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Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0, real or complex roots, discriminant, vertex and axis of symmetry, instantly.
HOW TO USE
- 1Enter the coefficients a, b and c of your equation ax² + bx + c = 0. The coefficient a cannot be zero, or the equation is not quadratic.
- 2The discriminant Δ = b² − 4ac decides the nature of the roots: positive gives two real roots, zero gives one repeated root, negative gives two complex conjugate roots.
- 3The vertex is the turning point of the parabola, and the axis of symmetry is the vertical line x = −b/(2a) it sits on.
- 4Complex roots are shown in the form a ± bi, where i is the imaginary unit.
EXAMPLE
INPUT
a = 1, b = −5, c = 6
OUTPUT
Δ = 1 · x₁ = 3, x₂ = 2 · vertex (2.5, −0.25) · axis x = 2.5
FAQ
How do I know whether the roots are real or complex?
Look at the discriminant Δ = b² − 4ac. If Δ > 0 there are two distinct real roots, if Δ = 0 there is one repeated real root, and if Δ < 0 there are two complex conjugate roots.
What does the vertex represent?
The vertex is the lowest point of a parabola that opens upward (a > 0) or the highest point of one that opens downward (a < 0). Its x-coordinate is −b/(2a), the same as the axis of symmetry.
Why are complex roots always a pair a ± bi?
The quadratic formula produces √Δ, and for a negative discriminant that is √(−Δ)·i. The plus and minus signs give two roots that mirror each other across the real number line.
What if a is zero?
Then the equation is linear, not quadratic, and the quadratic formula does not apply. The calculator flags this case instead of returning a misleading answer.
Is my data sent anywhere?
No. All calculations run entirely in your browser, your coefficients never leave your device.