Quadratic Formula Solver

How the quadratic formula solver works

A quadratic equation in standard form is ax² + bx + c = 0, where a ≠ 0. The quadratic formula gives the two roots directly:

x = (−b ± √(b² − 4ac)) / (2a)

The expression under the radical — b² − 4ac — is called the discriminant (Δ). Its sign determines what kind of roots you get:

• Δ > 0 — two distinct real roots. The parabola crosses the x-axis at two points.
• Δ = 0 — one repeated real root (a “double root”). The parabola just touches the x-axis at its vertex.
• Δ < 0 — two complex conjugate roots of the form p ± qi. The parabola never crosses the x-axis.

What this calculator shows

• Roots — written as "x = 1 or 2", "x = 2 (double)", or "x = −1 ± 2i" depending on the discriminant sign.
• Discriminant + sign banner — instant visual indicator of the root type.
• Vertex at (−b/2a, f(−b/2a)) — the parabola's minimum (if a > 0) or maximum (if a < 0).
• Axis of symmetry — the vertical line x = −b/2a.
• Factored form — shown whenever the roots are rational (e.g., (x − 2)(x − 1)). Irrational or complex roots don't factor nicely over the rationals, so this is omitted for those.
• Parabola plot — domain auto-fit around the vertex and roots, with vertex and root markers.
• Step-by-step — the full discriminant calc, the (−b ± √Δ)/(2a) substitution, and the simplification.

Common mistakes to avoid

• Forgetting the ± sign — a quadratic has two roots (possibly equal, possibly complex). The formula's ± captures both.
• Sign errors under the radical — it's b² − 4ac, and b is squared first. Two negatives inside (like c = −3) can flip the discriminant's sign if you drop a parenthesis.
• Treating a = 0 as quadratic — if a is 0, the equation is linear (bx + c = 0). This solver rejects a = 0 with an explicit error.
• Confusing vertex with roots — the vertex is at x = −b/2a (midpoint of the roots when they're real), not at a root.

Quick reference — discriminant → root type

• Δ = 25 (positive perfect square) → rational roots, clean factored form.
• Δ = 5 (positive non-square) → irrational real roots, no clean factored form.
• Δ = 0 → one repeated root (perfect-square trinomial like x² − 6x + 9 = (x − 3)²).
• Δ = −7 → complex conjugate pair, no real x-intercepts.