Quadratic equations were solved four thousand years ago
An equation with a squared unknown—standard form ax² + bx + c = 0 with a nonzero—has at most two roots. Count complex numbers and double roots carefully, and there are always two. Completing the square yields the famous formula.
The name comes from Latin for square. If a vanishes and b does not, the equation collapses to linear. Coefficients a, b, and c are the quadratic, linear, and constant terms. Solutions are roots of the associated quadratic function. With real coefficients one finds two distinct reals, a real double root, or a conjugate complex pair; allowing complexes and counting multiplicity, there are always two roots. Complex coefficients likewise always yield two complex roots, not necessarily distinct. Factoring into linear factors is equivalent when it works.
Methods vary. Factoring by inspection—finding numbers that sum to b and multiply to c when a = 1—is often students' first tool (Vieta's rule), but it mainly catches rational roots; most applied quadratics refuse neat factors. Completing the square follows a fixed algorithm: normalize, add the square of half the linear coefficient, rewrite as a square, then take plus-or-minus square roots. That procedure derives the quadratic formula expressing roots in terms of a, b, and c. Older texts sometimes rescale the linear coefficient, producing cosmetic variants of the same formula.
Problems reducible to quadratics were known by about 2000 BC. The formula remains the clean closed form for degree two; higher degrees need different stories, and some lack radical solutions entirely. From Babylonian tablets to modern calculators, the square term still marks the first genuinely nonlinear equation most people learn to finish.
Source: Quadratic equation
