Solving ax² + bx + c = 0
A quadratic equation has the form ax² + bx + c = 0 with a not equal to zero. Its graph is a parabola, and the solutions (roots) are the points where the parabola crosses the x-axis.
The quadratic formula always finds them. This calculator gives real or complex roots, the discriminant that decides which kind you get, the vertex of the parabola and its axis of symmetry.
How it is calculated
x = (−b ± √(b² − 4ac)) ÷ 2a
Discriminant D = b² − 4ac
Worked example
x² − 3x + 2 = 0
- Discriminant
- 1
- Roots
- 2 and 1
- Vertex
- (1.5, −0.25)
What the discriminant tells you
| Discriminant | Roots | Graph |
|---|---|---|
| D > 0 | Two different real roots | Crosses the x-axis twice |
| D = 0 | One repeated real root | Touches the x-axis once |
| D < 0 | Two complex roots | Never touches the x-axis |
Where quadratics come up
Anything following a parabolic path — a ball thrown in the air, the shape of a satellite dish, profit-vs-price curves in economics — is modelled by a quadratic. The roots are where that curve crosses zero: for a thrown ball, the two times it is at ground level.