for a quadratic equation.

In this article, we will derive the quadratic formula as a solution to the given equation. Deriving this formula is quite easy, but after we learn the formula, most people never bother to think about where is comes from, they just use it.

Let’s start with  and try to solve for the variable, . Our first step is to get the  by itself. To do that we just divide the equation by the coefficient  to get:

Then we take the resulting constant term to the other side of the equation by subtracting it from both sides

Now that we have this equation,

we are going to try to find a solution by “completing the square”. In this case we will add the same term to both sides of the equation. The term we are adding is one that makes the left hand side (LHS) look like the square of a single term.

The reason we chose  is because

So now we can write:

which gives us a single squared term on the LHS and only known coefficients on the right hand side (RHS).

A solution is fairly easy to arrive at from here since all we need to do is take the square root of both sides. Of course, when you take the square root you must allow for the fact that the thing that was squared could have been either positive or negative and you would have gotten the same value after squaring. To take that into account, we just realize that there are two possible values on the RHS, a positive one and a negative one.

Just to make things a little more aesthetically pleasing, we usually reverse the order of the terms under the radical so that it is written:

Of course, we really don’t want the solution to , we want the solution to , so we need to move the constant term on the LHS to the RHS leaving our variable by itself on the LHS.

Now we have the variable all alone, but we can clean up the LHS a bit.

Which finally reduces to:

Which is the only part most people bother to recall.