x = 3/4 or x = ((-1)^(1/3) (127 - 18 sqrt(43))^(2/3) - 13 (-1)^(2/3))/(3 (127 - 18 sqrt(43))^(1/3)) - 4/3 or x = 1/3 (13 (-1/(127 - 18 sqrt(43)))^(1/3) - (-1)^(1/3) (18 sqrt(43) - 127)^(1/3)) - 4/3 or x = (-(127 - 18 sqrt(43))^(2/3) - 13)/(3 (127 - 18 sqrt(43))^(1/3)) - 4/3
Explanation:
Solve for x over the real numbers:
4 x^4 + 13 x^3 - 8 x^2 + 21 x - 18 = 0
The left hand side factors into a product with two terms:
(4 x - 3) (x^3 + 4 x^2 + x + 6) = 0
Split into two equations:
4 x - 3 = 0 or x^3 + 4 x^2 + x + 6 = 0
Add 3 to both sides:
4 x = 3 or x^3 + 4 x^2 + x + 6 = 0
Divide both sides by 4:
x = 3/4 or x^3 + 4 x^2 + x + 6 = 0
Eliminate the quadratic term by substituting y = x + 4/3:
x = 3/4 or 14/3 + 4 (y - 4/3)^2 + (y - 4/3)^3 + y = 0
Expand out terms of the left hand side:
x = 3/4 or y^3 - (13 y)/3 + 254/27 = 0
Change coordinates by substituting y = z + λ/z, where λ is a constant value that will be determined later:
x = 3/4 or 254/27 - 13/3 (z + λ/z) + (z + λ/z)^3 = 0
Multiply both sides by z^3 and collect in terms of z:
x = 3/4 or z^6 + z^4 (3 λ - 13/3) + (254 z^3)/27 + z^2 (3 λ^2 - (13 λ)/3) + λ^3 = 0
Substitute λ = 13/9 and then u = z^3, yielding a quadratic equation in the variable u:
x = 3/4 or u^2 + (254 u)/27 + 2197/729 = 0
Find the positive solution to the quadratic equation:
x = 3/4 or u = 1/27 (18 sqrt(43) - 127)
Substitute back for u = z^3:
x = 3/4 or z^3 = 1/27 (18 sqrt(43) - 127)
Taking cube roots gives 1/3 (18 sqrt(43) - 127)^(1/3) times the third roots of unity:
x = 3/4 or z = 1/3 (18 sqrt(43) - 127)^(1/3) or z = -1/3 (-1)^(1/3) (18 sqrt(43) - 127)^(1/3) or z = 1/3 (-1)^(2/3) (18 sqrt(43) - 127)^(1/3)
Substitute each value of z into y = z + 13/(9 z):
x = 3/4 or y = 1/3 (18 sqrt(43) - 127)^(1/3) - (13 (-1)^(2/3))/(3 (127 - 18 sqrt(43))^(1/3)) or y = 13/3 ((-1)/(127 - 18 sqrt(43)))^(1/3) - 1/3 (-1)^(1/3) (18 sqrt(43) - 127)^(1/3) or y = 1/3 (-1)^(2/3) (18 sqrt(43) - 127)^(1/3) - 13/(3 (127 - 18 sqrt(43))^(1/3))
Bring each solution to a common denominator and simplify:
x = 3/4 or y = ((-1)^(1/3) (127 - 18 sqrt(43))^(2/3) - 13 (-1)^(2/3))/(3 (127 - 18 sqrt(43))^(1/3)) or y = 1/3 (13 ((-1)/(127 - 18 sqrt(43)))^(1/3) - (-1)^(1/3) (18 sqrt(43) - 127)^(1/3)) or y = (-(127 - 18 sqrt(43))^(2/3) - 13)/(3 (127 - 18 sqrt(43))^(1/3))
Substitute back for x = y - 4/3:
x = 3/4 or x = ((-1)^(1/3) (127 - 18 sqrt(43))^(2/3) - 13 (-1)^(2/3))/(3 (127 - 18 sqrt(43))^(1/3)) - 4/3 or x = 1/3 (13 (-1/(127 - 18 sqrt(43)))^(1/3) - (-1)^(1/3) (18 sqrt(43) - 127)^(1/3)) - 4/3 or x = (-(127 - 18 sqrt(43))^(2/3) - 13)/(3 (127 - 18 sqrt(43))^(1/3)) - 4/3
