y(y-6)3-27(y-6)2=0

Solution for y(y-6)3-27(y-6)2=0 equation:

Simplifying
y(y + -6) * 3 + -27(y + -6) * 2 = 0

Reorder the terms:
y(-6 + y) * 3 + -27(y + -6) * 2 = 0

Reorder the terms for easier multiplication:
3y(-6 + y) + -27(y + -6) * 2 = 0
(-6 * 3y + y * 3y) + -27(y + -6) * 2 = 0
(-18y + 3y2) + -27(y + -6) * 2 = 0

Reorder the terms:
-18y + 3y2 + -27(-6 + y) * 2 = 0

Reorder the terms for easier multiplication:
-18y + 3y2 + -27 * 2(-6 + y) = 0

Multiply -27 * 2
-18y + 3y2 + -54(-6 + y) = 0
-18y + 3y2 + (-6 * -54 + y * -54) = 0
-18y + 3y2 + (324 + -54y) = 0

Reorder the terms:
324 + -18y + -54y + 3y2 = 0

Combine like terms: -18y + -54y = -72y
324 + -72y + 3y2 = 0

Solving
324 + -72y + 3y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '3'.
3(108 + -24y + y2) = 0

Factor a trinomial.
3((6 + -1y)(18 + -1y)) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(6 + -1y)' equal to zero and attempt to solve:

Simplifying
6 + -1y = 0

Solving
6 + -1y = 0

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

Add '-6' to each side of the equation.
6 + -6 + -1y = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -1y = 0 + -6
-1y = 0 + -6

Combine like terms: 0 + -6 = -6
-1y = -6

Divide each side by '-1'.
y = 6

Simplifying
y = 6
Subproblem 2
Set the factor '(18 + -1y)' equal to zero and attempt to solve:

Simplifying
18 + -1y = 0

Solving
18 + -1y = 0

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

Add '-18' to each side of the equation.
18 + -18 + -1y = 0 + -18

Combine like terms: 18 + -18 = 0
0 + -1y = 0 + -18
-1y = 0 + -18

Combine like terms: 0 + -18 = -18
-1y = -18

Divide each side by '-1'.
y = 18

Simplifying
y = 18
Solution
y = {6, 18}