Quadratic Equation Solver

How it works

A quadratic equation has the form ax² + bx + c = 0 where a, b, and c are constants and a is not zero. The quadratic formula x = (−b ± √(b² − 4ac)) / 2a gives both roots in closed form. The expression under the square root, b² − 4ac, is called the discriminant and tells you immediately how many real solutions to expect.

If the discriminant is positive, there are two distinct real roots — the parabola crosses the x-axis at two points. If the discriminant is exactly zero, there is one repeated real root — the parabola touches the x-axis at exactly one point (tangent). If the discriminant is negative, there are two complex-conjugate roots and no real x-intercepts — the parabola sits entirely above or below the x-axis. This calculator handles all three cases and shows the appropriate output.

The quadratic formula always works, but factoring is faster when the roots are nice integers. Look for two numbers that multiply to (a·c) and add to b. If you find them, you can split the middle term and factor by grouping. If the numbers don't come quickly, the quadratic formula is more reliable.

For physics and engineering, quadratics show up in projectile motion (height as a function of time), kinematics with constant acceleration, optics, and electrical circuits. The two roots often correspond to two physically meaningful times or positions — e.g., the two times a thrown ball reaches a given height (going up and coming down).

Higher-degree polynomials don't have such tidy formulas. Cubic and quartic equations have closed-form solutions but they are messy; degree 5 and higher have no general closed-form solution (Abel-Ruffini theorem). Numerical methods like Newton's iteration find roots iteratively.