Puranapuranabhyam
By the completion or non-completion (pu · ra · na · pu · ra · na · bhyam)
What it means
Turn a messy equation with an x² term (a quadratic) into a perfect square — an expression like (x + something)² — by adding exactly the right amount to both sides. Once it's a perfect square, you take a square root and the answer pops out. This is the trick behind the "quadratic formula" you may see later — same idea, applied generally.
Worked example
Solve x² + 6x = 7. Take half of 6 → 3. Square it → 9. Add 9 to both sides: x² + 6x + 9 = 16. The left side is now (x + 3)² — a perfect square. Take the square root: x + 3 = ±4. So x = 1 or x = −7.