Quadratic Formula Calculator — Solve ax² + bx + c = 0
Enter the coefficients a, b, and c of ax² + bx + c = 0. The calculator applies the quadratic formula to find both roots (real or complex), computes the discriminant, locates the vertex of the parabola, and plots the curve.
x = (−b + √(b²−4ac)) / (2a) | larger root when a > 0
- 1
Discriminant Δ = b² − 4ac
-5² − 4 × 1 × 6 = 1Δ > 0 means two distinct real roots; Δ = 0 means one double root. - 2
√Δ
√1 = 1 - 3
2a
2 × 1 = 2 - 4
x₁ = (−b + √Δ) ÷ (2a)
(−-5 + 1) ÷ 2 = 3
How does this calculator work?
Enter a, b, c in ax² + bx + c = 0. The quadratic formula x = (−b ± √(b²−4ac))/(2a) gives both roots. Δ > 0 → two distinct real roots; Δ = 0 → one double root; Δ < 0 → two complex conjugate roots. Vertex at x = −b/(2a). Works for any a ≠ 0.
Formula
How this is calculated
The quadratic formula x = (−b ± √Δ)/(2a), where Δ = b²−4ac, solves any quadratic equation directly. The discriminant determines the nature of the solutions: Δ > 0 gives two distinct real roots; Δ = 0 gives one real root of multiplicity two (a double root at x = −b/(2a)); Δ < 0 gives two complex conjugate roots α ± βi, where α = −b/(2a) and β = √(−Δ)/(2a).
The quadratic y = ax² + bx + c describes a parabola. Its vertex — the minimum if a > 0 or maximum if a < 0 — is at x = −b/(2a) and y = c − b²/(4a). The roots are the x-intercepts of the parabola (where y = 0). When Δ < 0, the parabola lies entirely above or below the x-axis and has no x-intercepts.
The formula is derived by completing the square: rewrite x² + (b/a)x = −c/a, add (b/(2a))² to both sides to form a perfect square, then take the square root and rearrange. This is an exact closed-form solution — unlike iterative root-finding methods such as Newton-Raphson iteration, it introduces no approximation error beyond floating-point arithmetic.
Frequently asked questions
The discriminant encodes the number and type of roots. Δ > 0: two distinct real roots — the parabola crosses the x-axis at two points. Δ = 0: one repeated real root — the parabola touches the x-axis at exactly one point (the vertex). Δ < 0: no real roots — the parabola lies fully above or below the x-axis, and the roots are complex conjugates.
Setting a = 0 reduces the equation to bx + c = 0, which is linear, not quadratic. The quadratic formula is undefined for a = 0. The linear solution is x = −c/b (provided b ≠ 0). This calculator requires a ≠ 0.
When Δ < 0, the roots are x = α ± βi where α = −b/(2a) is the real part and β = √(−Δ)/(2a) is the magnitude of the imaginary part. For example, α = 1, β = 2 gives roots 1 + 2i and 1 − 2i. Complex conjugate roots always come in pairs and cannot appear in a real-coefficient polynomial one without the other.
Also known as
TG we-Calculate Editorial Team. (2026). Quadratic Formula Calculator — Solve ax² + bx + c = 0 [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/quadratic-formula-calculator
TG we-Calculate Editorial Team. "Quadratic Formula Calculator — Solve ax² + bx + c = 0." TG we-Calculate. 2026. https://we-calculate.com/calculator/quadratic-formula-calculator.
TG we-Calculate Editorial Team, "Quadratic Formula Calculator — Solve ax² + bx + c = 0," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/quadratic-formula-calculator
@misc{wecalculate_quadratic_formula_calculator, title = {Quadratic Formula Calculator — Solve ax² + bx + c = 0}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/quadratic-formula-calculator}}, year = {2026}, note = {TG we-Calculate} }
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