The general form of the 3rd degree equation (or Cubic) is: ax3 + bx2 + cx + d = 0
Cubics have 3 roots.
The 3 roots can be represented this way:
First root (of three):
Second root (of three):
Third root (of three):
The second and third formula are equal except for a "+ or -" sign at the
beginning, and another "+ or -" sign in the middle. Note that the second
and third formula contain the imaginary unit "i".
Now, the same three formulas in ASCII. The differences between the
second and third formula are highlighted here in yellow:
| x = -b/(3*a) -
(2^(1/3)*(-b^2 + 3*a*c))/(3*a*(-2*b^3 + 9*a*b*c - 27*a^2*d +
Sqrt[4*(-b^2 + 3*a*c)^3 + (-2*b^3 + 9*a*b*c - 27*a^2*d)^2])^(1/3)) +
(-2*b^3 + 9*a*b*c - 27*a^2*d + Sqrt[4*(-b^2 + 3*a*c)^3 + (-2*b^3 +
9*a*b*c - 27*a^2*d)^2])^(1/3)/(3*2^(1/3)*a) x = -b/(3*a) + ((1 + i*Sqrt[3])*(-b^2 + 3*a*c))/(3*2^(2/3)*a*(-2*b^3 + 9*a*b*c - 27*a^2*d + Sqrt[4*(-b^2 + 3*a*c)^3 + (-2*b^3 + 9*a*b*c - 27*a^2*d)^2])^(1/3)) - (1 - i*Sqrt[3])*(-2*b^3 + 9*a*b*c - 27*a^2*d + Sqrt[4*(-b^2 + 3*a*c)^3 + (-2*b^3 + 9*a*b*c - 27*a^2*d)^2])^(1/3)/(6*2^(1/3)*a) x = -b/(3*a) + ((1 - i*Sqrt[3])*(-b^2 + 3*a*c))/(3*2^(2/3)*a*(-2*b^3 + 9*a*b*c - 27*a^2*d + Sqrt[4*(-b^2 + 3*a*c)^3 + (-2*b^3 + 9*a*b*c - 27*a^2*d)^2])^(1/3)) - (1 + i*Sqrt[3])*(-2*b^3 + 9*a*b*c - 27*a^2*d + Sqrt[4*(-b^2 + 3*a*c)^3 + (-2*b^3 + 9*a*b*c - 27*a^2*d)^2])^(1/3)/(6*2^(1/3)*a) |
The formulas on top were obtained with Wolfram's
program Mathematica. There seemed not to be an option, so I copied
them by hand into ASCII.
josechu2004@gmail.com