Puranapuranabhyam Sutra

"Puranapuranabhyam," which means "By the Completion or Non-completion," is the foundation for the famous "completing the square" method to solve any quadratic equation.

The "Completing the Square" Rule 🔲

The goal is to transform one side of a quadratic equation into a perfect square trinomial, which can then be easily factored to solve for 'x'.

Example: Solve $x^2 + 4x - 5 = 0$

1. Isolate x terms: Move the constant to the other side: $x^2 + 4x = 5$.
2. Find completion term: Take half of the 'x' coefficient (4/2 = 2) and square it: $2^2 = 4$.
3. Add to both sides: $x^2 + 4x + 4 = 5 + 4$.
4. Factor and Simplify: The left side is a perfect square: $(x+2)^2 = 9$.
5. Solve for x: Take the square root of both sides: $x + 2 = \pm 3$.
   • First root: x + 2 = 3 → x = 1.
   • Second root: x + 2 = -3 → x = -5.