6(x+7)x=9

Solution for 6(x+7)x=9 equation:

Simplifying
6(x + 7) * x = 9

Reorder the terms:
6(7 + x) * x = 9

Reorder the terms for easier multiplication:
6x(7 + x) = 9
(7 * 6x + x * 6x) = 9
(42x + 6x2) = 9

Solving
42x + 6x2 = 9

Solving for variable 'x'.

Reorder the terms:
-9 + 42x + 6x2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 42x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-3 + 14x + 2x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-3 + 14x + 2x2)' equal to zero and attempt to solve:

Simplifying
-3 + 14x + 2x2 = 0

Solving
-3 + 14x + 2x2 = 0

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

Divide each side by '2'.
-1.5 + 7x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 7x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 7x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 7x + x2 = 0 + 1.5
7x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
7x + x2 = 1.5

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = 1.5 + 12.25

Reorder the terms:
12.25 + 7x + x2 = 1.5 + 12.25

Combine like terms: 1.5 + 12.25 = 13.75
12.25 + 7x + x2 = 13.75

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 13.75

Calculate the square root of the right side: 3.708099244

Break this problem into two subproblems by setting 
(x + 3.5) equal to 3.708099244 and -3.708099244.

Subproblem 1
x + 3.5 = 3.708099244

Simplifying
x + 3.5 = 3.708099244

Reorder the terms:
3.5 + x = 3.708099244

Solving
3.5 + x = 3.708099244

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = 3.708099244 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = 3.708099244 + -3.5
x = 3.708099244 + -3.5

Combine like terms: 3.708099244 + -3.5 = 0.208099244
x = 0.208099244

Simplifying
x = 0.208099244

Subproblem 2
x + 3.5 = -3.708099244

Simplifying
x + 3.5 = -3.708099244

Reorder the terms:
3.5 + x = -3.708099244

Solving
3.5 + x = -3.708099244

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = -3.708099244 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = -3.708099244 + -3.5
x = -3.708099244 + -3.5

Combine like terms: -3.708099244 + -3.5 = -7.208099244
x = -7.208099244

Simplifying
x = -7.208099244

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.208099244, -7.208099244}
Solution
x = {0.208099244, -7.208099244}