x = ((-5)^(1/3) (2140 - 9 sqrt(56235))^(2/3) - 17 (-5)^(2/3))/(15 (2140 - 9 sqrt(56235))^(1/3)) - 1/3 or x = 1/15 (17 5^(2/3) (-1/(2140 - 9 sqrt(56235)))^(1/3) - (-5)^(1/3) (9 sqrt(56235) - 2140)^(1/3)) - 1/3 or x = -1/3 - 17/(3 (10700 - 45 sqrt(56235))^(1/3)) - (2140 - 9 sqrt(56235))^(1/3)/(3 5^(2/3))
Step-by-step explanation:
Solve for x:
6/x^2 + (2 x - 8)/(x + 5) = 2/x - 3
Bring 6/x^2 + (2 x - 8)/(x + 5) together using the common denominator x^2 (x + 5). Bring 2/x - 3 together using the common denominator x:
(2 (x^3 - 4 x^2 + 3 x + 15))/(x^2 (x + 5)) = (2 - 3 x)/x
Cross multiply:
2 x (x^3 - 4 x^2 + 3 x + 15) = x^2 (2 - 3 x) (x + 5)
Expand out terms of the left hand side:
2 x^4 - 8 x^3 + 6 x^2 + 30 x = x^2 (2 - 3 x) (x + 5)
Expand out terms of the right hand side:
2 x^4 - 8 x^3 + 6 x^2 + 30 x = -3 x^4 - 13 x^3 + 10 x^2
Subtract -3 x^4 - 13 x^3 + 10 x^2 from both sides:
5 x^4 + 5 x^3 - 4 x^2 + 30 x = 0
Factor x from the left hand side:
x (5 x^3 + 5 x^2 - 4 x + 30) = 0
Split into two equations:
x = 0 or 5 x^3 + 5 x^2 - 4 x + 30 = 0
Eliminate the quadratic term by substituting y = x + 1/3:
x = 0 or 30 - 4 (y - 1/3) + 5 (y - 1/3)^2 + 5 (y - 1/3)^3 = 0
Expand out terms of the left hand side:
x = 0 or 5 y^3 - (17 y)/3 + 856/27 = 0
Divide both sides by 5:
x = 0 or y^3 - (17 y)/15 + 856/135 = 0
Change coordinates by substituting y = z + λ/z, where λ is a constant value that will be determined later:
x = 0 or 856/135 - 17/15 (z + λ/z) + (z + λ/z)^3 = 0
Multiply both sides by z^3 and collect in terms of z:
x = 0 or z^6 + z^4 (3 λ - 17/15) + (856 z^3)/135 + z^2 (3 λ^2 - (17 λ)/15) + λ^3 = 0
Substitute λ = 17/45 and then u = z^3, yielding a quadratic equation in the variable u:
x = 0 or u^2 + (856 u)/135 + 4913/91125 = 0
Find the positive solution to the quadratic equation:
x = 0 or u = 1/675 (9 sqrt(56235) - 2140)
Substitute back for u = z^3:
x = 0 or z^3 = 1/675 (9 sqrt(56235) - 2140)
Taking cube roots gives (9 sqrt(56235) - 2140)^(1/3)/(3 5^(2/3)) times the third roots of unity:
x = 0 or z = (9 sqrt(56235) - 2140)^(1/3)/(3 5^(2/3)) or z = -((-1)^(1/3) (9 sqrt(56235) - 2140)^(1/3))/(3 5^(2/3)) or z = ((-1)^(2/3) (9 sqrt(56235) - 2140)^(1/3))/(3 5^(2/3))
Substitute each value of z into y = z + 17/(45 z):
x = 0 or y = (9 sqrt(56235) - 2140)^(1/3)/(3 5^(2/3)) - (17 (-1)^(2/3))/(3 (5 (2140 - 9 sqrt(56235)))^(1/3)) or y = 17/3 ((-1)/(5 (2140 - 9 sqrt(56235^(1/3) - ((-1)^(1/3) (9 sqrt(56235) - 2140)^(1/3))/(3 5^(2/3)) or y = ((-1)^(2/3) (9 sqrt(56235) - 2140)^(1/3))/(3 5^(2/3)) - 17/(3 (5 (2140 - 9 sqrt(56235)))^(1/3))
Bring each solution to a common denominator and simplify:
x = 0 or y = ((-5)^(1/3) (2140 - 9 sqrt(56235))^(2/3) - 17 (-5)^(2/3))/(15 (2140 - 9 sqrt(56235))^(1/3)) or y = 1/15 (17 5^(2/3) ((-1)/(2140 - 9 sqrt(56235)))^(1/3) - (-5)^(1/3) (9 sqrt(56235) - 2140)^(1/3)) or y = -(2140 - 9 sqrt(56235))^(1/3)/(3 5^(2/3)) - 17/(3 (5 (2140 - 9 sqrt(56235)))^(1/3))
Substitute back for x = y - 1/3:
x = 0 or x = 1/15 (2140 - 9 sqrt(56235))^(-1/3) ((-5)^(1/3) (2140 - 9 sqrt(56235))^(2/3) - 17 (-5)^(2/3)) - 1/3 or x = 1/15 (17 5^(2/3) (-1/(2140 - 9 sqrt(56235)))^(1/3) - (-5)^(1/3) (9 sqrt(56235) - 2140)^(1/3)) - 1/3 or x = -1/3 - 1/3 5^(-2/3) (2140 - 9 sqrt(56235))^(1/3) - 17/3 (5 (2140 - 9 sqrt(56235)))^(-1/3)
5 (2140 - 9 sqrt(56235)) = 10700 - 45 sqrt(56235):
x = 0 or x = ((-5)^(1/3) (2140 - 9 sqrt(56235))^(2/3) - 17 (-5)^(2/3))/(15 (2140 - 9 sqrt(56235))^(1/3)) - 1/3 or x = 1/15 (17 5^(2/3) (-1/(2140 - 9 sqrt(56235)))^(1/3) - (-5)^(1/3) (9 sqrt(56235) - 2140)^(1/3)) - 1/3 or x = -1/3 - (2140 - 9 sqrt(56235))^(1/3)/(3 5^(2/3)) - 17/(3 (10700 - 45 sqrt(56235))^(1/3))
6/x^2 + (2 x - 8)/(x + 5) ⇒ 6/0^2 + (2 0 - 8)/(5 + 0) = ∞^~
2/x - 3 ⇒ 2/0 - 3 = ∞^~:
So this solution is incorrect
6/x^2 + (2 x - 8)/(x + 5) ≈ -3.83766
2/x - 3 ≈ -3.83766:
So this solution is correct
6/x^2 + (2 x - 8)/(x + 5) ≈ -2.44783 + 1.13439 i
2/x - 3 ≈ -2.44783 + 1.13439 i:
So this solution is correct
6/x^2 + (2 x - 8)/(x + 5) ≈ -2.44783 - 1.13439 i
2/x - 3 ≈ -2.44783 - 1.13439 i:
So this solution is correct
The solutions are:
 x = ((-5)^(1/3) (2140 - 9 sqrt(56235))^(2/3) - 17 (-5)^(2/3))/(15 (2140 - 9 sqrt(56235))^(1/3)) - 1/3 or x = 1/15 (17 5^(2/3) (-1/(2140 - 9 sqrt(56235)))^(1/3) - (-5)^(1/3) (9 sqrt(56235) - 2140)^(1/3)) - 1/3 or x = -1/3 - 17/(3 (10700 - 45 sqrt(56235))^(1/3)) - (2140 - 9 sqrt(56235))^(1/3)/(3 5^(2/3))
