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Instantly solve any second-degree polynomial equation. Calculate exact real and complex roots, the discriminant, and the vertex with step-by-step clarity. This free online tool allows you to solve quadratic equations quickly and accurately. No sign-up or installation required.
In algebra, a quadratic equation is any polynomial equation of the second degree. This simply means that the highest exponent on any variable in the equation is a 2 (e.g., x²). The standard form of a quadratic equation is universally written as: ax² + bx + c = 0.
While they may seem like abstract mathematical constructs, quadratic equations actually govern the physics of the real world. Any time an object is thrown into the air, its trajectory forms a perfect parabola shaped perfectly by gravity. The exact path of a baseball, an artillery shell, or a rocket launch can all be precisely calculated using a quadratic equation.
Before you even solve for the actual answers (the "roots") of a quadratic equation, you can instantly determine exactly what kind of answers you will get by looking at the Discriminant (Δ). The discriminant is the tiny piece of math sitting directly underneath the square root symbol in the quadratic formula: (b² - 4ac).
The equation will yield exactly two distinct real roots. If you graph this equation on a plane, the parabola will slice through the X-axis in exactly two different places.
The equation will yield exactly one real root. The parabola does not slice through the X-axis. Instead, the very tip of its curve simply kisses the X-axis at one precise point.
The equation yields two complex/imaginary roots. The parabola never touches the X-axis at all—it floats completely above it or below it in physical space.
A common point of confusion for students learning algebra is the name itself. The prefix "Quad" universally means "four" (e.g., quadriceps, quadrilateral). So why is an equation with a maximum exponent of "2" called a quadratic?
The name comes from the ancient Latin word quadratus, which means "square." Because the primary variable in the equation is squared (x²), the ancient mathematicians essentially referred to them as "square equations," which translates directly to "quadratic" in modern nomenclature.
Common questions regarding the quadratic formula, imaginary numbers, and polynomial roots.