Solve for x:
1/(x - 4) + x^2/(x - 1) - x/(x + 2) = 0
Bring 1/(x - 4) + x^2/(x - 1) - x/(x + 2) together using the common denominator (x - 4) (x - 1) (x + 2):
(x^4 - 3 x^3 - 2 x^2 - 3 x - 2)/((x - 4) (x - 1) (x + 2)) = 0
Multiply both sides by (x - 4) (x - 1) (x + 2):
x^4 - 3 x^3 - 2 x^2 - 3 x - 2 = 0
Eliminate the cubic term by substituting y = x - 3/4:
-2 - 3 (y + 3/4) - 2 (y + 3/4)^2 - 3 (y + 3/4)^3 + (y + 3/4)^4 = 0
Expand out terms of the left hand side:
y^4 - (43 y^2)/8 - (75 y)/8 - 1619/256 = 0
Subtract -1/8 i sqrt(1619) y^2 - (43 y^2)/8 - (75 y)/8 from both sides:
y^4 + 1/8 i sqrt(1619) y^2 - 1619/256 = 1/8 i sqrt(1619) y^2 + (43 y^2)/8 + (75 y)/8
y^4 + 1/8 i sqrt(1619) y^2 - 1619/256 = (y^2 + (i sqrt(1619))/16)^2:
(y^2 + (i sqrt(1619))/16)^2 = 1/8 i sqrt(1619) y^2 + (43 y^2)/8 + (75 y)/8
Add 2 (y^2 + (i sqrt(1619))/16) λ + λ^2 to both sides:
(y^2 + (i sqrt(1619))/16)^2 + 2 λ (y^2 + (i sqrt(1619))/16) + λ^2 = (75 y)/8 + 1/8 i sqrt(1619) y^2 + (43 y^2)/8 + 2 λ (y^2 + (i sqrt(1619))/16) + λ^2
(y^2 + (i sqrt(1619))/16)^2 + 2 λ (y^2 + (i sqrt(1619))/16) + λ^2 = (y^2 + (i sqrt(1619))/16 + λ)^2:
(y^2 + (i sqrt(1619))/16 + λ)^2 = (75 y)/8 + 1/8 i sqrt(1619) y^2 + (43 y^2)/8 + 2 λ (y^2 + (i sqrt(1619))/16) + λ^2
(75 y)/8 + 1/8 i sqrt(1619) y^2 + (43 y^2)/8 + 2 λ (y^2 + (i sqrt(1619))/16) + λ^2 = (2 λ + 43/8 + (i sqrt(1619))/8) y^2 + (75 y)/8 + 1/8 i sqrt(1619) λ + λ^2:
(y^2 + (i sqrt(1619))/16 + λ)^2 = y^2 (2 λ + 43/8 + (i sqrt(1619))/8) + (75 y)/8 + 1/8 i sqrt(1619) λ + λ^2
Complete the square on the right hand side:
(y^2 + (i sqrt(1619))/16 + λ)^2 = (y sqrt(2 λ + 43/8 + (i sqrt(1619))/8) + 75/(16 sqrt(2 λ + 43/8 + (i sqrt(1619))/8)))^2 + (4 (2 λ + 43/8 + (i sqrt(1619))/8) (λ^2 + 1/8 i sqrt(1619) λ) - 5625/64)/(4 (2 λ + 43/8 + (i sqrt(1619))/8))
To express the right hand side as a square, find a value of λ such that the last term is 0.
This means 4 (2 λ + 43/8 + (i sqrt(1619))/8) (λ^2 + 1/8 i sqrt(1619) λ) - 5625/64 = 1/64 (512 λ^3 + 96 i sqrt(1619) λ^2 + 1376 λ^2 + 172 i sqrt(1619) λ - 6476 λ - 5625) = 0.
Thus the root λ = -1/48 i (-43 i + 3 sqrt(1619) + 376 i (2/(27 sqrt(773) + 385))^(1/3) - 4 i 2^(2/3) (27 sqrt(773) + 385)^(1/3)) allows the right hand side to be expressed as a square.
(This value will be substituted later):
(y^2 + (i sqrt(1619))/16 + λ)^2 = (y sqrt(2 λ + 43/8 + (i sqrt(1619))/8) + 75/(16 sqrt(2 λ + 43/8 + (i sqrt(1619))/8)))^2
Take the square root of both sides:
y^2 + (i sqrt(1619))/16 + λ = y sqrt(2 λ + 43/8 + (i sqrt(1619))/8) + 75/(16 sqrt(2 λ + 43/8 + (i sqrt(1619))/8)) or y^2 + (i sqrt(1619))/16 + λ = -y sqrt(2 λ + 43/8 + (i sqrt(1619))/8) - 75/(16 sqrt(2 λ + 43/8 + (i sqrt(1619))/8))
Solve using the quadratic formula:
y = 1/8 (sqrt(2) sqrt(43 + i sqrt(1619) + 16 λ) + 4 sqrt(43/8 - i/8 sqrt(1619) - 2 λ + 75 1/sqrt(86 + 2 i sqrt(1619) + 32 λ))) or y = 1/8 (sqrt(2) sqrt(43 + i sqrt(1619) + 16 λ) - 4 sqrt(43/8 - i/8 sqrt(1619) - 2 λ + 75 1/sqrt(86 + 2 i sqrt(1619) + 32 λ))) or y = 1/8 (4 sqrt(43/8 - i/8 sqrt(1619) - 2 λ - 75 1/sqrt(86 + 2 i sqrt(1619) + 32 λ)) - sqrt(2) sqrt(43 + i sqrt(1619) + 16 λ)) or y = 1/8 (-sqrt(2) sqrt(43 + i sqrt(1619) + 16 λ) - 4 sqrt(43/8 - i/8 sqrt(1619) - 2 λ - 75 1/sqrt(86 + 2 i sqrt(1619) + 32 λ))) where λ = -1/48 i (-43 i + 3 sqrt(1619) + 376 i (2/(27 sqrt(773) + 385))^(1/3) - 4 i 2^(2/3) (27 sqrt(773) + 385)^(1/3))
Substitute λ = -1/48 i (-43 i + 3 sqrt(1619) + 376 i (2/(27 sqrt(773) + 385))^(1/3) - 4 i 2^(2/3) (27 sqrt(773) + 385)^(1/3)) and approximate:
y = -1.37923 or y = -0.823857 - 0.91438 i or y = -0.823857 + 0.91438 i or y = 3.02695
Substitute back for y = x - 3/4:
x - 3/4 = -1.37923 or y = -0.823857 - 0.91438 i or y = -0.823857 + 0.91438 i or y = 3.02695
Add 3/4 to both sides:
x = -0.629233 or y = -0.823857 - 0.91438 i or y = -0.823857 + 0.91438 i or y = 3.02695
Substitute back for y = x - 3/4:
x = -0.629233 or x - 3/4 = -0.823857 - 0.91438 i or y = -0.823857 + 0.91438 i or y = 3.02695
Add 3/4 to both sides:
x = -0.629233 or x = -0.0738574 - 0.91438 i or y = -0.823857 + 0.91438 i or y = 3.02695
Substitute back for y = x - 3/4:
x = -0.629233 or x = -0.0738574 - 0.91438 i or x - 3/4 = -0.823857 + 0.91438 i or y = 3.02695
Add 3/4 to both sides:
x = -0.629233 or x = -0.0738574 - 0.91438 i or x = -0.0738574 + 0.91438 i or y = 3.02695
Substitute back for y = x - 3/4:
x = -0.629233 or x = -0.0738574 - 0.91438 i or x = -0.0738574 + 0.91438 i or x - 3/4 = 3.02695
Add 3/4 to both sides:
x = -0.629233 or x = -0.0738574 - 0.91438 i or x = -0.0738574 + 0.91438 i or x = 3.77695
