Question

Condition of complex roots of a cubic equation

Answer 1

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Answer 2

Nature of Roots of A Cubic Equation:

As, we calculate the discriminant of quadratic, we also calculate the discriminant of the cubics.

$\triangle = - 27 {a}^{2} {d}^{2} + 18bcd - 4 a{c}^{3} - 4 {b}^{3} d + {b}^{2} {c}^{2}$

Here, Δ is the cubic discriminant for :

$f(x) = a {x}^{3} + b {x}^{2} + cx + d$

1. If, Δ > 0 ⇒x₁,₂,₃ ∈ ℝ

2. If, Δ = 0 ⇒x₁,₂,₃ ∈ ℝ

• Then, if ; bc = 9ad ⇒ x₁ = x₂ = x₃
• And, if ; bc ≠ 9ad ⇒x₁ ≠ (x₂ = x₃)

3. If, Δ < 0 ⇒x₁ ∈ ℝ and , x₂,₃ ∉ ℝ

Therefore, for two complex conjugate complex roots to exist for a cubic,

–27a²d²+18abcd–4b³d–4ac³+b²c² < 0