x = (2^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) - 10 3^(1/3))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)) or x = -((-1)^(1/3) (2^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) + 10 (-3)^(1/3)))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)) or x = ((-1)^(1/3) ((-2)^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) + 10 3^(1/3)))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))
Step-by-step explanation:
Solve for x:
y = x^3 + 5 x + 1
y = x^3 + 5 x + 1 is equivalent to x^3 + 5 x + 1 = y:
x^3 + 5 x + 1 = y
Subtract y from both sides:
1 + 5 x + x^3 - y = 0
Change coordinates by substituting x = z + λ/z, where λ is a constant value that will be determined later:
1 - y + 5 (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 + 5) + z^3 (1 - y) + z^2 (3 λ^2 + 5 λ) + λ^3 = 0
Substitute λ = -5/3 and then u = z^3, yielding a quadratic equation in the variable u:
-125/27 + u^2 - u (y - 1) = 0
Find the positive solution to the quadratic equation:
u = 1/18 (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))
Substitute back for u = z^3:
z^3 = 1/18 (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))
Taking cube roots gives (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)/(2^(1/3) 3^(2/3)) times the third roots of unity:
z = (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)/(2^(1/3) 3^(2/3)) or z = -((-1/2)^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))/3^(2/3) or z = ((-1)^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))/(2^(1/3) 3^(2/3))
Substitute each value of z into x = z - 5/(3 z):
x = (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)/(2^(1/3) 3^(2/3)) - (5 (2/3)^(1/3))/(-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3) or x = -(5 (-1)^(2/3) (2/3)^(1/3))/(-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3) - (((-1)/2)^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))/3^(2/3) or x = (5 ((-2)/3)^(1/3))/(-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3) + ((-1)^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))/(2^(1/3) 3^(2/3))
Bring each solution to a common denominator and simplify:
 x = (2^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) - 10 3^(1/3))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)) or x = -((-1)^(1/3) (2^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) + 10 (-3)^(1/3)))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)) or x = ((-1)^(1/3) ((-2)^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) + 10 3^(1/3)))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))
