Formulae to Solve Polynomial Equations


The Linear Formula

x = (-b)/a

The linear formula gives the solution of a*x+b=0 for real numbers a, b with a!=0.


The Quadratic Formula

x = (-b+s*sqrt(b^2-4*a*c))/(2*a)

The linear formula gives the solutions of a*x^2+b*x+c=0 for real numbers a, b, c with a!=0.

Directions: Take s=-1,1.


The Cubic Formula

x = (-2*b+((-1+sqrt(-3))/2)^n*cbrt(4*(-2*b^3+9*a*b*c-27*a^2*d+sqrt((-2*b^3+9*a*b*c-27*a^2*d)^2-4*(b^2-3*a*c)^3)))+((-1-sqrt(-3))/2)^n*cbrt(4*(-2*b^3+9*a*b*c-27*a^2*d-sqrt((-2*b^3+9*a*b*c-27*a^2*d)^2-4*(b^2-3*a*c)^3))))/(6*a)

The cubic formula gives the solutions of a*x^3+b*x^2+c*x+d=0 for real numbers a, b, c, d with a!=0.

Directions: Take n=0,1,2.  Use real cube roots if possible, and principal roots otherwise.


The Quartic Formula

x = (-3*b+s*(sqrt(3*(3*b^2-8*a*c+2*a*cbrt(4*(2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e+sqrt((2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e)^2-4*(c^2-3*b*d+12*a*e)^3)))+2*a*cbrt(4*(2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e-sqrt((2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e)^2-4*(c^2-3*b*d+12*a*e)^3)))))+t*sqrt(3*(3*b^2-8*a*c+2*a*(-1+sqrt(-3))/2*cbrt(4*(2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e+sqrt((2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e)^2-4*(c^2-3*b*d+12*a*e)^3)))+2*a*(-1-sqrt(-3))/2*cbrt(4*(2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e-sqrt((2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e)^2-4*(c^2-3*b*d+12*a*e)^3))))))+t*sgn((sgn(-b^3+4*a*b*c-8*a^2*d)-1/2)*(sgn(max((2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e)^2-4*(c^2-3*b*d+12*a*e)^3,min(3*b^2-8*a*c,3*b^4+16*a^2*c^2+16*a^2*b*d-16*a*b^2*c-64*a^3*e)))-1/2))*sqrt(3*(3*b^2-8*a*c+2*a*(-1-sqrt(-3))/2*cbrt(4*(2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e+sqrt((2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e)^2-4*(c^2-3*b*d+12*a*e)^3)))+2*a*(-1+sqrt(-3))/2*cbrt(4*(2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e-sqrt((2*c^3-9*b*c*d+27*a*d^2+27*b^2*e-72*a*c*e)^2-4*(c^2-3*b*d+12*a*e)^3))))))/(12*a)

The quartic formula gives the solutions of a*x^4+b*x^3+c*x^2+d*x+e=0 for real numbers a, b, c, d, e with a!=0.

Directions: Take s=-1,1 and t=-1,1.  Use real cube roots if possible, and principal roots otherwise.