functions Quadratic Equation Solver star

Solving ax² + bx + c = 0

Every quadratic equation of the form ax² + bx + c = 0 (with a ≠ 0) can be solved with the same formula: x = (-b ± √(b² - 4ac)) / 2a. The expression under the square root, b² - 4ac, is called the discriminant (often written D or Δ), and its sign alone tells you what kind of roots to expect before you finish the calculation. If D is positive, the ± gives two distinct real roots. If D is exactly zero, there is one repeated real root, x = -b/2a. If D is negative, the two roots are a complex conjugate pair instead, written as p ± qi where p = -b/2a and q = √|D| / 2a.

The same coefficients also describe a parabola, y = ax² + bx + c, and x = -b/2a is not a coincidence — it is the parabola's axis of symmetry, the vertical line the curve mirrors around. Plugging that x back into the equation gives the vertex, the parabola's single highest or lowest point (a minimum if a > 0, a maximum if a < 0). This is why the discriminant's sign has such a clean geometric meaning: it describes exactly how many times the parabola crosses the x-axis relative to its vertex — twice, once, or not at all.

From Babylonian Geometry to Imaginary Numbers

Methods for solving what we would now call quadratic equations go back roughly 4,000 years, to Babylonian scribes working on clay tablets around 2000 BCE. But their approach was entirely geometric, not algebraic — there was no symbolic formula, no letters standing in for unknowns, and no notion of negative numbers to accept or reject. Problems were solved with step-by-step recipes for manipulating lengths and areas, methods we now recognize as equivalent to "completing the square," expressed purely as instructions for cutting and rearranging shapes. A general, symbolic formula — one that reliably returned positive real solutions for any equation of this form — was not clearly written down until over 2,600 years later, by the 7th-century Indian mathematician Brahmagupta, who also pioneered treating zero and negative numbers as legitimate quantities.

Even then, one case stayed off the table: a negative discriminant. For centuries, mathematicians who hit a negative number under the square root simply concluded there was no solution. That changed in the 16th century, when Italian mathematicians Gerolamo Cardano and Rafael Bombelli began systematically working with square roots of negative numbers while solving cubic equations — quantities earlier generations dismissed outright as "impossible" or "imaginary." Bombelli worked out consistent rules for adding and multiplying them, effectively founding what we now call complex numbers, even though the skeptical name "imaginary" stuck around as a reminder of how contested they once were. Today a negative discriminant simply means the parabola never touches the x-axis — a complete answer, not a dead end.