Polynomials

Roots of Polynomials

quad(a, b, c)

Computes the real roots of the quadratic equation

\[ax^2 + bx + c\]

>>> quad(1, -7, 12)
= (x1 = 3, x2 = 4)
>>> quad(1, 7, 11)
= ()

quadc(a, b, c)

Computes the complex roots of the quadratic equation

\[az^2 + bz + c\]

>>> quadc(1, -7, 12)
= (z1 = 3, z2 = 4)
>>> quadc(1, 7, 11)
= (z1 = -4.618033988749895, z2 = -2.381966011250105)

cubic(a, b, c, d)

Computes the real roots of the cubic equation

\[ax^3 + bx^2 + cx + d\]

>>> cubic(1, 4, 7, 11)
= (x1 = -2.894558248242799)
>>> cubic(1, -3, -7, 12)
= (x1 = -2.302775637731995, x2 = 1.302775637731995, x3 = 4)

cubicc(a, b, c)

Computes the complex roots of the cubic equation

\[az^3 + bz^2 + cz + d\]

>>> cubicc(1, 4, 7, 11)
= (z1 = -2.894558248242799, z2 = -0.5527208758786003-1.869420790744347i, z3 = -0.5527208758786003+1.869420790744347i)
>>> cubic(1, -3, -7, 12)
= (z1 = -2.302775637731995, z2 = 1.302775637731995, z3 = 4)