Vedic Mathematics | Puranapuranabhyam
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.
Solve the following using purana – apuranabhyam.
1.
x3 – 6x2 + 11x – 6 = 0
2.
x3 + 9x2 + 23x + 15 = 0
3.
x2 + 2x – 3 = 0
4.
x4 + 4x3 + 6x2 + 4x – 15 = 0