Complex Roots Calculator
Solve a quadratic ax^2 + bx + c for its two roots, including complex conjugate roots when the discriminant is negative. Enter a, b, and c to see both roots. Free, runs in the browser, no sign-up.
What this calculator does
The Complex Roots Calculator estimates your result instantly from the values you enter, using standard math & conversions formulas. Solve a quadratic ax^2 + bx + c for its two roots, including complex conjugate roots when the discriminant is negative. Enter a, b, and c to see both roots. Free, runs in the browser, no sign-up. Calculations run in your browser, so nothing you type is sent to a server, and the output is an estimate rather than a certified or official figure.
Result
About Complex Roots Calculator
The Complex Roots Calculator solves a x² + b x + c = 0 using the quadratic formula. It first computes the discriminant d = b² − 4ac.
When d is zero or positive the two roots are real; when d is negative the roots form a complex conjugate pair, reported around a real part of −b ÷ (2a) with an imaginary part of √(−d) ÷ (2a).
Formula & Mathematical Calculation
Root 1 = f(a, b, c)
Calculated according to standard math formulas implemented in the calculator core.
Calculation Example
For a = 1, b = 2, and c = 5, the discriminant is 2² − 4 × 1 × 5 = 4 − 20 = −16, which is negative. The real part is −2 ÷ 2 = −1 and the imaginary part is √16 ÷ 2 = 2, so the tool shows Root 1 = -1 + 2i and Root 2 = -1 - 2i.
Key Benefits
- Detects from the discriminant whether the roots are real or complex.
- Reports complex roots as a clear conjugate pair with real and imaginary parts.
- Handles real-root quadratics too, showing both real solutions.
- Free and runs in the browser with no sign-up.
How to Use this Calculator
-
Enter coefficient a
Type the coefficient of x squared into the a field; it cannot be zero. -
Enter coefficient b
Type the coefficient of x into the b field. -
Enter coefficient c
Type the constant term into the c field. -
Calculate
Press Calculate to see Root 1 and Root 2.
Assumptions & Limitations
What the calculation assumes
- Every value you enter is treated as an exact, known amount.
- Any rate or interval you enter is assumed to hold for the whole period unless the form asks for it separately.
- Results use standard published formulas and account only for the values entered above.
- Rounding is applied for display; the underlying calculation uses your full precision.
- Results are returned in the same units you enter, unless the tool states the output unit.
What it does not cover
- Complex Roots Calculator produces a calculated result for reference and learning, not a certified or official measurement.
- Real-world outcomes differ when tolerances, rounding rules or local standards apply.
- Definitions and standards vary by region and change over time; verify the convention that applies to your case.
- Figures are only as accurate as the inputs you supply — check them before acting on the result.
- For work where the result carries real consequences, confirm it against an authoritative source.
Frequently Asked Questions
Free to use, no account needed. Calculations run in your browser, so nothing you type is sent to a server. Results are estimates — how we build and check our calculators.