✖️ Quadratic Equation Solver
Enter a, b and c from your equation to get both roots, the discriminant and the vertex of the parabola.
Roots
How the Quadratic Equation Solver works
The quadratic formula gives both solutions. The discriminant b² − 4ac tells you how many real roots there are: two if positive, one if zero, none (two complex) if negative.
x = (−b ± √(b² − 4ac)) ÷ 2a
Discriminant D = b² − 4ac
Vertex = (−b ÷ 2a, c − b² ÷ 4a)
Step-by-step calculation
- 1Enter a, b and c.
- 2Calculate the discriminant b² − 4ac.
- 3Apply the quadratic formula.
- 4Find the vertex.
Worked example
x² − 3x + 2 = 0 (a = 1, b = −3, c = 2): D = 9 − 8 = 1, so x = (3 ± 1) ÷ 2 → x = 2 or x = 1. The vertex is (1.5, −0.25).
Quadratic Equation Solver examples
| a | Roots |
|---|---|
| 1 | x = 2 or x = 1 |
| 2 | x = 0.75 ± 0.661438i |
| 5 | x = 0.3 ± 0.556776i |
| 10 | x = 0.15 ± 0.421307i |
| 100 | x = 0.015 ± 0.140624i |
Other inputs kept at the calculator's default values. Results calculated by this calculator.
Why use a Quadratic Equation Solver?
Solve quadratic equations with full steps.
Important factors that can affect your results
- a cannot be 0.
- Complex roots when D < 0.
- Rounding of irrational roots.
Frequently asked questions
What if a is 0?
Then it is a linear equation bx + c = 0, with one solution x = −c ÷ b.
What are complex roots?
When D is negative, the roots contain i = √−1, for example 1 ± 2i.
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Last reviewed November 2026 · How we build and check our tools · Report a problem