Solve for x:
x^4-6 x^2-7 x-6 = 0
The left hand side factors into a product with three terms:
(x-3) (x+2) (x^2+x+1) = 0
Split into three equations:
x-3 = 0 or x+2 = 0 or x^2+x+1 = 0
Add 3 to both sides:
x = 3 or x+2 = 0 or x^2+x+1 = 0
Subtract 2 from both sides:
x = 3 or x = -2 or x^2+x+1 = 0
Subtract 1 from both sides:
x = 3 or x = -2 or x^2+x = -1
Add 1/4 to both sides:
x = 3 or x = -2 or x^2+x+1/4 = -3/4
Write the left hand side as a square:
x = 3 or x = -2 or (x+1/2)^2 = -3/4
Take the square root of both sides:
x = 3 or x = -2 or x+1/2 = (i sqrt(3))/2 or x+1/2 = 1/2 (-i) sqrt(3)
Subtract 1/2 from both sides:
x = 3 or x = -2 or x = (i sqrt(3))/2-1/2 or x+1/2 = 1/2 (-i) sqrt(3)
Subtract 1/2 from both sides:
Answer: | x = 3 or x = -2 or x = (i sqrt(3))/2-1/2 or x = 1/2 (-i) sqrt(3)-1/2
