Python Root Finder for Linear, Quadratic, Cubic, and Quartic Equations by James Pate Williams, Jr.

z1 = complex(0, 0)
z2 = complex(0, 0)
z3 = complex(0, 0)
z4 = complex(0, 0)

def complex_quadratic_formula(a, b, c):
    
    import cmath

    z1 = complex((-b + cmath.sqrt(b * b - 4 * a * c)) / (2 * a))
    z2 = complex((-b - cmath.sqrt(b * b - 4 * a * c)) / (2 * a))
    
    print("z1 = ", z1)
    print("z2 = ", z2)

def real_quadratic_formula(a, b, c):
    if b * b - 4 * a * c >= 0:
        
        import math
    
        z1 = float((-b + math.sqrt(b * b - 4 * a * c)) / (2 * a))
        z2 = float((-b - math.sqrt(b * b - 4 * a * c)) / (2 * a))

    else:
        
        import cmath

        z1 = complex((-b + cmath.sqrt(b * b - 4 * a * c)) / (2 * a))
        z2 = complex((-b - cmath.sqrt(b * b - 4 * a * c)) / (2 * a))
    
    print("z1 = ", z1)
    print("z2 = ", z2)

def cubic_equation(A, B, C, D):
    # https://en.wikipedia.org/wiki/Cubic_function
    # http://www.wolframalpha.com/widgets/view.jsp?id=3f4366aeb9c157cf9a30c90693eafc55

    import cmath
    import math

    A2 = float(A * A)
    B2 = float(B * B)
    B3 = float(B * B * B)
    C2 = float(C * C)
    C3 = float(C * C * C)
    D2 = float(D * D)
    Delta = float(18 * A * B * C * D - 4.0 * B3 * D
                + B2 * C2 - 4.0 * A * C3 - 27.0 * A2 * D2)
    Delta0 = float(B2 - 3.0 * A * C)
    Delta1 = float(2.0 * B3 - 9.0 * A * B * C + 27.0 * A2 * D)
    r = cmath.sqrt(-27.0 * A2 * Delta)
    c = complex(0.5 * (Delta1 + r)) ** (1.0 / 3.0)
    d = complex(0.5 * (Delta1 - r)) ** (1.0 / 3.0)

    if c.real == 0 and c.imag == 0:
        c = d

    i = complex(0.0, 1.0)
    chi = complex(-0.5, 0) + i * 0.5 * math.sqrt(3)

    import array

    ar = array.array('f', [0,0,0])
    ai = array.array('f', [0,0,0])

    for k in range(0, 3, 1):
        chik = chi ** k
        z = -(B + chik * c + Delta0 / (chik * c)) / (3.0 * A)
        ar[k] = z.real
        ai[k] = z.imag;

    z1 = ar[0] + ai[0] * complex(0, 1);
    z2 = ar[1] + ai[1] * complex(0, 1);
    z3 = ar[2] + ai[2] * complex(0, 1);
   
    print("z1 = ", z1)
    print("z2 = ", z2)
    print("z3 = ", z3)

def quartic_equation(A, B, C, D, E):
    # https://en.wikipedia.org/wiki/Quartic_function
    # https://keisan.casio.com/exec/system/1181809416

    import cmath

    A2 = float(A * A)
    A3 = float(A * A2)
    A4 = float(A2 * A2)
    B2 = float(B * B)
    B3 = float(B * B2)
    B4 = float(B2 * B2)
    C2 = float(C * C)
    C3 = float(C * C2)
    C4 = float(C2 * C2)
    D2 = float(D * D)
    D3 = float(D * D2)
    D4 = float(D2 * D2)
    E2 = float(E * E)
    E3 = float(E * E2)
    E4 = float(E2 * E2)
    Delta = float(256 * A3 * E3 - 192 * A2 * B * D * E2
                  - 128 * A2 * C2 * E2 + 144 * A2 * C * D2 * E
                  - 27 * A2 * D4 + 144 * A * B2 * C * E2
                  - 6 * A * B2 * D2 * E - 80 * A * B * C2 * D * E
                  + 18 * A * B * C * D3 + 16 * A * C4 * E
                  - 4 * A * C3 * D2 - 27 * B4 * E2 + 18 * B3 * C * D * E
                  - 4 * B3 * D3 - 4 * B2 * C3 * E + B2 * C2 * D2)
    Delta0 = float(C2 - 3 * B * D + 12 * A * E)
    Delta1 = float(2 * C3 - 9 * B * C * D
                   + 27 * B2 * E + 27 * A * D2 - 72 * A * C * E)
    p = float((8 * A * C - 3 * B2) / (8 * A2))
    q = float((B3 - 4 * A * B * C + 8 * A2 * D) / (8 * A3))
    R = cmath.sqrt(Delta1 * Delta1 - 4 * Delta0 * Delta0 * Delta0)
    P = cmath.sqrt(-27 * Delta)
    Q = complex(0.5 * (Delta1 + P)) ** complex(1.0 / 3.0)
    S = 0.5 * cmath.sqrt(((-2.0 * p) / 3.0)
                         + (Q + Delta0 / Q) / (3 * A))
    a = complex(-B / (4 * A), 0)
    s = 0.5 * cmath.sqrt(-4 * S * S - 2 * p + q / S)
    z1 = a - S + s
    z2 = a - S - s
    s = 0.5 * cmath.sqrt(-4 * S * S - 2 * p - q / S)
    z3 = a + S + s
    z4 = a + S - s

    print("z1 = ", z1)
    print("z2 = ", z2)
    print("z3 = ", z3)
    print("z4 = ", z4)

option = int(1)

print("Menu")
print("1 Solve a linear equation")
print("2 Solve a quadratic eqution")
print("3 Solve a cubic equation")
print("4 Solve a quartic equation")
print("5 Quit")

option = int(input("Option 1 to 5: "))

while option != 5:
    if option >= 1:
        a = float(input("a = "))
        b = float(input("b = "))
    if option >= 2:
        c = float(input("c = "))
    if option >= 3:
        d = float(input("d = "))
    if option == 4:
        e = float(input("e = "))

    if option == 1:
        z1 = -b / a
        print("z1 = ", z1)

    elif option == 2:
        real_quadratic_formula(a, b, c)

    elif option == 3:
        cubic_equation(a, b, c, d)

    elif option == 4:
        quartic_equation(a, b, c, d, e)

    option = int(input("Option 1 to 5: "))