Completing the Square

Before the quadratic formula, mathematicians solved quadratics by completing the square. To solve x² + 6x − 7 = 0: take half of 6, square it (9), add and subtract: x² + 6x + 9 − 9 − 7 = 0. The first three terms form a perfect square: (x + 3)² = 16. Take the square root: x + 3 = ±4, so x = 1 or x = −7.

This method is not just historical — it is the engine behind the quadratic formula itself. Derive the formula by completing the square on the general ax² + bx + c = 0, and the famous (−b ± √(b² − 4ac)) / 2a falls out naturally. The formula is completing-the-square packaged for reuse.