Solve any quadratic equation instantly — roots, discriminant, vertex, and more
Coefficients
Enter the coefficients of ax² + bx + c = 0
Roots
x₁
3
x₂
2
Quadratic formula
x = (−b ± √(b² − 4ac)) / (2a)
x = (−(-5) ± √(1)) / (2 × 1)
x₁ = 3, x₂ = 2
Vertex x
2.5
Vertex y
-0.25
Axis of symmetry
x = 2.5
Opens
Upward (min)
Sum of roots (−b/a)
5
Product of roots (c/a)
6
Standard form
1x² − 5x + 6 = 0
Vertex form
1(x − 2.5)² − 0.25 = 0
Factored form
1(x − 3)(x − 2) = 0
Try an example
The quadratic equation solver calculates exact solutions to any equation in the form ax² + bx + c = 0. Enter your three coefficients and get the discriminant, both roots (real or complex), vertex coordinates, axis of symmetry, and the equation rewritten in standard, vertex, and factored forms — all in one step, with no sign-up required.
The discriminant (Δ = b² − 4ac) reveals the nature of the roots without fully solving the equation. If Δ > 0 there are two distinct real roots; if Δ = 0 there is exactly one repeated real root; and if Δ < 0 the two roots are complex conjugates (no real solutions).
When the discriminant is negative the solver computes the complex conjugate pair using √(−Δ) for the imaginary part and displays them as a ± bi. The imaginary unit i is shown explicitly so the result is unambiguous.
The vertex is the turning point of the parabola — its minimum when a > 0 or maximum when a < 0. Its x-coordinate is −b/(2a) and its y-coordinate is c − b²/(4a). It is also the axis of symmetry and the optimal or worst-case value in many real-world optimization problems.
Yes. The solver accepts any real number for a, b, and c including decimals like 1.5 or −0.25. Results are displayed up to six significant decimal places with trailing zeros removed for readability.