Collatz Sequence Calculator
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).
- 1For ax² + bx + c = 0: roots = (−b ± √(b²−4ac)) / 2a
- 2When b²−4ac < 0: roots are complex
- 3√(negative) = i × √(|negative|)
- 4Complex roots always appear as conjugate pairs
x² + 4 = 0=x = ±2i
x² − 2x + 5 = 0=x = 1 ± 2i
| 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 |
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