The Universal Solver

For any quadratic ax² + bx + c = 0, the solutions come from the quadratic formula:

x = (−b ± √(b² − 4ac)) / 2a

Plug in a, b, c and compute. The ± sign produces two solutions, since a quadratic can have up to two real roots. The piece under the root, b² − 4ac, is the discriminant: positive means two real solutions, zero means one repeated solution, negative means two complex solutions.

The ideas are over 1,000 years old. The 9th-century scholar al-Khwarizmi worked them out in words in Baghdad, and his book's title gave us the term "algebra." The compact symbolic formula came centuries later.