Solve equations
Solve a quadratic that won’t factor neatly
Reaching for the quadratic formula is fine, but completing the square shows the roots’ structure directly.
Spot it when
A quadratic with no obvious integer factors, or you need the vertex/turning point.
Watch it solved
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!
When you’ll actually hit this
Real moments — exams, homework, competitions — where this shortcut earns its keep.
A projectile’s height reads h = x² + 6x − 4 and you want the turning point and the roots together.
Add the missing corner 3² = 9 → (x+3)² = 13, so x = −3 ± √13 and the vertex falls out.
An exam quadratic resists factoring; completing the square is faster than fumbling the formula.
Complete the square (or read the parts) → x = 2 and x = −5.
A function question wants the minimum value of y = x² + 6x + 5.
(x+3)² − 4, so the minimum is −4 at x = −3 — read straight off the completed square.
The method behind it
पूरणापूरणाभ्याम्
Pūraṇāpūraṇābhyāṃ
“By the completion or non-completion”
Complete the square (or cube) by adding and subtracting the missing term — Vedic style completion method.
5 steps.
3 intuitive steps.
Use when
- • Quadratic equations, cube root extraction
Avoid when
- • Linear equations