व्यष्टिसमष्टि
Vyaṣṭisamaṣṭhi
"Part and whole"
What this sutra solves
Factor a quadratic quickly — the basis for solving, simplifying, and finding dimensions.
A rectangular garden has area x² + 5x + 6. You want its two side lengths.
Two numbers with product 6 and sum 5 are 2 and 3 → sides (x+2) and (x+3).
Vyaṣṭisamaṣṭhi balances the "individual" (the parts) with the "whole." For x² + bx + c, we need two parts that fit both the product and the sum.
Factor the trinomial
Vyaṣṭisamaṣṭhi balances the "individual" (the parts) with the "whole." For x² + bx + c, we need two parts that fit both the product and the sum.
⚡ Speed Advantage
See how few steps Vedic method needs!
Best for
- • Factoring quadratic and higher-degree polynomials
Use when
- • Quadratic trinomials, factorization problems
Avoid when
- • Irreducible polynomials over integers
Intuition
Factor a quadratic by finding two numbers whose sum is the middle coefficient and product is the last — the Vedic way to factor trinomials.
Story Mode
From Whole to Parts
Every trinomial is a product hiding its factors. The 'whole' (samasthi) is the expression; the 'parts' (vyashti) are the two root-factors. Sum and product are the two clues. Find what multiplies to the constant and adds to the middle — the factors reveal themselves.
Vedic vs conventional
trial and error factoring — variable steps.
systematic sum-and-product recognition.
Applications
Factoring quadratic trinomials
Factor x²+bx+c by finding two numbers summing to b with product c.
Common Mistakes to Avoid
Using product but not sum to choose factors
Wrong approach
Correct approach
Why this happens
💡 Students latch onto the constant term and forget the middle coefficient.
Dropping the leading coefficient in non-monic quadratics
Wrong approach
Correct approach
Why this happens
💡 The beginner pattern x²+bx+c is over-applied to harder quadratics.
Why It Works
Start from two linear factors:
Expand:
Read the two clues:
Factoring reverses expansion: find two parts whose sum gives the middle coefficient and whose product gives the constant term.
x²+()x+ab = ()(). The sutra recognizes that the 'individual' factors (vyashti) combine into the 'whole' (samasthi) through their sum and product.