Quadratic Equation Solver
Enter a, b and c to get both roots — real or complex — along with the discriminant, the vertex, the axis of symmetry and the steps.
How it works
The quadratic formula solves any equation of the form ax² + bx + c = 0. The discriminant b² − 4ac decides what kind of answer you get before you finish the calculation: positive gives two real roots, zero gives one repeated root, negative gives a conjugate pair of complex roots.
The vertex is where the parabola turns, at x = −b ÷ 2a. Together with the sign of a — which decides whether the curve opens upwards or downwards — it tells you the shape without plotting anything.
The formula
Quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
Discriminant
Δ = b² − 4ac
Vertex
x = −b ÷ 2a, then substitute for y
Sum and product of roots
sum = −b ÷ a · product = c ÷ a
Worked examples
| Scenario | Working | Result |
|---|---|---|
| x² − 3x + 2 = 0 | Δ = 9 − 8 = 1 | x = 2 and x = 1 |
| x² − 4x + 4 = 0 | Δ = 0 | x = 2, a repeated root |
| x² + x + 1 = 0 | Δ = −3 | Complex roots −0.5 ± 0.866i |
When you'd use it
- Checking homework before handing it in
- Finding where a projectile's path crosses the ground
- Solving for break-even in a quadratic cost model
- Getting the vertex to sketch a parabola
Common questions
What does the discriminant tell me?
How many real roots there are, before you finish solving. Positive means two, zero means one repeated root, negative means none — the parabola never meets the x-axis and both roots are complex.
Why must a be non-zero?
With a = 0 there is no x² term, so the equation is linear — bx + c = 0 — and the quadratic formula divides by zero. Solve it as x = −c ÷ b instead.
Can it handle complex roots?
Yes. When the discriminant is negative both roots are shown in a ± bi form rather than reporting no solution, which is what most school answers expect.