x = (68 2^(1/3) + (27 i sqrt(591) + 445)^(2/3))/(27 (1/2 (27 i sqrt(591) + 445))^(1/3)) + 26/27 or x = (68 (-2)^(2/3) - (-2)^(1/3) (27 i sqrt(591) + 445)^(2/3))/(27 (27 i sqrt(591) + 445)^(1/3)) + 26/27 or x = 1/27 ((-2)/(27 i sqrt(591) + 445))^(1/3) ((-1)^(1/3) (27 i sqrt(591) + 445)^(2/3) - 68 2^(1/3)) + 26/27
Step-by-step explanation:
Solve for x over the real numbers:
14 x^3 - 53 x^2 + 41 x - 4 = -4 x^3 - x^2 + x + 4
Subtract -4 x^3 - x^2 + x + 4 from both sides:
18 x^3 - 52 x^2 + 40 x - 8 = 0
Factor constant terms from the left hand side:
2 (9 x^3 - 26 x^2 + 20 x - 4) = 0
Divide both sides by 2:
9 x^3 - 26 x^2 + 20 x - 4 = 0
Eliminate the quadratic term by substituting y = x - 26/27:
-4 + 20 (y + 26/27) - 26 (y + 26/27)^2 + 9 (y + 26/27)^3 = 0
Expand out terms of the left hand side:
9 y^3 - (136 y)/27 - 1780/2187 = 0
Divide both sides by 9:
y^3 - (136 y)/243 - 1780/19683 = 0
Change coordinates by substituting y = z + λ/z, where λ is a constant value that will be determined later:
-1780/19683 - 136/243 (z + λ/z) + (z + λ/z)^3 = 0
Multiply both sides by z^3 and collect in terms of z:
z^6 + z^4 (3 λ - 136/243) - (1780 z^3)/19683 + z^2 (3 λ^2 - (136 λ)/243) + λ^3 = 0
Substitute λ = 136/729 and then u = z^3, yielding a quadratic equation in the variable u:
u^2 - (1780 u)/19683 + 2515456/387420489 = 0
Find the positive solution to the quadratic equation:
u = (2 (445 + 27 i sqrt(591)))/19683
Substitute back for u = z^3:
z^3 = (2 (445 + 27 i sqrt(591)))/19683
Taking cube roots gives 1/27 2^(1/3) (445 + 27 i sqrt(591))^(1/3) times the third roots of unity:
z = 1/27 2^(1/3) (445 + 27 i sqrt(591))^(1/3) or z = -1/27 (-2)^(1/3) (445 + 27 i sqrt(591))^(1/3) or z = 1/27 (-1)^(2/3) 2^(1/3) (445 + 27 i sqrt(591))^(1/3)
Substitute each value of z into y = z + 136/(729 z):
y = (68 2^(2/3))/(27 (27 i sqrt(591) + 445)^(1/3)) + 1/27 (2 (27 i sqrt(591) + 445))^(1/3) or y = (68 (-2)^(2/3))/(27 (27 i sqrt(591) + 445)^(1/3)) - 1/27 (-2)^(1/3) (27 i sqrt(591) + 445)^(1/3) or y = 1/27 (-1)^(2/3) (2 (27 i sqrt(591) + 445))^(1/3) - (68 (-1)^(1/3) 2^(2/3))/(27 (27 i sqrt(591) + 445)^(1/3))
Bring each solution to a common denominator and simplify:
y = (2^(1/3) ((27 i sqrt(591) + 445)^(2/3) + 68 2^(1/3)))/(27 (445 + 27 i sqrt(591))^(1/3)) or y = (68 (-2)^(2/3) - (-2)^(1/3) (27 i sqrt(591) + 445)^(2/3))/(27 (445 + 27 i sqrt(591))^(1/3)) or y = 1/27 2^(1/3) (-1/(445 + 27 i sqrt(591)))^(1/3) ((-1)^(1/3) (27 i sqrt(591) + 445)^(2/3) - 68 2^(1/3))
Substitute back for x = y + 26/27:
x = (68 2^(1/3) + (27 i sqrt(591) + 445)^(2/3))/(27 (1/2 (27 i sqrt(591) + 445))^(1/3)) + 26/27 or x = (68 (-2)^(2/3) - (-2)^(1/3) (27 i sqrt(591) + 445)^(2/3))/(27 (27 i sqrt(591) + 445)^(1/3)) + 26/27 or x = 1/27 ((-2)/(27 i sqrt(591) + 445))^(1/3) ((-1)^(1/3) (27 i sqrt(591) + 445)^(2/3) - 68 2^(1/3)) + 26/27
