How does the Complex Roots Calculator work?

For ax² + bx + c = 0: roots = (−b ± √(b²−4ac)) / 2a Then: When b²−4ac < 0: roots are complex Then: √(negative) = i × √(|negative|) Then: Complex roots always appear as conjugate pairs.

Can you give an example of how to use the Complex Roots Calculator?

Example: Input x² + 4 = 0 gives a result of x = ±2i.

Is the Complex Roots Calculator free to use?

Yes — completely free with no registration, download, or subscription required. All calculations happen instantly in your browser.

How accurate is the Complex Roots Calculator?

Our Complex Roots Calculator uses verified mathematical formulas and is accurate to multiple decimal places. Results are calculated in real-time using the same methods used by professionals.

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Starting Number (positive integer ≤ 1,000,000)

Complex roots occur when a polynomial equation has no real solutions — the discriminant is negative. The roots come in conjugate pairs: a + bi and a − bi, where i = √(−1).

Symbol	Name	Unit	Description
a, b, c	quadratic coefficients	—	—
Δ	discriminant (b²−4ac)	—	determines if roots are real or complex
i	imaginary unit	—	where i² = −1

Equation	Discriminant	Roots
x²+4=0	−16	±2i
x²−2x+5=0	−16	1±2i
x²+2x+5=0	−16	−1±2i
x²+x+1=0	−3	−½±(√3/2)i

References

›Complex numbers — Khan Academy

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