Prior to Calculus: Difference of Squares

Welcome to the most important topic you learn in all of algebra! This is used so often that it is completely necessary for you to master this technique and look for ways to use it all the time. Sometimes it can be more difficult to spot than others, but I will help you recognize those situations!

What does “difference of squares” mean?

Difference of squares describes a situation where you have two terms, preferably both perfect squares, where one is being subtracted from the other. It is a factoring technique that will take any situation that I previously described and separate it into smaller parts. It is the most useful thing you learn in algebra. Here are a few examples that can be factored using this technique.

You can factor literally anything that is two terms being subtracted from each other. It wouldn’t always be useful, so in addition to teaching you the technique, I’ll teach you the trick to knowing if it is going to be useful or not.

What is the difference of squares formula?

When A and B are any two mathematical expressions,

It is important to know that the square root sign really implies the “principle” square root. This means only the positive square root. For example:

When you write the square root symbol, you need to explicitly state plus or minus in front of the symbol if you intend for it to be either the positive or negative square root. Here, I only mean the positive square root, so I just use the symbol for the “principle” square root.

It’s important to know that A and B can be anything at all. For now, I’ll stick with simple A’s and B’s, but then I’ll show you some complicated examples. It is a great trick to know how to do.

How do we use it?

This is, again, one of the things you simply look to the formula for. We use the formula exactly, and it is as simple as that. This is the easy part, and I’ll give you many examples. The hard part is knowing when it is a good time to use it. I’ll get to that later. For now, let’s start with a simple example.

I showed all the steps, because I wanted you to see exactly how I did everything. You can skip the step in your head where you take the square root. For just about any example, you can do that step in your head. It’s really not something you have to show your work on.

Now, I’ll show you an example that isn’t quite as obvious. The A and B does not have to be as simple as the first example.

Again, I showed you all the steps. The examples are going to get harder, so I am going to keep showing the steps to you. Remember that you don’t have to how the steps. You can just take the square root in your head, and not show the middle step.

Ready for another example? I want to show you that A and B don’t have to be perfect squares. They can be anything, and it works the exact same way. This isn’t always useful, but sometimes it can be! It’s very good to know this.

I know this doesn’t seem too useful, but sometimes you get a problem where you really need to know that this is possible. I’ll show you one more example. You don’t have to use simple variables and coefficients. This next one is as complicated as I can possibly make it, but you can still use this formula the exact same way!

So now you know how to do it, but that is really the easy part. How do you know when it’s a good idea to use this? When is it going to just make things worse? I’ll show you a few examples of the times you should be looking to use this technique.

You will come across fractions in many different situations. You could find them in integrals, after taking a derivative, or when you are just trying to simplify an expression. Fractions are a good time to be on the lookout! If you have a difference of two squares in the numerator or the denominator, check the opposite place to see if there could be something that will cancel out. It is pretty easy to see what the difference of to squares will factor to. Will one of the factors cancel with something else already in the problem? I’ll show you an example. If you have a fraction that you want to simplify, it is always good to check if you can use this method to cancel out something. I’ll show you an example.

This technique can also be used when solving an algebraic equation. It really can make things a lot easier, and I’ll show you an example of this. Solve for x in the following equation.

These are just two situations out of many in which this can arise. The knowledge and use of this formula is very powerful! You should always be on the lookout for this phenomenon, and try to apply the formula. This factoring technique comes up all the time. It is always good to try it, and see if it helps you out. If not, you can always erase that bit of your work and try something else.

So, I encourage you to please practice this. I would never lie to you! I use this all the time. It is extremely important. Don’t underestimate it! If you have any questions, please ask away! admin@calculustricks.com is the best way to contact me. Like me on facebook and follow me on twitter!

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  1. Richard says:

    The problem following this : This next one is as complicated as I can possibly make it, but you can still use this formula the exact same way

    is not consistent you’re supposed to have (x-2) not (x-5) you ran with the wrong numbers from the beginning while calculating.

    you start out with (x-2) and you end with (x-5) which is suppose to end in (x-2) unless I’m wrong.

    Excellent tutorial otherwise, I love it.

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