First of all, don't be menaced by the word algebra in the next sentence, this is very easy. In this instructable I will show you how to multiply using algebraic identities. It is a simple method and very easy once you get the concepts down.
Step 1: Solving Using the First Identity
You can use the first identity when you have a number squared. for the sake of simplicity I will be using 25 squared to show you how to do this. You can move on to bigger and harder numbers once you learn it.The basic idea is that we'll take a number,in this case 25, and split it into two easier to multiply numbers which are being added together. So we get (20+5)squared. If you look at the first identity you can see that 20=a and 5=b. Now, using the first algebraic identity we can determine that this will equal 20squared+5squared+2(20x5). Do the multiplication and you get 400+25+200. This equals 625.
Step 2: Solving Using the Second Identity
This is very similar to using identity one to solve a problem but it is used under different circumstances. For example, if you had 195 squared and you were using method one, you would end up with 100 + 95, meaning you would have to square 95 which is pretty hard. In this method instead of writing 195 as 100+95, we will write it as 200-5. The main thing to remember is that in this method we will end up with negative 2ab, meaning that we will subtract 2ab, not add it.
Step 3: Solving Using the Third Identity
This method is probably the simplest in concept but is, in some cases, hard to apply. The example which I will be using makes it look fairly simple because I use easy numbers (this is just to help you understand). In this method you will solve a problem in which two different numbers are being multiplied. For example, 55x45. Now the trick is to find a number which is in between the two numbers that is an equal distance away from the two numbers. Now you're probably confused about what I just said but just take a look at the example, 55x45, the number that is an equal distance away from these two numbers is 50. This is because it is 5 less than 55 and 5 more than 45. Once you have your number, you can split up both of the numbers you are multiplying into easier to multiply numbers which in this case would make our expression (50+5)(50-5). If you look at the third identity you will see that here 50=a and 5=b. Now using the third identity we can determine that our expression will equal 50 squared - 5 squared. That equals 2500-25 which equals 2475.