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Discriminant Calculator (b² − 4ac)

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 discriminant card put first: enter a, b and c and the tool computes D = b² − 4ac with your numbers substituted into the formula, exactly and step by step. For x² − 3x + 2 = 0 it shows D = (−3)² − 4·1·2 = 9 − 8 = 1.

The sign of the discriminant classifies the equation before any root is computed: D > 0 means two distinct real roots, D = 0 means one repeated (double) root, and D < 0 means a pair of complex conjugate roots. The solver states the classification explicitly on every run, so 'determine the nature of the roots' exercises answer themselves.

Because the full solution is computed anyway, you also get the roots that the discriminant predicts — as reduced fractions when D is a perfect square, as simplified radicals like (1 ± √5)/2 otherwise — plus the vertex and, when possible, the factored form, all from the same three coefficients.

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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.