Quadratic Formula Solver

The Quadratic Formula Solver solves any equation of the form ax² + bx + c = 0 for x. Enter the three coefficients — a, b, and c — and it instantly returns the roots, whether they're two distinct real numbers, one repeated real number, or a pair of complex conjugates, along with a step-by-step derivation showing exactly how the answer was reached.

It's built for algebra students checking homework, teachers preparing worked examples, and anyone — engineers, developers, hobbyist mathematicians — who needs to solve a quadratic without doing the arithmetic by hand.

How to use it

1. Enter the coefficient a (the number multiplying x²). It must not be zero.
2. Enter the coefficient b (the number multiplying x).
3. Enter the coefficient c (the constant term).
4. Read the Roots field for the solution, the Discriminant (Δ) field for its value, and Step-by-Step Solution for the full derivation.

Results update automatically as you type. For example, entering a = 1, b = -5, c = 6 (for x² − 5x + 6 = 0) returns roots x₁ = 3 and x₂ = 2.

What is the quadratic formula?

For any equation ax² + bx + c = 0 with a ≠ 0, the solutions are given by:

x = (−b ± √(b² − 4ac)) / (2a)

The expression under the square root, b² − 4ac, is called the discriminant (Δ). Its sign determines what kind of roots the equation has.

What does the discriminant tell you?

• Δ > 0 — the equation has two distinct real roots. The parabola y = ax² + bx + c crosses the x-axis at two points.
• Δ = 0 — the equation has exactly one real root, repeated (a "double root"). The parabola just touches the x-axis at its vertex.
• Δ < 0 — the equation has no real roots. Instead it has two complex conjugate roots of the form p + qi and p − qi, where i is the imaginary unit (√−1). The parabola never crosses the x-axis.

Why can't a be zero?

If a = 0, the x² term disappears and the equation reduces to bx + c = 0 — a linear equation, not a quadratic one, with at most a single solution found by simple division rather than the quadratic formula. The solver flags a = 0 as invalid input rather than silently switching to linear-equation logic, since a genuine quadratic requires a nonzero leading coefficient.

How are complex roots formatted?

Complex roots are shown in standard a + bi notation, for example -0.5 + 0.866025i and its conjugate -0.5 - 0.866025i. The real part comes from −b / (2a) and the imaginary part from √|Δ| / (2a); the two roots always share the same real part and differ only in the sign of the imaginary part.

Privacy

This calculator runs entirely in your browser. Your coefficients are never uploaded, logged, or stored — the computation happens locally on your device.