Fast Squares Ending in 9

Easily square numbers in your head like 109² and 69².  I will show you how and you will impress your friends. 

How would you like to be able to do squares really fast in your head?  This is the quick trick and it is really easy.  Lets say someone asks you what 69 times 69 is and you immediately say 4761.  You might be asking, "How did you do that?"

Okay here we go: 70²−2(70)+1

Okay, I heard some groans after that.  I will explain this methodology and it is really easy.  I promise.

Steps for squaring a number that ends in 9: 
Add 1 to the number so that the number ends in a 0.
Lets go back to 69.  1+69=70

Square the number we added 1 to.
70²=4900

Multiply by 2 the number we added 1 to.
2(70)=140

Subtract the multiplied number from the squared number and add 1.
4900-140+1=4761

This works with any number that ends in 9.  Here is why it works.

If we have a number 70², we know that 69x70 is 70 less than 70².  69x69 is 69 less than 69x70 so if we wanted to know what 69² is we could simply take 70² and subtract 70 and subtract 69.  That is kind of hard to do in our head so lets do the following:

-(69+70)=-(70 +70 -1) This can be re-written as -2(70) +1
So Voila we have 69²=70²−2(70)+1

Lets try 109²

We add 1 and have 110. 
110²=12100
2(110)=220
12100-220+1=11881

Phew!  Pretty nice isn't it?  But what about numbers that end in an 8 or 1 or what about all the other numbers?  We will learn how to square all the numbers in following posts.