Answer:
x = -2.98079 or x = -1.15272 or x = 0.892002 or x = 4.24151
Step-by-step explanation:
Solve for x:
-x^2 + x + 14 + 2/x - 13/x^2 = 0
Bring -x^2 + x + 14 + 2/x - 13/x^2 together using the common denominator x^2:
(-x^4 + x^3 + 14 x^2 + 2 x - 13)/x^2 = 0
Multiply both sides by x^2:
-x^4 + x^3 + 14 x^2 + 2 x - 13 = 0
Multiply both sides by -1:
x^4 - x^3 - 14 x^2 - 2 x + 13 = 0
Eliminate the cubic term by substituting y = x - 1/4:
13 - 2 (y + 1/4) - 14 (y + 1/4)^2 - (y + 1/4)^3 + (y + 1/4)^4 = 0
Expand out terms of the left hand side:
y^4 - (115 y^2)/8 - (73 y)/8 + 2973/256 = 0
Add (sqrt(2973) y^2)/8 + (115 y^2)/8 + (73 y)/8 to both sides:
y^4 + (sqrt(2973) y^2)/8 + 2973/256 = (sqrt(2973) y^2)/8 + (115 y^2)/8 + (73 y)/8
y^4 + (sqrt(2973) y^2)/8 + 2973/256 = (y^2 + sqrt(2973)/16)^2:
(y^2 + sqrt(2973)/16)^2 = (sqrt(2973) y^2)/8 + (115 y^2)/8 + (73 y)/8
Add 2 (y^2 + sqrt(2973)/16) λ + λ^2 to both sides:
(y^2 + sqrt(2973)/16)^2 + 2 λ (y^2 + sqrt(2973)/16) + λ^2 = (73 y)/8 + (sqrt(2973) y^2)/8 + (115 y^2)/8 + 2 λ (y^2 + sqrt(2973)/16) + λ^2
(y^2 + sqrt(2973)/16)^2 + 2 λ (y^2 + sqrt(2973)/16) + λ^2 = (y^2 + sqrt(2973)/16 + λ)^2:
(y^2 + sqrt(2973)/16 + λ)^2 = (73 y)/8 + (sqrt(2973) y^2)/8 + (115 y^2)/8 + 2 λ (y^2 + sqrt(2973)/16) + λ^2
(73 y)/8 + (sqrt(2973) y^2)/8 + (115 y^2)/8 + 2 λ (y^2 + sqrt(2973)/16) + λ^2 = (2 λ + 115/8 + sqrt(2973)/8) y^2 + (73 y)/8 + (sqrt(2973) λ)/8 + λ^2:
(y^2 + sqrt(2973)/16 + λ)^2 = y^2 (2 λ + 115/8 + sqrt(2973)/8) + (73 y)/8 + (sqrt(2973) λ)/8 + λ^2
Complete the square on the right hand side:
(y^2 + sqrt(2973)/16 + λ)^2 = (y sqrt(2 λ + 115/8 + sqrt(2973)/8) + 73/(16 sqrt(2 λ + 115/8 + sqrt(2973)/8)))^2 + (4 (2 λ + 115/8 + sqrt(2973)/8) (λ^2 + (sqrt(2973) λ)/8) - 5329/64)/(4 (2 λ + 115/8 + sqrt(2973)/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 + 115/8 + sqrt(2973)/8) (λ^2 + (sqrt(2973) λ)/8) - 5329/64 = 1/64 (512 λ^3 + 96 sqrt(2973) λ^2 + 3680 λ^2 + 460 sqrt(2973) λ + 11892 λ - 5329) = 0.
Thus the root λ = 1/48 (-3 sqrt(2973) - 115) + 1/12 (-i sqrt(3) + 1) ((3 i sqrt(10705335) - 8327)/2)^(1/3) + (173 (i sqrt(3) + 1))/(3 2^(2/3) (3 i sqrt(10705335) - 8327)^(1/3)) allows the right hand side to be expressed as a square.
(This value will be substituted later):
(y^2 + sqrt(2973)/16 + λ)^2 = (y sqrt(2 λ + 115/8 + sqrt(2973)/8) + 73/(16 sqrt(2 λ + 115/8 + sqrt(2973)/8)))^2
Take the square root of both sides:
y^2 + sqrt(2973)/16 + λ = y sqrt(2 λ + 115/8 + sqrt(2973)/8) + 73/(16 sqrt(2 λ + 115/8 + sqrt(2973)/8)) or y^2 + sqrt(2973)/16 + λ = -y sqrt(2 λ + 115/8 + sqrt(2973)/8) - 73/(16 sqrt(2 λ + 115/8 + sqrt(2973)/8))
Solve using the quadratic formula:
y = 1/8 (sqrt(2) sqrt(16 λ + 115 + sqrt(2973)) + sqrt(2) sqrt((10252 - 32 sqrt(2973) λ - 256 λ^2 + 292 sqrt(2) sqrt(16 λ + 115 + sqrt(2973)))/(16 λ + 115 + sqrt(2973)))) or y = 1/8 (sqrt(2) sqrt(16 λ + 115 + sqrt(2973)) - sqrt(2) sqrt((10252 - 32 sqrt(2973) λ - 256 λ^2 + 292 sqrt(2) sqrt(16 λ + 115 + sqrt(2973)))/(16 λ + 115 + sqrt(2973)))) or y = 1/8 (sqrt(2) sqrt((10252 - 32 sqrt(2973) λ - 256 λ^2 - 292 sqrt(2) sqrt(16 λ + 115 + sqrt(2973)))/(16 λ + 115 + sqrt(2973))) - sqrt(2) sqrt(16 λ + 115 + sqrt(2973))) or y = 1/8 (-sqrt(2) sqrt(16 λ + 115 + sqrt(2973)) - sqrt(2) sqrt((10252 - 32 sqrt(2973) λ - 256 λ^2 - 292 sqrt(2) sqrt(16 λ + 115 + sqrt(2973)))/(16 λ + 115 + sqrt(2973)))) where λ = 1/48 (-3 sqrt(2973) - 115) + 1/12 (-i sqrt(3) + 1) ((3 i sqrt(10705335) - 8327)/2)^(1/3) + (173 (i sqrt(3) + 1))/(3 2^(2/3) (3 i sqrt(10705335) - 8327)^(1/3))
Substitute λ = 1/48 (-3 sqrt(2973) - 115) + 1/12 (-i sqrt(3) + 1) ((3 i sqrt(10705335) - 8327)/2)^(1/3) + (173 (i sqrt(3) + 1))/(3 2^(2/3) (3 i sqrt(10705335) - 8327)^(1/3)) and approximate:
y = -3.23079 or y = -1.40272 or y = 0.642002 or y = 3.99151
Substitute back for y = x - 1/4:
x - 1/4 = -3.23079 or y = -1.40272 or y = 0.642002 or y = 3.99151
Add 1/4 to both sides:
x = -2.98079 or y = -1.40272 or y = 0.642002 or y = 3.99151
Substitute back for y = x - 1/4:
x = -2.98079 or x - 1/4 = -1.40272 or y = 0.642002 or y = 3.99151
Add 1/4 to both sides:
x = -2.98079 or x = -1.15272 or y = 0.642002 or y = 3.99151
Substitute back for y = x - 1/4:
x = -2.98079 or x = -1.15272 or x - 1/4 = 0.642002 or y = 3.99151
Add 1/4 to both sides:
x = -2.98079 or x = -1.15272 or x = 0.892002 or y = 3.99151
Substitute back for y = x - 1/4:
x = -2.98079 or x = -1.15272 or x = 0.892002 or x - 1/4 = 3.99151
Add 1/4 to both sides:
Answer: x = -2.98079 or x = -1.15272 or x = 0.892002 or x = 4.24151
