Difference of two squares

(I saw this post from six years ago, barely a month after I published 25 Math Short Cuts. This may partly explain how my first printing of 3,500 was sold out in one month).

A sponsored post by Brilliant.org asks “What is 99^2 – 98^2 (99 squared minus 98 squared)?

Many will not even try to solve this problem without using a calculator because they anticipate a very long solution using the conventional methods taught in schools.

We can offer at least three simpler solutions, all of which can be found in our book, 25 Math Short Cuts.

1) We can use Math Short Cut # 20 -Squaring Numbers near a Power of Ten to quickly determine the squares of 99 and 98. Using the Vedic sutra (word formula) “whatever the deficiency, lessen by that amount and set up the square of the deficiency. Now, the deficiency of 99 from 100 is 1. So we have 99 – 1 = 98 and 1 x 1 is 01(must have the same number of digits as the number of zeroes in the base. The final answer for 99 squared is 9801. For 98 squared, it is 98 -2 = 96| 2 x 2 = 04 or 9604.

We can now easily find the difference 9801-9604 = 200-3 = 197 using subtraction by parts, which is discussed in Math Short Cut #3 – Subtraction without borrowing.

2) A simpler solution involves the application of the binomial expansion which was used several times in the book to prove many shortcuts. In page 70, we have (x + a)^2 = x^2 + 2(x)(a) + a^2. If we subtract x^2 from this we will get 2(x)(a) + a. If a = 1, the expression will be 2x + 1 or simply (x+ 1) + x. Now if we let x = 98, then x+1 = 99 and 99^2 – 98^2 = 99 + 98 = 197. Similarly 87^2 – 86^2 = 87 + 86 = 173)

3) The simplest solution is to apply DOTS:(The Difference Of Two Squares is the product of the sum and the difference) which is discussed in Math Short Cut # 24.

a^2 – b^2 = (a + b) ( a -b). So 99^2 – 98^2 = (99+98)(99-98) = 197(1) = 197. Using this method 251^2 – 249^2 is easily found to be (251+ 249)(251-249) = 500(2) = 1000.