x² + 1 = 0
- Input
- a=1, b=0, c=1
- Expected output
- x = ±i
Δ = −4; roots ±i (imaginary units).
complex roots quadratic equation
Complex Roots of Quadratic Equation
When Δ < 0, √Δ does not exist in the reals. The roots are complex conjugates: x = (−b ± i√|Δ|) / (2a).
Complex roots in the complex plane
- Complex roots always appear as conjugate pairs (a+bi and a−bi) when the equation has real coefficients. Their moduli and arguments have geometric symmetry.
Examples
Safe use
- Input
- context + tool result
- Expected output
- interpreted with limits and next steps
Use the result as technical or educational support, keeping the tool limits explicit in the workflow.
Full tool FAQ
The discriminant Δ = b²−4ac determines the nature of roots: Δ>0 → two distinct real roots; Δ=0 → double root; Δ<0 → no real roots.
Frequently asked questions
What is an imaginary number?
i = √(−1). Complex numbers have a real part and an imaginary part: a + bi.
Does this page replace official or professional review?
No. It helps explain the scenario and use the tool more safely, but real decisions should consider official sources, full context and qualified guidance when needed.