Quadratic Factoring Calculator
Factor ax² + bx + c and solve it: discriminant, real or complex roots with exact surd form, vertex, axis of symmetry, and the sum and product of the roots.
Interactive calculator · Method QUADRATIC-FACTORING-1.0 · Last reviewed: 2026-11-04
Calculate your scenario
Discriminant D = b² − 4ac.
Your inputs
Default worked example. Edit the inputs to calculate your scenario.
Default worked example
D = 1² − 4 × 2 × -6 = 49; x = (−b ± √D) ÷ 2a = (-1 ± √49) ÷ 4; vertex x = −b ÷ 2a = -0.25.
How to use the calculator
Write the equation as ax² + bx + c = 0 and enter a, b and c, including signs. Missing terms have coefficient 0.
What the result means
2x² + x − 6 has discriminant 49, a perfect square, so it factors as (2x − 3)(x + 2) with roots 1.5 and −2; the parabola’s vertex is at (−0.25, −6.125).
Formula and units
Discriminant D = b² − 4ac. Roots x = (−b ± √D) ÷ 2a: two real roots when D > 0, one repeated root when D = 0, complex roots −b/2a ± (√−D / 2|a|) i when D < 0. Vertex h = −b ÷ 2a, k = c − b² ÷ 4a; axis x = h. Sum of roots = −b/a, product = c/a. With integer a, b, c and a perfect-square D the quadratic factors over the integers as k(q₁x − p₁)(q₂x − p₂); otherwise it is written as a(x − r₁)(x − r₂).
Worked examples
2x² + x − 6 (factors over the integers): a (coefficient of x², not 0): 2 dimensionless; b (coefficient of x): 1 dimensionless; c (constant term): -6 dimensionless.
Example result: Roots: x = 1.5, x = -2; Factored form: (2x − 3)(x + 2); Discriminant b² − 4ac: 49 (two real roots); Vertex (h, k): (-0.25, -6.125); Axis of symmetry: x = -0.25; Sum and product of roots: sum -0.5, product -3.
x² + 2x + 5 (complex roots): a (coefficient of x², not 0): 1 dimensionless; b (coefficient of x): 2 dimensionless; c (constant term): 5 dimensionless.
Example result: Roots: x = -1 ± 2i; Factored form: (x − (-1 + 2i))(x − (-1 − 2i)); no real factors; Discriminant b² − 4ac: -16 (two complex conjugate roots); Vertex (h, k): (-1, 4); Axis of symmetry: x = -1; Sum and product of roots: sum -2, product 5.
Limitations and common mistakes
Quadratics only (degree 2, a ≠ 0). Integer factoring is attempted only for whole-number coefficients; decimals give a factored form with roots rounded to 6 places (multiply through by a power of 10 first for an exact integer factorisation). The exact surd form is shown for whole-number coefficients when D is not a perfect square.
Dropping the sign of b or c: for x² − 5x + 6 enter b = −5 and c = 6.
Sources and assumptions
Method QUADRATIC-FACTORING-1.0; last formula review 2026-11-04. Sources and their supported claims. Calculation methodology.