The Quadratic Formula

Why does this page even exist? Well, if you are simply looking for the Quadratic Formula, you have found it here. (Look over to your right near the top of the column... it is hard to miss.) But while you are here, if you would like to follow along, you can learn how the formula was derived and why it works.

The Quadratic Formula is a derivation of the general form of the equation:
where          

This equation, when solved for , ultimately will yield two solutions. Once the formula is known, it is very much easier to use than running through the calculations we show here to derive it.

Specific Example of a Quadratic Equation

Let's start with an example where we show how to solve a typical quadratic equation. The equation we'll solve for will be:

A cursory look here shows that the equation is not in the standard form that we need it to be to proceed. Let's start by subtracting 60 from both sides.
Then, let's divide everything through by 9 to get the coefficient on the first term to be 1.


Now reduce the coefficient on the second term to its lowest terms and leave it as an improper fraction:

Completing The Square

Next we apply a technique called "completing the square". What this means is that we are going to add some amount to both sides of the equation that will allow us to collect the terms on the left side in the form . The "?" in this term will be our missing "square" number.

But what number do we need to add? The answer exemplifies the name, "completing the square". We start by looking at the coefficient on the second term. We take one half of that number, then square it. The result is what we add to our equation on both sides. This will "complete the square" and allow us to very easily summarize the left side.



Now expand the squared fraction on the right side of the equation.



Then we do our summarization and collection.



Take the square root of both sides.



And then we simplify. Notice the plus/minus sign on the right side. This is because the left hand number in either sign when squared will give a positive result.



Subtract the numeric term remaining on the left from both sides (once with the positive right term and once with the negative right term.)





Collect the numbers and we have our answers!

Checking Our Work

So now that we have our answers, we need to check our work. Let's put the answers back into the original equation and see if it balances...







Yes! The first solution does balance the equation. How about the second solution?











Yes! Again we have a working solution.

Derive The Generalized Quadratic Formula

OK now that we have demonstrated how to complete the square, and how that works with a specific example of the Quadratic Equation, let's take the generalized form of that equation as shown at the beginning of this page and solve it for in much the same manner.

Once again, here is the general form of the Quadratic Equation:

Subtract the constant from both sides, just as we did in the example.


And as in the example, now we divide both sides by the value of the coefficient on the first term.

Completing The Square

At this point, we complete the square, just as before.



Then we expand out the right hand term and collect the pieces.





Next, we take the square root of each side.



Simplify...



Now subtract off the non term from both sides, leaving by itself on the left. And it leaves us with a quite recognizable equation. It is the Quadratic Formula!



The last thing is to break the equation into the positive and negative pieces, clearly demonstrating that there are two solutions.

Using Our Equation to Check Our Work

To wrap it up, let's look again at the quadratic equation we started with and parse out the pieces that will go into the Quadratic Formula we just derived. If we did everything correctly, after we enter our numbers we should come up with the same answers we calculated.
The equation we started with as our initial example was



The model for the generalized equation is



This makes our variables:

And The Answer is...

And when we use those numbers in our Quadratic Formula, see what we get.







It works!