Square Root Progression
The methods outlined below are intended as aids to mental arithmetic in multiplication.
The simple rule for progressing from one square root to another is as follows:
 Double the Starting Square. [Result 1].
 Add the difference between the Starting and Ending Square to Result 1 [Result 2].
 Multiply Result 2 by the difference. [Result 3]
 Add Result 3 to the original square to get the final square root.
Example 1 : To move from the square of 9 to the square of 15.
 Difference between 15 and 9 = 6.
 Double starting square: 9+9 and add difference (6) giving us 24.
 Multiply 24 by difference 6 = 144.
 The square of 9 = 81 + 144 = 225 (the square of 15).
Example 2 : The simple case of moving from the square of 11 to the square of 12.
 The square of 11 is 121
 Add 11+11+1= 23 to 121 = 144 (the square of 12)
Knowledge of square root progression could be important in determining the factors of some numbers.
For instance, say you had a number like 391. On the face of it, it isn't obvious that 391 has prime factors. But we know that if you add 9 (a perfect square) to 391 we get 400 (another perfect square).
Those facts tell us 391 must have prime factors. See Prime Factors Of A Number. Given 20 is the square root of 400 and 3 is the square root of 9, the answer lies 20 ± 3 which gives 17 and 23  the factors of 391.
Of course, the smaller the numbers the easier the mental maths!
Input the appropriate figures below to explore other numbers.
As an extension of the application of the above method please see Mental Multiplication.


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