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Lopanasthapanabhyam: By Alternate Elimination and Retention

Lopanasthapanabhyam is the corollary to Vyashtisamanstih, the eleventh sutra of Vedic mathematics.

This corollary is used primarily to factorise algebraic equations into quadratic equations by alternately setting the value of two of the variables at zero.

Example:
Factorise the following equation:
3x2 + 2y2 + 6z2 + 7xy + 7yz + 11xz

Step One: With lopanasthapanabhyam ("By Elimination and Retention"), we would start by eliminating one of the variables while retaining the others. Let's simplify the problem by saying z = zero. In other words, we eliminate z while retaining x and y
If z = 0 the equation becomes: 3x2 + 2y2 + 7xy
This can be rewritten as the quadratic equation: (3x + y)(x + 2y)

Step Two: Now factorise the original equation, but set the value of one of the other variables to zero instead. Let's eliminate y while retaining x and z
If y = 0 the equation becomes: 3x2 + 6z2 + 11xz
This can be rewritten as the quadratic equation: (3x + 2z)(x + 3z)

Step Three: Fill in the gaps by joining the two bolded equations in the previous steps:
(3x + y)(x + 2y) combined with (3x + 2z)(x + 3z) becomes: (3x + y + 2z)(x + 2y + 3z)

"By Elimination and Retention" can also be used to find the highest common factor of algebraic expressions. To learn more about this upsutra, check out the resources below:

RESOURCES:
Vedic Mathematics by Sri Bharati Krisna Tirthaji
http://www.vedamu.org/Mathematics/course.asp
http://www.sanalnair.org/articles/vedmath/intro.htm
http://www.vedicganita.org/ganitsutras.htm