+2440 \(35*35=1225\)
I believe you want an easier method to work out problems like these where you square a number. Here is an idea for you.
\(35^2=?\)
When you square a number, this is a trick you can use! Round the number to the nearest multiple of 10. In this case, that is 30. Now, go up by the distance you went down. In this example, you went down by 5 to get to 30. Now, go up 5 to get to 40. Now multiply them together.
This should be much simpler now. \(30*40=1200\). All you need to do now is to add the square of the distance. In this case, it is \(5^2=25\).
\(1200+25=1225\)
That's much easier, don't you think? In fact, this is usually so easy that you may be able to do it mentally.
But why does it work? Well, that is actually easier to explan than you may think. Let's call the number we want to square A. Let's call the distance we want to go up and down d. Therefore, we get the following equation:
| \(A^2=(A-d)(A+d)+d^2\) | Now, expand the multiplcation of 2 binomials. |
| \(A^2=A^2-d^2+d^2\) | Now, simplify. |
| \(A^2=A^2\) | |
This means that no matter what distance you pick, you will eventually make it the original number squared. That's pretty cool!
When squaring a number that ends in "5", such as this example, there is a very simple and beautiful trick that you can use to get the answer very quickly. This is what you do: 35 x 35 = Multiply the two "5's" by each other and you, of course, get: 5 x 5 = 25. The next digit is "3" and multiply it by the next higher digit or "4" in this case and, of course, you would get: 3 x 4 =12. Then put the two results together and you have: 1,225 !. So, 45 x 45 =2,025 and 55 x 55 =3,025, and 75 x 75 =5,625...and so on.
