Quadratic Formula Calculator
Find the roots of any quadratic equation ax² + bx + c = 0 with the quadratic formula. Enter a, b and c, and the calculator shows the discriminant and both roots, real or complex, rounded to 2 decimal places. Real roots are where the parabola crosses the x-axis. The coefficient a can't be 0, because then the equation is not quadratic.
Solve ax² + bx + c = 0
How to use the Quadratic Formula Calculator
- Enter your numbers in the Quadratic Formula Calculator above.
- Press the Solve button to see the result.
- To try other numbers, change any value and press the button again.
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.