(6x+9)(9x+6)2=9

Solution for (6x+9)(9x+6)2=9 equation:

Simplifying
(6x + 9)(9x + 6) * 2 = 9

Reorder the terms:
(9 + 6x)(9x + 6) * 2 = 9

Reorder the terms:
(9 + 6x)(6 + 9x) * 2 = 9

Reorder the terms for easier multiplication:
2(9 + 6x)(6 + 9x) = 9

Multiply (9 + 6x) * (6 + 9x)
2(9(6 + 9x) + 6x * (6 + 9x)) = 9
2((6 * 9 + 9x * 9) + 6x * (6 + 9x)) = 9
2((54 + 81x) + 6x * (6 + 9x)) = 9
2(54 + 81x + (6 * 6x + 9x * 6x)) = 9
2(54 + 81x + (36x + 54x2)) = 9

Combine like terms: 81x + 36x = 117x
2(54 + 117x + 54x2) = 9
(54 * 2 + 117x * 2 + 54x2 * 2) = 9
(108 + 234x + 108x2) = 9

Solving
108 + 234x + 108x2 = 9

Solving for variable 'x'.

Reorder the terms:
108 + -9 + 234x + 108x2 = 9 + -9

Combine like terms: 108 + -9 = 99
99 + 234x + 108x2 = 9 + -9

Combine like terms: 9 + -9 = 0
99 + 234x + 108x2 = 0

Factor out the Greatest Common Factor (GCF), '9'.
9(11 + 26x + 12x2) = 0

Ignore the factor 9.
Subproblem 1
Set the factor '(11 + 26x + 12x2)' equal to zero and attempt to solve:

Simplifying
11 + 26x + 12x2 = 0

Solving
11 + 26x + 12x2 = 0

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

Divide each side by '12'.
0.9166666667 + 2.166666667x + x2 = 0

Move the constant term to the right:

Add '-0.9166666667' to each side of the equation.
0.9166666667 + 2.166666667x + -0.9166666667 + x2 = 0 + -0.9166666667

Reorder the terms:
0.9166666667 + -0.9166666667 + 2.166666667x + x2 = 0 + -0.9166666667

Combine like terms: 0.9166666667 + -0.9166666667 = 0.0000000000
0.0000000000 + 2.166666667x + x2 = 0 + -0.9166666667
2.166666667x + x2 = 0 + -0.9166666667

Combine like terms: 0 + -0.9166666667 = -0.9166666667
2.166666667x + x2 = -0.9166666667

The x term is 2.166666667x.  Take half its coefficient (1.083333334).
Square it (1.173611113) and add it to both sides.

Add '1.173611113' to each side of the equation.
2.166666667x + 1.173611113 + x2 = -0.9166666667 + 1.173611113

Reorder the terms:
1.173611113 + 2.166666667x + x2 = -0.9166666667 + 1.173611113

Combine like terms: -0.9166666667 + 1.173611113 = 0.2569444463
1.173611113 + 2.166666667x + x2 = 0.2569444463

Factor a perfect square on the left side:
(x + 1.083333334)(x + 1.083333334) = 0.2569444463

Calculate the square root of the right side: 0.506896879

Break this problem into two subproblems by setting 
(x + 1.083333334) equal to 0.506896879 and -0.506896879.

Subproblem 1
x + 1.083333334 = 0.506896879

Simplifying
x + 1.083333334 = 0.506896879

Reorder the terms:
1.083333334 + x = 0.506896879

Solving
1.083333334 + x = 0.506896879

Solving for variable 'x'.

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

Add '-1.083333334' to each side of the equation.
1.083333334 + -1.083333334 + x = 0.506896879 + -1.083333334

Combine like terms: 1.083333334 + -1.083333334 = 0.000000000
0.000000000 + x = 0.506896879 + -1.083333334
x = 0.506896879 + -1.083333334

Combine like terms: 0.506896879 + -1.083333334 = -0.576436455
x = -0.576436455

Simplifying
x = -0.576436455

Subproblem 2
x + 1.083333334 = -0.506896879

Simplifying
x + 1.083333334 = -0.506896879

Reorder the terms:
1.083333334 + x = -0.506896879

Solving
1.083333334 + x = -0.506896879

Solving for variable 'x'.

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

Add '-1.083333334' to each side of the equation.
1.083333334 + -1.083333334 + x = -0.506896879 + -1.083333334

Combine like terms: 1.083333334 + -1.083333334 = 0.000000000
0.000000000 + x = -0.506896879 + -1.083333334
x = -0.506896879 + -1.083333334

Combine like terms: -0.506896879 + -1.083333334 = -1.590230213
x = -1.590230213

Simplifying
x = -1.590230213

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.576436455, -1.590230213}
Solution
x = {-0.576436455, -1.590230213}