Cubic and Quartic 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.

Quartic 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.

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