Fun with math - Part Deux: Squaring 2 digit numbers in your head.

Here is another cool math trick that could help you impress some people and be the life of the next party you attend (there are more and more of these recently as the country reopens). How do you square any 2 digit number in your head really quickly in 4 easy steps. 

In these steps we will use 2 examples: 43 squared and 78 squared.

1. First find the distance to the nearest multiple of 10 (either below or above our number).  In the case of 43 that would be 3 because 43 - 40 = 3. In the case of 78 that would be 2, because 80 - 78 = 2.
2. Make the original number the midpoint between the multiple of 10 and the number the same distance away from it. For 43 you would use 46 because 43 is 3 away from 40 and 46. For 78 you would use 76 because 78 is 2 away from 80 and from 76.
3. Multiply the multiple of 10 by the number you found in step 2. As a tip, if you want to do this in your head, break it down into the 10s and the 1s which will make it easier. For 43 you would take 40x46 = 1840 (in my head I would have done 40x40 = 1600 and 6x40 = 240, 40x46 = 1600 + 240 = 1840). For 78 you would do 76x80 = 6080 (mentally you could do 70*80 = 5600 and 6*80 = 480, add the two to get 6080)... on to the final step:
4. Square the distance calculated in step 2 and add that to the number found in step 3. That will be your final answer. So for 43 you would add 3 squared = 9 to 1840, so 43 squared = 1849. For 78 squared you would add 4 (which is the 2x2) to 6080 and arrive at 78 squared = 6084.

Voila! As simple as that. Try it out and please let me know how it goes.

P.S. The most difficult part to do mentally is step 3, but as mentioned above if you break it down into the 10s and the 1s (otherwise known as using the distributive property), you can get pretty good at this relatively quickly.

P.P.S. If you think this is some magical hocus pocus, it really isn't. What we are doing here is using the distributive property with binomial expressions, which is something that Algebra 1 students would study at some point in their mathematics careers at school.

P.P.P.S. For any 2 digit number ending in 5, such as 25 squared or 65 squared, etc..., it's actually even easier. Take the next number up from the original 10s value and multiply those 2 digits together and tack on 25 to the end. That's it. So for 25 squared you would multiply 2x3 = 6 and tack on 25, so 25 squared = 625. For 65 squared you would multiply 6x7 = 42 and so your final answer would be 65 squared = 4225.