पूरणापूरणाभ्याम्
Pūraṇāpūraṇābhyāṃ
"By the completion or non-completion"
What this sutra solves
Rewrite any quadratic as a perfect square to find its peak, minimum, or roots.
A ball’s height follows h = x² + 6x − 4 (shifted form). You want to reshape it to read off the turning point.
Add the missing corner 3² = 9 → (x+3)² = 13, so the vertex and roots fall out at once.
Pūraṇāpūraṇābhyāṃ means "by completion and non-completion." The idea: add exactly the piece that turns the left side into a perfect square.
A quadratic that will not factor neatly
Pūraṇāpūraṇābhyāṃ means "by completion and non-completion." The idea: add exactly the piece that turns the left side into a perfect square.
⚡ Speed Advantage
See how few steps Vedic method needs!
Best for
- • Quadratic equations, completing the square
Use when
- • Quadratic equations, cube root extraction
Avoid when
- • Linear equations
Intuition
Complete the square (or cube) by adding and subtracting the missing term — Vedic style completion method.
Story Mode
Making It Whole
An incomplete square is like a frame missing its corner. Add the right piece — ()² — and suddenly the frame is perfect. The equation transforms from an irregular puzzle into a perfect square, and the answer reveals itself.
Vedic vs conventional
5 steps.
3 intuitive steps.
Applications
Completing the square
Solve quadratic equations by recognizing the missing perfect-square piece.
Common Mistakes to Avoid
Adding b/2 instead of (b/2)^2
Wrong approach
Correct approach
Why this happens
💡 Students stop after finding half the coefficient and forget the square-completion step.
Completing only one side of the equation
Wrong approach
Correct approach
Why this happens
💡 The visual pattern of a perfect square is remembered, but equation balance is overlooked.
Why It Works
Start with a quadratic:
Add the missing square term to both sides:
The left side becomes a perfect square:
Completion works because the middle term of (x+h)^2 is 2hx, so h must be b/2.
To solve x²+bx=c: recognize , add ()² to both sides → ()²=². Vedic framing sees this as 'completing what is missing to make a perfect square'.