Solve for x by completing the square:
2 x^10 + 3 x^9 - 5 x^8 - 9 x^7 - x^6 + 3 x^5 + 7 x^4 + 9 x^3 - x^2 - 6 x - 2 = 0
The left hand side factors into a product with five terms:
(x - 1)^2 (x + 1)^3 (2 x + 1) (x^2 - 2) (x^2 + 1) = 0
Split into five equations:
(x - 1)^2 = 0 or (x + 1)^3 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0
Take the square root of both sides:
x - 1 = 0 or (x + 1)^3 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0
Add 1 to both sides:
x = 1 or (x + 1)^3 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0
Taking cube roots gives 0 times the third roots of unity:
x = 1 or x + 1 = 0 or x + 1 = 0 or x + 1 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0
Subtract 1 from both sides:
x = 1 or x = -1 or x + 1 = 0 or x + 1 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0
Subtract 1 from both sides:
x = 1 or x = -1 or x = -1 or x + 1 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0
Subtract 1 from both sides:
x = 1 or x = -1 or x = -1 or x = -1 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0
Subtract 1 from both sides:
x = 1 or x = -1 or x = -1 or x = -1 or 2 x = -1 or x^2 - 2 = 0 or x^2 + 1 = 0
Divide both sides by 2:
x = 1 or x = -1 or x = -1 or x = -1 or x = -1/2 or x^2 - 2 = 0 or x^2 + 1 = 0
Add 2 to both sides:
x = 1 or x = -1 or x = -1 or x = -1 or x = -1/2 or x^2 = 2 or x^2 + 1 = 0
Take the square root of both sides:
x = 1 or x = -1 or x = -1 or x = -1 or x = -1/2 or x = sqrt(2) or x = -sqrt(2) or x^2 + 1 = 0
Subtract 1 from both sides:
x = 1 or x = -1 or x = -1 or x = -1 or x = -1/2 or x = sqrt(2) or x = -sqrt(2) or x^2 = -1
Take the square root of both sides:
x = 1 or x = -1 or x = -1 or x = -1 or x = -1/2 or x = sqrt(2) or x = -sqrt(2) or x = i or x = -i
There are {2} duplicate solutions:
Answer: x = -1 or x = 1 or x = -1/2 or x = -i or x = i or x = sqrt(2) or x = -sqrt(2)
