Quadratic Equation Calculator

A quadratic equation is any equation that can be rearranged into the form ax² + bx + c = 0, where a is not zero. The defining feature is the x² term — it is what makes the graph of the equation a parabola, a symmetric curve that opens either upward or downward, rather than the straight line produced by an equation with only x.

Solving the equation means finding the value or values of x where that curve crosses the horizontal axis — the points where the expression equals exactly zero. Depending on the shape of the parabola, there can be two such crossing points, one, or none at all, and the discriminant is what tells you which case you are in before you finish the calculation.

The shape of a parabola turns out to describe a surprising number of real, physical and financial situations.

The expression under the square root in the quadratic formula, b² − 4ac, is called the discriminant, and its sign alone tells you what kind of answer to expect. A positive discriminant means the parabola crosses the x-axis twice, giving two distinct real roots. A discriminant of exactly zero means the parabola just touches the axis at a single point — its vertex — giving one repeated root. A negative discriminant means the parabola never reaches the axis at all, and the two roots become a complex conjugate pair rather than real numbers.

For a problem describing something physical, a negative discriminant is usually a meaningful result in itself: it says the scenario in question never actually happens, at least not within the real numbers being modeled — a ball thrown too gently to reach a given height, for instance, rather than a mistake in the calculation.

Two shortcuts, known as Vieta’s formulas, let you sanity-check a pair of roots without redoing the whole calculation. The two roots always add up to −b/a, and they always multiply to c/a. After solving x² − 3x − 10 = 0 to get roots of 5 and −2, it takes a moment to confirm: 5 + (−2) = 3, which matches −b/a = −(−3)/1 = 3, and 5 × (−2) = −10, which matches c/a = −10/1. If either check fails, an arithmetic slip happened somewhere upstream.

Problems that amount to quadratic equations are far older than the algebraic notation used to solve them today. Babylonian mathematicians were solving problems equivalent to quadratics using geometric methods on clay tablets roughly 4,000 years ago, long before anyone wrote an equation using symbols like x, + or =.

The systematic method taught in schools today — essentially "completing the square" turned into a formula — was formalized much later, and the word "algebra" itself comes from "al-jabr," a term used in the title of a 9th-century mathematical treatise. Modern symbolic notation, with letters standing in for unknown and known quantities, is more recent still, mostly a product of 16th- and 17th-century European mathematics.