Puranapuranabhyam

Meaning: "By the completion or non-completion (of the square or other forms)". This sutra often relates to the process of completing the square in quadratic equations or manipulating expressions to achieve a 'complete' or recognizable form.

Primary Applications:

• Solving Quadratic Equations: This is the most direct application, where the method of "completing the square" is used. For an equation like ax² + bx + c = 0, you manipulate it to form a perfect square trinomial on one side.
  Example: x² + 6x = 7. To "complete" x² + 6x into a square, take half the coefficient of x (6/2 = 3) and square it (3² = 9). Add this to both sides: x² + 6x + 9 = 7 + 9. This becomes (x+3)² = 16. Now it's easily solvable.
• Simplifying Expressions: It can be used to simplify algebraic expressions by recognizing or forcing parts of them into perfect squares or other "complete" forms. This can make factorization or further manipulation easier.
• Calculus (Integration): In integration, sometimes algebraic manipulation using completion of the square is necessary to transform an integrand into a standard form for which an integration formula is known. For example, integrals involving expressions like 1/(ax² + bx + c).
• Geometric Interpretations: The idea of "completing" can be visualized geometrically, for instance, by adding smaller squares or rectangles to an existing shape to form a larger, complete square.
• "Non-completion" aspect: This can refer to situations where an expression is close to a complete form but isn't quite. Recognizing this "non-completion" might lead to strategies like "difference of squares" factorization. For example, x² - 7 can be seen as "non-complete" for x² - a², but by thinking of 7 as (√7)², we can factor it as (x - √7)(x + √7).

Core Idea: The sutra emphasizes transforming expressions into more manageable or standard forms by adding or subtracting necessary components to achieve "completion," or by leveraging how close an expression is to a "complete" form.