Puranapuranabhyam

The Sutra can be taken as Purana - Apuranabhyam which means by the

completion or non - completion. Purana is well known in the present

system. We can see its application in solving the roots for general form of

quadratic equation.

We have : ax2 + bx + c = 0

x2 + (b/a)x + c/a = 0 ( dividing by a )

x2 + (b/a)x = - c/a

completing the square ( i.e.,, purana ) on the L.H.S.

x2 + (b/a)x + (b2/4a2) = -c/a + (b2/4a2)

[x + (b/2a)]2 = (b2 - 4ac) / 4a2
________

- b ± √ b2 – 4ac

Proceeding in this way we finally get x = _______________

2a



Now we apply purana to solve problems.

Example 1. x3 + 6x2 + 11 x + 6 = 0.

Since (x + 2 )3 = x3 + 6x2 + 12x + 8

Add ( x + 2 ) to both sides

We get x3 + 6x2 + 11x + 6 + x + 2 = x + 2

i.e.,, x3 + 6x2 + 12x + 8 = x + 2

i.e.,, ( x + 2 )3 = ( x + 2 )

this is of the form y3 = y for y = x + 2

solution y = 0, y = 1, y = - 1

i.e.,, x + 2 = 0,1,-1

which gives x = -2,-1,-3