Quadratic Equation Solver

The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It gives the exact roots of any quadratic equation ax² + bx + c = 0 by evaluating the expression under the radical — the discriminant — to determine how many and what type of solutions exist.

When the discriminant D = b² − 4ac is negative, the square root of a negative number is required, which yields imaginary results. The equation therefore has no real roots — instead it has two complex conjugate roots of the form x = (−b ± i√|D|) / 2a, where i is the imaginary unit (√−1). This means the corresponding parabola does not cross the x-axis at any point.

TheCalcPro's quadratic engine detects a negative discriminant automatically and switches into complex-number mode. It expresses each root in the standard form a ± bi, where the real part a = −b / 2a and the imaginary part b = √|D| / 2a are displayed separately. No plugins or manual mode switching are needed — the solver handles real, repeated, and complex roots transparently within a single workflow.

Yes. The solver exposes every intermediate step: coefficient extraction, discriminant computation (D = b² − 4ac), the ± branch evaluation, and the final simplified root form. This makes it suitable for verifying homework proofs, cross-checking textbook derivations, or understanding how the completing-the-square method leads to the quadratic formula.