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                 x2 + 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