0=x^4-6x^3+9x^2

Solution for 0=x^4-6x^3+9x^2 equation:

Simplifying
0 = x4 + -6x3 + 9x2

Reorder the terms:
0 = 9x2 + -6x3 + x4

Solving
0 = 9x2 + -6x3 + x4

Solving for variable 'x'.
Remove the zero:
-9x2 + 6x3 + -1x4 = 9x2 + -6x3 + x4 + -9x2 + 6x3 + -1x4

Reorder the terms:
-9x2 + 6x3 + -1x4 = 9x2 + -9x2 + -6x3 + 6x3 + x4 + -1x4

Combine like terms: 9x2 + -9x2 = 0
-9x2 + 6x3 + -1x4 = 0 + -6x3 + 6x3 + x4 + -1x4
-9x2 + 6x3 + -1x4 = -6x3 + 6x3 + x4 + -1x4

Combine like terms: -6x3 + 6x3 = 0
-9x2 + 6x3 + -1x4 = 0 + x4 + -1x4
-9x2 + 6x3 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-9x2 + 6x3 + -1x4 = 0

Factor out the Greatest Common Factor (GCF), 'x2'.
x2(-9 + 6x + -1x2) = 0

Factor a trinomial.
x2((-3 + x)(3 + -1x)) = 0

Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve:

Simplifying
x2 = 0

Solving
x2 = 0

Move all terms containing x to the left, all other terms to the right.

Simplifying
x2 = 0

Take the square root of each side:
x = {0}
Subproblem 2
Set the factor '(-3 + x)' equal to zero and attempt to solve:

Simplifying
-3 + x = 0

Solving
-3 + x = 0

Move all terms containing x to the left, all other terms to the right.

Add '3' to each side of the equation.
-3 + 3 + x = 0 + 3

Combine like terms: -3 + 3 = 0
0 + x = 0 + 3
x = 0 + 3

Combine like terms: 0 + 3 = 3
x = 3

Simplifying
x = 3
Subproblem 3
Set the factor '(3 + -1x)' equal to zero and attempt to solve:

Simplifying
3 + -1x = 0

Solving
3 + -1x = 0

Move all terms containing x to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + -1x = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -1x = 0 + -3
-1x = 0 + -3

Combine like terms: 0 + -3 = -3
-1x = -3

Divide each side by '-1'.
x = 3

Simplifying
x = 3
Solution
x = {0, 3, 3}