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Factoring Quadratics Calculator

Solve ax² + bx + c = 0 with the full quadratic formula worked out step by step: exact roots like (3 ± √5)/2 or −1 ± 2i, the discriminant, the vertex and the factored form.

Solves ax² + bx + c = 0 (a ≠ 0). Decimals and negative numbers are allowed.

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Reading coefficients
Computing the discriminant
Applying the quadratic formula
Done

Result

This page opens the quadratic equation solver with the factored-form card put first: enter a, b and c and, whenever the roots are rational, the trinomial is rewritten as a product of integer factors. The default example x² − 3x + 2 factors as (x − 1)(x − 2) because its roots are x = 1 and x = 2.

The factoring comes straight from the quadratic formula: the solver computes the discriminant, checks that it is a perfect square, finds the roots as reduced fractions p/q, and assembles factors of the form (qx − p) — so 6x² − 5x + 1 with roots 1/2 and 1/3 becomes (2x − 1)(3x − 1), and a double root gives a perfect square like (x − 2)².

When the discriminant is not a perfect square the tool says so honestly — the trinomial does not factor over the rational numbers — and shows the exact radical roots instead, which is precisely the boundary between 'factor it' and 'use the formula' exercises. Either way the full step list documents the decision.

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Accepted Formats & Rules

  • Handles all three cases: two real roots, one repeated root and complex conjugate roots.
  • Simplifies radicals (√48 → 4√3) and cancels common factors with 2a for the cleanest exact form.
  • Also reports the vertex (−b/2a, −D/4a) with the vertex form and factors the trinomial when the roots are rational.

Key Benefits

✓ Every equation is solved the way a teacher writes it: discriminant, substitution into the formula, radical simplification and the final roots as numbered steps.
✓ Exact symbolic answers — rational roots come out as reduced fractions, irrational ones as simplified radicals like (1 ± √5)/2, complex ones as a ± bi.
✓ Runs entirely in your browser: no sign-up, no uploads, instant results on any device.

Use Cases

  • • Solve any ax² + bx + c = 0 for homework with the full quadratic-formula working: discriminant, substitution, simplified radical and exact roots like (1 ± √5)/2.
  • • Check whether a trinomial factors: rational roots produce the integer factorization, such as (2x − 1)(3x − 1) for 6x² − 5x + 1.
  • • Find the vertex and axis of symmetry of a parabola exactly — h = −b/2a and k = −D/4a as reduced fractions with the vertex form written out.