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:

3x

^{2} + 2y

^{2} + 6z

^{2} + 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: 3x

^{2} + 2y

^{2} + 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: 3x

^{2} + 6z

^{2} + 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

http://hinduism.about.com/library/weekly/aa062901a.htm

http://www.vedicmaths.org/

http://www.hinduism.co.za/vedic.htm

*Mathemagics* by Arthur Benjamin and Michael B. Shermer

http://en.wikipedia.org/wiki/Vedic_math

http://www.tifr.res.in/~vahia/dani-vmsm.pdf

http://www.sacw.net/DC/CommunalismCollection/ArticlesArchive/NoVedic.html