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PŨRANĀPŨRAŅĀBHYĀM
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 Example 2: x3 + 8x2 + 17x + 10 = 0 We know ( x + 3 )3 = x3 + 9x2 + 27x + 27 So adding on the both sides, the term ( x2 + 10x + 17 ), we get x3 + 8x2 + 17x + x2 + 10x + 17 = x2 + 10x + 17 i.e.,, x3 + 9x2 + 27x + 27 = x2 + 6x + 9 + 4x + 8 i.e.,, ( x + 3 )3 = ( x + 3 )2 + 4 ( x + 3 ) – 4 y3 = y2 + 4y – 4 for y = x + 3 y = 1, 2, -2. Hence x = -2, -1, -5 Thus purana is helpful in factorization. Further purana can be applied in solving Biquadratic equations also. 1. x3 – 6x2 + 11x – 6 = 0 2. x3 + 9x2 + 23x + 15 = 0 3. x2 + 2x – 3 = 0 4. x4 + 4x3 + 6x2 + 4x – 15 = 0 |
