The quadratic formula
Every quadratic equation has the form ax² + bx + c = 0, with a ≠ 0. The formula solves x by first computing the discriminant: Δ = b² − 4ac. The roots are then x = (−b ± √Δ) / (2a).
The three possible cases
| Discriminant | Result |
|---|---|
| Δ > 0 | Two different real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | Two complex (conjugate) roots |
Worked example
For x² − 5x + 6 = 0: a=1, b=−5, c=6. Δ = 25 − 24 = 1. Roots: x₁ = (5+1)/2 = 3 and x₂ = (5−1)/2 = 2.
Where quadratic equations show up
Besides schoolwork, quadratic equations model projectile trajectories, area calculations, profit optimization in economics problems, and many physics and engineering problems.
Frequently asked questions
What is the discriminant (Δ)?
It is b² − 4ac, which determines how many roots the equation has and what type: two real, one repeated, or two complex.
What happens if a = 0?
The equation stops being quadratic (it becomes linear), so a must be different from zero.
What are complex roots?
When the discriminant is negative, there are no real roots — the solutions involve the imaginary unit i (the square root of −1).
Does this calculator accept negative and decimal numbers?
Yes, coefficients a, b and c can be positive, negative or decimal.