Answer:
x = ((18 sqrt(755833) - 17050)^(1/3) - (284 (-1)^(2/3))/(8525 - 9 sqrt(755833))^(1/3))/(15 2^(2/3)) + 1/3 or x = 1/3 + 142/15 ((-2)/(8525 - 9 sqrt(755833)))^(1/3) - 1/15 ((-1)/2)^(1/3) (9 sqrt(755833) - 8525)^(1/3) or x = 1/3 - (2^(1/3) (8525 - 9 sqrt(755833))^(2/3) + 284)/(15 2^(2/3) (8525 - 9 sqrt(755833))^(1/3))
Step-by-step explanation:
Solve for x over the real numbers:
1.56 = ((x + 0) (x + 0) (1 - x))/(2 - x)
1.56 = 39/25 and ((x + 0) (x + 0) (1 - x))/(2 - x) = (x^2 (1 - x))/(2 - x):
39/25 = (x^2 (1 - x))/(2 - x)
39/25 = ((1 - x) x^2)/(2 - x) is equivalent to ((1 - x) x^2)/(2 - x) = 39/25:
(x^2 (1 - x))/(2 - x) = 39/25
Cross multiply:
25 x^2 (1 - x) = 39 (2 - x)
Expand out terms of the left hand side:
25 x^2 - 25 x^3 = 39 (2 - x)
Expand out terms of the right hand side:
25 x^2 - 25 x^3 = 78 - 39 x
Subtract 78 - 39 x from both sides:
-25 x^3 + 25 x^2 + 39 x - 78 = 0
Multiply both sides by -1:
25 x^3 - 25 x^2 - 39 x + 78 = 0
Eliminate the quadratic term by substituting y = x - 1/3:
78 - 39 (y + 1/3) - 25 (y + 1/3)^2 + 25 (y + 1/3)^3 = 0
Expand out terms of the left hand side:
25 y^3 - (142 y)/3 + 1705/27 = 0
Divide both sides by 25:
y^3 - (142 y)/75 + 341/135 = 0
Change coordinates by substituting y = z + λ/z, where λ is a constant value that will be determined later:
341/135 - 142/75 (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 - 142/75) + (341 z^3)/135 + z^2 (3 λ^2 - (142 λ)/75) + λ^3 = 0
Substitute λ = 142/225 and then u = z^3, yielding a quadratic equation in the variable u:
u^2 + (341 u)/135 + 2863288/11390625 = 0
Find the positive solution to the quadratic equation:
u = (9 sqrt(755833) - 8525)/6750
Substitute back for u = z^3:
z^3 = (9 sqrt(755833) - 8525)/6750
Taking cube roots gives (9 sqrt(755833) - 8525)^(1/3)/(15 2^(1/3)) times the third roots of unity:
z = (9 sqrt(755833) - 8525)^(1/3)/(15 2^(1/3)) or z = -1/15 (-1/2)^(1/3) (9 sqrt(755833) - 8525)^(1/3) or z = ((-1)^(2/3) (9 sqrt(755833) - 8525)^(1/3))/(15 2^(1/3))
Substitute each value of z into y = z + 142/(225 z):
y = 1/15 ((9 sqrt(755833) - 8525)/2)^(1/3) - 142/15 (-1)^(2/3) (2/(8525 - 9 sqrt(755833)))^(1/3) or y = 142/15 ((-2)/(8525 - 9 sqrt(755833)))^(1/3) - 1/15 ((-1)/2)^(1/3) (9 sqrt(755833) - 8525)^(1/3) or y = 1/15 (-1)^(2/3) ((9 sqrt(755833) - 8525)/2)^(1/3) - 142/15 (2/(8525 - 9 sqrt(755833)))^(1/3)
Bring each solution to a common denominator and simplify:
y = ((18 sqrt(755833) - 17050)^(1/3) - (284 (-1)^(2/3))/(8525 - 9 sqrt(755833))^(1/3))/(15 2^(2/3)) or y = 142/15 ((-2)/(8525 - 9 sqrt(755833)))^(1/3) - 1/15 ((-1)/2)^(1/3) (9 sqrt(755833) - 8525)^(1/3) or y = -(2^(1/3) (8525 - 9 sqrt(755833))^(2/3) + 284)/(15 2^(2/3) (8525 - 9 sqrt(755833))^(1/3))
Substitute back for x = y + 1/3:
Answer: x = ((18 sqrt(755833) - 17050)^(1/3) - (284 (-1)^(2/3))/(8525 - 9 sqrt(755833))^(1/3))/(15 2^(2/3)) + 1/3 or x = 1/3 + 142/15 ((-2)/(8525 - 9 sqrt(755833)))^(1/3) - 1/15 ((-1)/2)^(1/3) (9 sqrt(755833) - 8525)^(1/3) or x = 1/3 - (2^(1/3) (8525 - 9 sqrt(755833))^(2/3) + 284)/(15 2^(2/3) (8525 - 9 sqrt(755833))^(1/3))
