Mrs. “anon for this”,
Here’s how it works,
Instead of x and x + 2, let’s use (x – 1) and (x + 1). Now FOIL (Firsts, Outers, Inners and Lasts) gives us (x – 1)*(x + 1) = x^2 + x – x – 1^2 = x^2 -1.
So 6 x 8 = (7 – 1)*(7 + 1) = 7^2 – 1 = 48.
This will also work for (x – 2)*(x + 2) = x^2 – 4 and so on.
Here’s a trick I figured out for squaring an integer where the units digit is 5. Let’s square 75. Take all the digits besides for the last one (in this case it’s just the number 7) multiply it by by that number + 1 (7 * 8 = 56) and attach the numbers 25 at the end (5625). Please check that 75 * 75 = 5625.
Proof (10x + 5)^2 = 100x^2 + 50x + 50x + 25 = 100x^2 + 100x + 25 = 100(x^2 + x) + 25 = 100 * x * (x + 1) + 25.
Now try it for 45 and move the decimal two spots to the left for 4.5 ^ 2. You should get 20.25.