Solve any quadratic equation ax² + bx + c = 0 in one click. This free quadratic formula calculator gives exact roots (surds, fractions or complex numbers), decimals, the discriminant, vertex, factored form and a graph, with the full working shown step by step.
The quadratic formula
Worked example: 2x² − 4x − 6 = 0
- Identify a = 2, b = −4, c = −6.
- Discriminant: D = (−4)² − 4(2)(−6) = 16 + 48 = 64.
- √64 = 8, so x = (4 ± 8) / 4.
- x₁ = 12/4 = 3 and x₂ = −4/4 = −1.
- Factored form: 2(x − 3)(x + 1). Vertex at (1, −8).
Other ways to solve a quadratic
Factoring is fastest when the roots are whole numbers. Completing the square turns the equation into vertex form a(x − h)² + k = 0. The quadratic formula always works, which is why this calculator uses it and then simplifies the answer to its exact form.
Frequently asked questions
How do I use the quadratic equation solver?
Rearrange your equation so one side is 0, then enter the coefficients a, b and c. Press Solve to see the roots, discriminant, vertex, factored form, graph and step-by-step working.
What does the discriminant tell you?
The discriminant b² − 4ac tells you how many real solutions there are: positive means two real roots, zero means one repeated root, and negative means two complex roots with no x-intercepts.
Can it solve quadratics with complex roots?
Yes. When the discriminant is negative, the solver gives both complex roots in a ± bi form, exactly where possible, e.g. 1 ± 2i.
Does it give exact answers?
Yes. For whole-number or fraction coefficients it simplifies square roots and fractions, e.g. (3 ± √5)/2, and also shows the decimal values.
What if a = 0?
Then the equation is linear (bx + c = 0), not quadratic. The solver tells you and gives the single solution x = −c/b.
Related tools
Free educational tool by HawkInc. Results are calculated in your browser and rounded for display; check critical work by hand or with a second method.