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!

Calculus I: The Quotient Rule

Have you adequately studied the previous lesson about the Product Rule? It’s an important one, so make sure you do!

What’s the quotient rule? What does that even mean? Is it difficult to use? All your questions will be answered! Remember the product rule, and what it was used for? The quotient rule is very similar, but has a different formula. The quotient rule can be used when you have a function of x, divided by another function of x. Basically, that means a fraction. Here are a couple of examples:

See? If you have a fraction with x’s in both the numerator and denominator, this is the time for the quotient rule. Keep an eye out for these types.

What is the quotient rule?


(Like I showed you above, this just means if you have x’s on the top and bottom)

The quotient rule says:

“y prime equals the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared.” This is a mouthful. You really should say this out loud over and over again, until you fully understand it. You should do your best to memorize that sentence, and the formula.

Yes, you should do your best to memorize that long sentence. I know that it is kind of boring and hard to remember, but it’s very important. However, this site wouldn’t be called CalculusTricks.com if I didn’t have a trick for you. I took calculus I in high school, and then I took it again in college as a refresher. The second time I took it, my teacher gave us an awesome little trick that helped us out a lot. Please bear with me, as it’s not going to seem like that big of a deal at first, but it is extemely helpful!

Say the following: “Low Dee High minus High Dee Low, all over Low squared.”

You have to say it with some rhythm. The “Dee” means derivative of. Like we say “dee-why-dee-exx” for . “Low” and “High” refer to the denominator and the numerator respectively. This is a much shorter sentence to remember, and feels much easier to say. This is something you would say to yourself, in your head, while you are taking a test. If you repeat this over and over, you won’t mess up the formula when you need to use it! This is the one you NEED to memorize, if you want to score well on your tests.

So how do we use it?

Let’s start with a very simple example.

All that I did here was follow the formula from above. You have to recognize what part is the denominator, and which is the numerator (this is very self explanatory). Then, say the trick out loud a couple of times as you take the derivative. “Low Dee High minus High Dee Low, all over Low squared.” I didn’t leave a single step out when I was simplifying, so that you could see everything I did very clearly. Do you see how everything I did matched the forumla exactly? I’ll show you a couple more examples, in case you are still confused.

Something a little bit harder, perhaps? Remember, apply the forumla exactly as we did above.

Simplify the most that you can, but don’t worry if it still looks a bit messy. Some of them just come out that way. I used the quotient rule the same way that I did in the first example. This time, the fraction was just a bit more difficult. We can handle this, right?

Okay, I think it might be time to step it up a notch. For my final example, I’m going to show you something you need to be on the lookout for. What if you need to use the product rule inside of the quotient rule?!? Did you even know that that could be possible? Other text books will try to trick you with a problem like this, but I’ll be straight forward with you and warn you about these. In the example below, I’ll do the most difficult problem you can get with the quotient rule. Ready?

You need the product rule for the numerator!!! This is going to be really messy. All I ask is to please try to follow me. This is as hard as it gets. Remember and apply the formula!

 

For this problem, I did it on the fly (I did the product rule directly inside the quotient rule as I was going through the problem. I did not stop and do the product rule off on the side. You can do that if you’d like to, though). This is exactly how I would have solved this problem, if it were assigned to me. If you wanted to, you could have broken it down into smaller parts to make it easier for yourself. You could say f(x) equals what? g(x) equals what? And the same thing for f ‘(x) and g ‘(x). Then, you can insert those answers directly into the formula. I did a similar process on the last example of my Product Rule post. You should try to factor as best as you can, in order to simplify the problem. Occasionally, you will get an answer where something in the numerator can cancel out something in the denominator. That is rare, but most text books have a few problems set up specifically to do that.

I hope this lesson has been helpful to you. Try some problems, use my trick, and practice some more. If you then have any specific questions, be sure to email me at admin@calculustricks.com. I check my throughout the day, and I’ll respond at my earliest convenience.

Good luck!

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