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# By Alternative Elimination and Retention

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This sutra is helpful in case of factorization of quadratic equation.

It is very difficult to factorise the long quadratic (2x2+ 6y2 + 3z2 + 7xy + 11yz + 7zx). But "Lopana-Sthapana" or By Alternative Elimination and Retention removes the difficulty

Example 1
Lets assume E =(2x2 + 6y2 + 3z2 + 7xy + 11yz + 7zx)
By "Lopana-Sthapana" we eliminate z by putting z = 0.

Hence the given expression becomes ,
E = 2x2 + 6v+ 7xy = (x+2y) (2x+3y)

Similarly, if y=0, then,
E = 2x2+ 3z2 + 7zx = (x+3z) (2x+z)
We get , E = (x+2y+3z) (2x+3y+z)

Now,
Factorizing we get
2x2 + 2y2+ 5xy + 2x- 5y –12 = (x+3) (2x-4) and (2y+3) (y-4)

Hence Result , E = (x+2y+3) (2x+y-4)

* This "Lopana-sthapana" method (of alternate elimination and retention) will be found highly useful in HCF, in Solid Geometry and in Co-ordinate Geometry of the straight line, the Hyperbola, the conjugate Hyperbola, the Asymptotes etc.

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