Quadratic Equation Solver
Solve ax² + bx + c = 0 step by step: discriminant, real and complex roots, vertex, axis of symmetry — plus a graph of the parabola.
Coefficients
Enter the coefficients of ax² + bx + c = 0. Decimals and fractions (as decimals) are fine.
How it works
Every quadratic is solved with the quadratic formula x = (−b ± √(b² − 4ac)) / 2a (derived by completing the square). The discriminant Δ = b² − 4ac decides the nature of the roots before any are computed: Δ > 0 gives two distinct real roots, Δ = 0 one repeated root, and Δ < 0 a pair of complex conjugates.
The vertex sits at x = −b / 2a — exactly midway between the two roots — and the parabola is symmetric about the vertical line through it, the axis of symmetry. Complex roots never touch the x-axis, so the graph shows the curve floating entirely above or below it.
Frequently asked questions
What does the discriminant tell me?
It classifies the roots without solving: Δ > 0 → two distinct real roots (the parabola crosses the x-axis twice); Δ = 0 → one repeated root (it just touches the axis); Δ < 0 → two complex conjugate roots (it never reaches the axis).
Why do complex roots always come in pairs?
Because the coefficients are real numbers. The quadratic formula gives −b/2a ± i·√|Δ|/2a — whatever the "±" produces, the two values are mirror images across the real axis, i.e. complex conjugates.
What happens if a = 0?
Then it is not a quadratic at all — it collapses to the linear equation bx + c = 0 with the single root x = −c/b. The solver rejects a = 0 and tells you so; for general linear work the equation is trivial to solve by hand.
Where is the vertex relative to the roots?
Exactly halfway between them: the vertex x-coordinate −b/2a is the average of the two roots (real or complex). That is why the axis of symmetry x = −b/2a mirrors one root onto the other.