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Quadratic Equation with Complex Roots

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.

Processing... 0%
Reading coefficients
Computing the discriminant
Applying the quadratic formula
Done

Result

This page opens the quadratic equation solver preloaded with a = 1, b = 2, c = 5 — an equation whose discriminant is negative: D = 2² − 4·1·5 = −16. No real number squares to −16, so the parabola never crosses the x-axis and the two roots are complex conjugates.

The solver continues the quadratic formula into the complex numbers: √(−16) = 4i, so x = (−2 ± 4i)/2 = −1 ± 2i. The radical under the negative sign is simplified exactly like a real one (√(−12) becomes 2i√3), common factors are cancelled with 2a, and the result is stated in the standard a ± bi form with decimal approximations alongside.

Replace the prefilled example with any coefficients of your own: whenever D < 0 you will get the exact conjugate pair, and whenever D ≥ 0 the same steps produce the real roots instead — so the page doubles as a general check for 'solve over the complex numbers' homework.

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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.