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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).
Formula
For ax² + bx + c = 0: x = (−b ± √(b²−4ac))/(2a); √(negative) = i√(|negative|)
- a, b, c
- quadratic coefficients
- Δ
- discriminant (b²−4ac) — determines if roots are real or complex
- i
- imaginary unit — where i² = −1
Guida passo passo
- 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
Esempi risolti
Ingresso
x² + 4 = 0
Risultato
x = ±2i
Ingresso
x² − 2x + 5 = 0
Risultato
x = 1 ± 2i
Domande frequenti
What is the complex conjugate of a root?
If 3 + 2i is a root, then 3 − 2i is its conjugate. They always appear in pairs for real polynomials.
When do complex roots occur?
When the discriminant b²−4ac is negative. This means the parabola doesn't cross the x-axis.
Can I visualize complex roots on a number line?
No, complex roots require a 2D plane (complex plane): real part on x-axis, imaginary part on y-axis.
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