5x(3x+3)x+6=x+6

Solution for 5x(3x+3)x+6=x+6 equation:

Simplifying
5x(3x + 3) * x + 6 = x + 6

Reorder the terms:
5x(3 + 3x) * x + 6 = x + 6

Reorder the terms for easier multiplication:
5x * x(3 + 3x) + 6 = x + 6

Multiply x * x
5x2(3 + 3x) + 6 = x + 6
(3 * 5x2 + 3x * 5x2) + 6 = x + 6
(15x2 + 15x3) + 6 = x + 6

Reorder the terms:
6 + 15x2 + 15x3 = x + 6

Reorder the terms:
6 + 15x2 + 15x3 = 6 + x

Add '-6' to each side of the equation.
6 + 15x2 + -6 + 15x3 = 6 + -6 + x

Reorder the terms:
6 + -6 + 15x2 + 15x3 = 6 + -6 + x

Combine like terms: 6 + -6 = 0
0 + 15x2 + 15x3 = 6 + -6 + x
15x2 + 15x3 = 6 + -6 + x

Combine like terms: 6 + -6 = 0
15x2 + 15x3 = 0 + x
15x2 + 15x3 = x

Solving
15x2 + 15x3 = x

Solving for variable 'x'.

Reorder the terms:
-1x + 15x2 + 15x3 = x + -1x

Combine like terms: x + -1x = 0
-1x + 15x2 + 15x3 = 0

Factor out the Greatest Common Factor (GCF), 'x'.
x(-1 + 15x + 15x2) = 0

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

Simplifying
x = 0

Solving
x = 0

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

Simplifying
x = 0
Subproblem 2
Set the factor '(-1 + 15x + 15x2)' equal to zero and attempt to solve:

Simplifying
-1 + 15x + 15x2 = 0

Solving
-1 + 15x + 15x2 = 0

Begin completing the square.  Divide all terms by
15 the coefficient of the squared term: 

Divide each side by '15'.
-0.06666666667 + x + x2 = 0

Move the constant term to the right:

Add '0.06666666667' to each side of the equation.
-0.06666666667 + x + 0.06666666667 + x2 = 0 + 0.06666666667

Reorder the terms:
-0.06666666667 + 0.06666666667 + x + x2 = 0 + 0.06666666667

Combine like terms: -0.06666666667 + 0.06666666667 = 0.00000000000
0.00000000000 + x + x2 = 0 + 0.06666666667
x + x2 = 0 + 0.06666666667

Combine like terms: 0 + 0.06666666667 = 0.06666666667
x + x2 = 0.06666666667

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 0.06666666667 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 0.06666666667 + 0.25

Combine like terms: 0.06666666667 + 0.25 = 0.31666666667
1.25 + x2 = 0.31666666667

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.31666666667

Calculate the square root of the right side: 0.562731434

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.562731434 and -0.562731434.

Subproblem 1
x + 0.5 = 0.562731434

Simplifying
x + 0.5 = 0.562731434

Reorder the terms:
0.5 + x = 0.562731434

Solving
0.5 + x = 0.562731434

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 0.562731434 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.562731434 + -0.5
x = 0.562731434 + -0.5

Combine like terms: 0.562731434 + -0.5 = 0.062731434
x = 0.062731434

Simplifying
x = 0.062731434

Subproblem 2
x + 0.5 = -0.562731434

Simplifying
x + 0.5 = -0.562731434

Reorder the terms:
0.5 + x = -0.562731434

Solving
0.5 + x = -0.562731434

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -0.562731434 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.562731434 + -0.5
x = -0.562731434 + -0.5

Combine like terms: -0.562731434 + -0.5 = -1.062731434
x = -1.062731434

Simplifying
x = -1.062731434

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.062731434, -1.062731434}
Solution
x = {0, 0.062731434, -1.062731434}