How to Compute Fast Any Square

An algebraic identity extremely useful for computing squares is a well known one: a² - b² = (a - b)(a + b). For our purposes, I'll rewrite it differently:

a² = (a - b)(a + b) + b².

For example, find 112². I'll take a = 100, for 100 is a number simple to manipulate and close to 112. 100 is obtained from 112 by subtracting 12 - set b = 12. Thus we have

112² = (112 - 12)(112 + 12) + 12² = 12400 + 144 = 12544.

As another example, what is 198²? Setting a = 198, b = 2, gives

198² = (198 - 2)(198 + 2) + 2² = 200·196 + 4 = 20000 + 18000 + 1200 + 4 = 39204.


Related material
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  • Multiplication by 9, 99, 999, (Multiply + Subtract) etc.
  • Squaring 2-Digit Numbers
  • Division by 5
  • Multiplication by 2
  • Multiplication by 5
  • Multiplication by 9, 99, 999, etc. (Something Special)
  • Product of 10a + b and 10a + c where b + c = 10
  • Product of numbers close to 100
  • Product of two one-digit numbers greater than 5
  • Product of 2-digit numbers
  • Squaring Numbers in Range 26-50
  • Squaring Numbers in Range 51-100
  • Squares of Numbers That End in 5
  • Squares Can Be Computed Squentially
  • Adding a Long List of Numbers
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