Cubic and Quatric Equations

Cubic Equation x³ + a₁x² + a₂x + a₃ = 0

Let Q = (3a₂ - a₁²)/9, R = (9a₁a₂ - 27a₃ - 2a₁³)/54, S = , T =

Solutions:

If a₁, a₂, a₃ are real and if D = Q³ + R² is the discriminant, then
(i)     one root is real and two complex conjugate if D > 0
(ii)     all roots are real and at least two are equal if D = 0
(iii)     all roots are real and unequal if D < 0.
If D < 0, computation is simplyfied by use of trigonometry.

Solutions if D < 0 :

where cosθ = -R / √ -Q³

x₁ + x₂ + x₃ = -a₁, x₁x₂ + x₂x₃ + x₃x₁ = a₂, x₁x₂x₃ = - a₃
where x₁, x₂ , x₃ are the three roots.

Quatric equation x⁴ + a₁x³ + a₂x² + a₃x + a₄ = 0.

Let y₁ be a real root of the cubic equation
(1) y³ - a₂y² + (a₁a₃ - 4a₄)y + (4a₂a₄ - a₃² - a₁²a₄) = 0.

Solutions: The roots of z² + 1/2(a₁ ± √a₁² - 4a₂ + 4y₁)z + 1/2(y₁ ± √y₁² - 4a₄) = 0.

If all roots of (1) are real, computation is simplified by using that particular real root which produces all real coefficients in the quadratic equation.

where x₁,x₂,x₃,x₄ are the four roots.