Quadratic Solver

A quadratic equation is any polynomial equation of degree two, written in the standard form ax² + bx + c = 0 where a ≠ 0. Quadratics appear everywhere in science and engineering — projectile motion, optimization, area problems, parabolic mirrors and antenna design, financial modeling and more.

The quadratic formula x = (−b ± √(b² − 4ac)) / 2a gives the two solutions. The expression under the square root, b² − 4ac, is called the discriminant and tells you what kind of solutions to expect: positive means two distinct real roots, zero means one repeated real root (the parabola just touches the x-axis), and negative means a pair of complex conjugate roots.

Geometrically, every quadratic graphs as a parabola. The sign of the leading coefficient a determines whether it opens upward (a > 0) or downward (a < 0). The vertex sits at x = −b/2a, and the y-intercept is simply c. The roots are where the parabola crosses (or touches) the x-axis.

Use this calculator to solve homework problems, check your work or visualize how changing coefficients reshapes the parabola. Try setting a = 1, b = 0 and varying c — you'll see the parabola shift up or down. Setting b = 0 makes the parabola symmetric about the y-axis. Experiment freely to build intuition.

Completing the square is the algebraic technique behind the quadratic formula and remains a useful skill long after you have learned to plug numbers into the formula. Rewriting ax² + bx + c in the form a(x − h)² + k immediately reveals the vertex (h, k) and the axis of symmetry x = h. This calculator does the arithmetic for you, but understanding the underlying derivation makes the formula memorable and shows why every quadratic curve looks the way it does.

Quadratics describe projectile motion under constant gravity, which is why anyone studying physics ends up solving them constantly. The height of a thrown ball as a function of time follows h(t) = −½gt² + v₀t + h₀, a perfect quadratic. Solving h(t) = 0 with this calculator gives you the moment the ball hits the ground; finding the vertex gives you the peak height and the time it occurs. The same equations govern artillery, rocketry, video-game physics and amusement-park ride design.

In business and economics, quadratics often appear as cost or revenue functions. A monopolist's revenue R(p) = p × D(p) where D is a linear demand curve produces a quadratic in price p. The price that maximises revenue is the vertex of that parabola, which this calculator can locate. Profit functions add a linear cost term but stay quadratic, so the same vertex analysis tells you the profit-maximising quantity. Real businesses extend this with cubic or piecewise models, but the quadratic is a powerful first approximation.

Complex roots appear whenever the discriminant b² − 4ac is negative. They come in conjugate pairs of the form p ± qi where i² = −1 and represent points where the parabola never crosses the x-axis. While complex roots seem abstract, they are crucial in electrical engineering, signal processing and quantum mechanics. This calculator returns them when needed, so you never have to take a square root of a negative number by hand. Treat them as honest mathematical objects, not as errors.