Solve for x:
27 x^5+8 x^2 = 0
The left hand side factors into a product with three terms:
x^2 (3 x+2) (9 x^2-6 x+4) = 0
Split into three equations:
x^2 = 0 or 3 x+2 = 0 or 9 x^2-6 x+4 = 0
Take the square root of both sides:
x = 0 or 3 x+2 = 0 or 9 x^2-6 x+4 = 0
Subtract 2 from both sides:
x = 0 or 3 x = -2 or 9 x^2-6 x+4 = 0
Divide both sides by 3:
x = 0 or x = -2/3 or 9 x^2-6 x+4 = 0
Divide both sides by 9:
x = 0 or x = -2/3 or x^2-(2 x)/3+4/9 = 0
Subtract 4/9 from both sides:
x = 0 or x = -2/3 or x^2-(2 x)/3 = -4/9
Add 1/9 to both sides:
x = 0 or x = -2/3 or x^2-(2 x)/3+1/9 = -1/3
Write the left hand side as a square:
x = 0 or x = -2/3 or (x-1/3)^2 = -1/3
Take the square root of both sides:
x = 0 or x = -2/3 or x-1/3 = i/sqrt(3) or x-1/3 = -i/sqrt(3)
Add 1/3 to both sides:
x = 0 or x = -2/3 or x = 1/3+i/sqrt(3) or x-1/3 = -i/sqrt(3)
Add 1/3 to both sides:
Answer: |x = 0 or x = -2/3 or x = 1/3+i/sqrt(3) or x = 1/3-i/sqrt(3)