Quadratic Formula Calculator
Find the roots (x-intercepts) of any quadratic equation ax² + bx + c = 0 using the quadratic formula. Handles real and complex roots.
Solve ax² + bx + c = 0
How to use the Quadratic Formula Calculator
- Enter your inputs into the Quadratic Formula Calculator above.
- Results update instantly as you type — no submit button needed.
- Adjust any value to see how the result changes in real time.
The quadratic formula
x = (−b ± √(b² − 4ac)) / 2a
The discriminant Δ = b² − 4ac determines the type of roots: Δ > 0 → two real roots; Δ = 0 → one repeated real root; Δ < 0 → two complex conjugate roots.
Worked example
2x² + 5x − 3 = 0: a=2, b=5, c=−3. Discriminant: 25 + 24 = 49. x = (−5 ± 7)/4. Two real roots: x = 2/4 = 0.5 and x = −12/4 = −3.
Frequently asked questions
What if the discriminant is negative?
The equation has two complex roots: x = (−b ± i√|Δ|) / 2a. They come as complex conjugates (a + bi and a − bi).
Can I solve any quadratic this way?
Yes — the formula works for every quadratic equation. Factoring or completing the square are sometimes faster for simple cases but never more general.
What does the discriminant tell me geometrically?
It tells you how the parabola intersects the x-axis. Two intersections (Δ>0), tangent at one point (Δ=0), or no intersection (Δ<0). The parabola opens up if a>0, down if a<0.