Use the intersect method to solve the equation. 14x^3-53x^2+41x-4=-4x^3-x^2+1x+4

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:

Answer: 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