LOPANASTHAPANABHYAM

Lopana sthapanabhyam means 'by alternate elimination and retention'.

Consider the case of factorization of quadratic equation of type ax2

+ by2 + cz2 + dxy + eyz + fzx This is a homogeneous equation of

second degree in three variables x, y, z. The sub-sutra removes the

difficulty and makes the factorization simple. The steps are as

follows:

i) Eliminate z by putting z = 0 and retain x and y and factorize

thus obtained a quadratic in x and y by means of ‘adyamadyena’

sutra.;

ii) Similarly eliminate y and retain x and z and factorize the

quadratic in x and z.

iii) With these two sets of factors, fill in the gaps caused by

the elimination process of z and y respectively. This gives actual

factors of the expression.

Example 1: 3x2 + 7xy + 2y2 + 11xz + 7yz + 6z2.

Step (i): Eliminate z and retain x, y; factorize
3x2 + 7xy + 2y2 = (3x + y) (x + 2y)

Step (ii): Eliminate y and retain x, z; factorize
3x2 + 11xz + 6z2 = (3x + 2z) (x + 3z)

Step (iii): Fill the gaps, the given expression
= (3x + y + 2z) (x + 2y + 3z)

Example 2: 12x2 + 11xy + 2y2 - 13xz - 7yz + 3z2.

Step (i): Eliminate z i.e., z = 0; factorize
12x2 + 11xy + 2y2 = (3x + 2y) (4x + y)

Step (ii): Eliminate y i.e., y = 0; factorize
12x2 - 13xz + 3z2 = (4x -3z) (3x – z)

Step (iii): Fill in the gaps; the given expression
= (4x + y – 3z) (3x + 2y – z)

Example 3: 3x2+6y2+2z2+11xy+7yz+6xz+19x+22y+13z+20

Step (i): Eliminate y and z, retain x and independent term
i.e., y = 0, z = 0 in the expression (E).
Then E = 3x2 + 19x + 20 = (x + 5) (3x + 4)

Step (ii): Eliminate z and x, retain y and independent term
i.e., z = 0, x = 0 in the expression.
Then E = 6y2 + 22y + 20 = (2y + 4) (3y + 5)

Step (iii): Eliminate x and y, retain z and independent term
i.e., x = 0, y = 0 in the expression.
Then E = 2z2 + 13z + 20 = (z + 4) (2z + 5)

Step (iv): The expression has the factors (think of

independent terms)
= (3x + 2y + z + 4) (x + 3y + 2z + 5).

In this way either homogeneous equations of second degree or general

equations of second degree in three variables can be very easily

solved by applying ‘adyamadyena’ and ‘lopanasthapanabhyam’ sutras.