Quadratic Equation Calculator

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. It is a second-degree polynomial equation, meaning the highest power of the variable x is 2. Quadratic equations have at most two solutions (roots) and graph as parabolas.

The discriminant D = b² − 4ac determines the nature of the roots without computing them. If D > 0, there are two distinct real roots. If D = 0, there is one repeated real root. If D < 0, there are two complex conjugate roots. The discriminant is the value under the square root in the quadratic formula.

Complex roots occur when the discriminant is negative, meaning the parabola never crosses the x-axis. The roots take the form p ± qi, where p is the real part, q is the imaginary part, and i = √(−1). Complex roots always come in conjugate pairs. For example, x² + 1 = 0 has roots x = ±i.

The vertex of the parabola y = ax² + bx + c is at the point (h, k) where h = −b/(2a) and k = f(h) = a(h)² + b(h) + c. The vertex represents the minimum point (when a > 0) or maximum point (when a < 0) of the parabola. This calculator computes both vertex coordinates automatically.

Yes. When the discriminant is negative, the calculator returns complex conjugate roots in the form p ± qi. The real and imaginary parts are shown separately in the results panel. The graph will show the parabola without x-axis crossings.

Factoring splits the quadratic into two linear terms (e.g., x² − 5x + 6 = (x − 2)(x − 3)) and is fast when the roots are rational integers. The quadratic formula x = (−b ± √D) / (2a) works for every quadratic equation, including those with irrational or complex roots. Use the quadratic formula when factoring is not obvious or when the discriminant is not a perfect square.

Substitute each root back into the original equation ax² + bx + c = 0. If the result is zero (within rounding tolerance), the solution is correct. You can also use Vieta’s formulas: the sum of roots should equal −b/a and the product should equal c/a.

If a = 0, the equation is no longer quadratic — it becomes a linear equation bx + c = 0 with at most one solution x = −c/b (assuming b ≠ 0). This calculator validates that a ≠ 0 before solving. If you need to solve a linear equation, rearrange it directly: x = −c/b.

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