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Tuesday, 26 March 2013

Vedic Mathematical Formulae part 2


VILOKANAM
The Sutra 'Vilokanam' means 'Observation'. Generally we come across problems which can be solved by mere observation. But we follow the same conventional procedure and obtain the solution. But the hint behind the Sutra enables us to observe the problem completely and find the pattern and finally solve the problem by just observation.
Let us take the equation x + ( 1/x ) = 5/2 Without noticing the logic in the problem, the conventional process tends us to solve the problem in the following way.

                       1        5
               x  + __  =  __
                       x        2

                x+ 1         5
                _____   =   __
                    x            2

                2x2 + 2 = 5x
                2x2 – 5x + 2 = 0
                2x2 – 4x – x + 2 = 0
                2x (x – 2) – (x – 2) = 0
                   (x – 2) (2x – 1) = 0
                               x – 2 = 0 gives x = 2
                             2x – 1 = 0 gives x = ½

        But by Vilokanam i.e.,, observation

                          1        5
                   x + __  =  __    can be viewed as
                          x        2

                          1             1
                   x + __  =  2 + __     giving x = 2 or ½.
                          x             2
Consider some examples.
Example 1 :
                        x        x + 2        34
                  ____  +  _____  =   ___
                     x + 2        x           15
In the conventional process, we have to take L.C.M, cross-multiplication. simplification and factorization. But Vilokanam gives

                   34      9 + 25       3        5
                   __  =  _____   = __  +  __
                   15       5 x 3        5        3

                        x         x + 2        3        5
                     ____  +  _____  =  __  +__
                     x + 2         x           5        3

        gives
                  x            3                     5
               _____  =  __         or      __
               x + 2         5                     3

            5x = 3x + 6    or     3x = 5x + 10
            2x = 6           or    -2x = 10
              x = 3           or        x = -5
Example 2 :

               x + 5      x + 6         113
               ____  +  _____   =    ___
               x + 6      x + 5          56

        Now,
               113       49 + 64        7          8
               ___   =  _______  =___  + ___
                56          7 x 8          8          7

               x + 5       7               x+5          8
               ____   =  __        or   ____  =   __
               x + 6       8               x+6          7

            8x + 40 = 7x+ 42        7x + 35 = 8x + 48
                                      or
            x = 42 - 40 =2            -x = 48 –35 = 13
                x = 2               or    x = -13.
Example 3:

                   5x + 9      5x – 9        82
                   _____  +  _____  =  2 ___
                   5x - 9      5x + 9        319
At first sight it seems to a difficult problem.
But careful observation gives

                82        720      841 - 121        29       11
             2___   =  ___   =  ________    =___  -  __
               319       319         11 x 29         11       29
(Note: 292 = 841, 112 = 121)

            5x + 9     29        -11
            _____  = __   or  ___
            5x -9      11         29
(Note: 29 = 20 + 9 = 5 x 4 + 9 ; 11 = 20 – 9 = 5 x 4 – 9 )
i.e.,
         x = 4  or
                     5x + 9        -11
                     _____   =    ___
                     5x - 9          29

            145x + 261 =-55x + 99
            145x + 55x =  99  – 261
                   200x = -162

             -162          -81
        x =  ____   =  ____
               200          100
Simultaneous Quadratic Equa
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