Sopaantyadvayamantyam

Meaning: "The ultimate (last term) and twice the penultimate (second to last term)".

Primary Application: Solving Specific Algebraic Equations

This sutra is applied to solve certain types of algebraic equations, particularly those that can be structured or recognized to fit a specific pattern where the relationship between the "ultimate" (last term) and "twice the penultimate" (twice the second to last term) is key. It often involves setting an expression or a combination of terms to zero.

Common Scenarios:

1. Equations with Fractions: Consider an equation with two fractions on one side summing to zero, like:
   M / (ax+b) + N / (cx+d) = 0
   If M(cx+d) + N(ax+b) = 0, and if we expand this to (Mc+Na)x + (Md+Nb) = 0.
   Here, (Md+Nb) is the "ultimate" term (constant term), and (Mc+Na)x involves the "penultimate" concept (coefficient of x). If the sutra's condition, related to the sum of these being zero in a specific way, is met, it leads to a solution. A more direct interpretation is if Md+Nb = 0 and Mc+Na = 0 if M and N are related in a certain way, implying x is indeterminate or specific constraints.
2. Relationship between coefficients in products: If an equation of the form (x+a)(x+b) = (x+c)(x+d) is given, and if a+b = c+d (sum of independent terms in factors on LHS = sum on RHS), then this implies 2x + (a+b) = 0 if another condition (ab=cd implies 0=0) is met. However, a more direct application of Sopaantyadvayamantyam is seen when a specific structure of coefficients in a single polynomial equated to zero is present. For instance, if you have a polynomial P(x) = 0, and P(x) can be factored into Q(x) * R(x) = 0, then the sutra might apply to the structure of Q(x) or R(x) if they fit the "ultimate and twice penultimate" pattern leading to a root.
3. Specific condition for quadratic roots: For a quadratic equation Ax² + Bx + C = 0, if B = A+C (twice the coefficient of the middle term related to the sum of the first and last), this doesn't directly use the "twice the penultimate" wording. The sutra is more about structuring the equation or parts of it. If an expression like P(x) = (some factor related to 'ultimate') + 2 * (some factor related to 'penultimate') = 0, then this structure itself is the solution method.

Key Idea: The sutra is used when an algebraic expression can be set up or observed such that if the last term (Antyam) and twice the second to last term (Sopaantyadvayam) sum up to zero (or satisfy a similar specific relationship), then the value of the variable that makes this true is the solution. It's a pattern-recognition technique for specialized equation forms.