6x+6=9x(6-5x)

Solution for 6x+6=9x(6-5x) equation:

Simplifying
6x + 6 = 9x(6 + -5x)

Reorder the terms:
6 + 6x = 9x(6 + -5x)
6 + 6x = (6 * 9x + -5x * 9x)
6 + 6x = (54x + -45x2)

Solving
6 + 6x = 54x + -45x2

Solving for variable 'x'.

Combine like terms: 6x + -54x = -48x
6 + -48x + 45x2 = 54x + -45x2 + -54x + 45x2

Reorder the terms:
6 + -48x + 45x2 = 54x + -54x + -45x2 + 45x2

Combine like terms: 54x + -54x = 0
6 + -48x + 45x2 = 0 + -45x2 + 45x2
6 + -48x + 45x2 = -45x2 + 45x2

Combine like terms: -45x2 + 45x2 = 0
6 + -48x + 45x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(2 + -16x + 15x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(2 + -16x + 15x2)' equal to zero and attempt to solve:

Simplifying
2 + -16x + 15x2 = 0

Solving
2 + -16x + 15x2 = 0

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

Divide each side by '15'.
0.1333333333 + -1.066666667x + x2 = 0

Move the constant term to the right:

Add '-0.1333333333' to each side of the equation.
0.1333333333 + -1.066666667x + -0.1333333333 + x2 = 0 + -0.1333333333

Reorder the terms:
0.1333333333 + -0.1333333333 + -1.066666667x + x2 = 0 + -0.1333333333

Combine like terms: 0.1333333333 + -0.1333333333 = 0.0000000000
0.0000000000 + -1.066666667x + x2 = 0 + -0.1333333333
-1.066666667x + x2 = 0 + -0.1333333333

Combine like terms: 0 + -0.1333333333 = -0.1333333333
-1.066666667x + x2 = -0.1333333333

The x term is -1.066666667x.  Take half its coefficient (-0.5333333335).
Square it (0.2844444446) and add it to both sides.

Add '0.2844444446' to each side of the equation.
-1.066666667x + 0.2844444446 + x2 = -0.1333333333 + 0.2844444446

Reorder the terms:
0.2844444446 + -1.066666667x + x2 = -0.1333333333 + 0.2844444446

Combine like terms: -0.1333333333 + 0.2844444446 = 0.1511111113
0.2844444446 + -1.066666667x + x2 = 0.1511111113

Factor a perfect square on the left side:
(x + -0.5333333335)(x + -0.5333333335) = 0.1511111113

Calculate the square root of the right side: 0.388730127

Break this problem into two subproblems by setting 
(x + -0.5333333335) equal to 0.388730127 and -0.388730127.

Subproblem 1
x + -0.5333333335 = 0.388730127

Simplifying
x + -0.5333333335 = 0.388730127

Reorder the terms:
-0.5333333335 + x = 0.388730127

Solving
-0.5333333335 + x = 0.388730127

Solving for variable 'x'.

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

Add '0.5333333335' to each side of the equation.
-0.5333333335 + 0.5333333335 + x = 0.388730127 + 0.5333333335

Combine like terms: -0.5333333335 + 0.5333333335 = 0.0000000000
0.0000000000 + x = 0.388730127 + 0.5333333335
x = 0.388730127 + 0.5333333335

Combine like terms: 0.388730127 + 0.5333333335 = 0.9220634605
x = 0.9220634605

Simplifying
x = 0.9220634605

Subproblem 2
x + -0.5333333335 = -0.388730127

Simplifying
x + -0.5333333335 = -0.388730127

Reorder the terms:
-0.5333333335 + x = -0.388730127

Solving
-0.5333333335 + x = -0.388730127

Solving for variable 'x'.

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

Add '0.5333333335' to each side of the equation.
-0.5333333335 + 0.5333333335 + x = -0.388730127 + 0.5333333335

Combine like terms: -0.5333333335 + 0.5333333335 = 0.0000000000
0.0000000000 + x = -0.388730127 + 0.5333333335
x = -0.388730127 + 0.5333333335

Combine like terms: -0.388730127 + 0.5333333335 = 0.1446032065
x = 0.1446032065

Simplifying
x = 0.1446032065

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.9220634605, 0.1446032065}
Solution
x = {0.9220634605, 0.1446032065}