Quadratic Formula Calculator — Solve ax² + bx + c = 0

Coefficients

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Degree

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Discriminant

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Root Form

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Sample Table

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Coefficients

Enter a, b, c for ax² + bx + c = 0. Results refresh live as you type. Any of a, b, c may be negative or fractional; use zero for a to switch to the linear equation bx + c = 0.

a — quadratic Set to 0 for a linear fallback. b — linear Any real number. c — constant Any real number.

Solution

Discriminant, roots, vertex, and axis of symmetry for the current equation.

Equation 1x² − 5x + 6 = 0

Discriminant Δ—

Vertex (h, k)—

Axis of Symmetry—

Roots—

y-value table around the roots

Nine (x, y) points sampled around the roots. Rows landing exactly on a root are flagged; y ≈ 0 there confirms the root checks out.

x	y = f(x)	Note

Step-by-step derivation

Every printed result, derived from your coefficients in order.

How the math works

Discriminant: Δ = b² − 4ac. Δ > 0 → two distinct real roots; Δ = 0 → one double root; Δ < 0 → two complex conjugate roots.
Roots: x = (−b ± √Δ) / 2a, applied exactly as printed in the derivation. Complex roots are shown as real part ± imaginary part × i.
Vertex: h = −b / 2a, k = f(h) = c − b² / 4a. The axis of symmetry is the vertical line x = h, which sits halfway between the two roots.
a = 0 falls back to the linear solve bx + c = 0 → x = −c / b; a = b = 0 means either no solution (c ≠ 0) or every x is a solution (c = 0).
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