9+4[1+2(2x-4)]=6x(x+9)

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Solution for 9+4[1+2(2x-4)]=6x(x+9) equation: 


Simplifying
9 + 4[1 + 2(2x + -4)] = 6x(x + 9)

Reorder the terms:
9 + 4[1 + 2(-4 + 2x)] = 6x(x + 9)
9 + 4[1 + (-4 * 2 + 2x * 2)] = 6x(x + 9)
9 + 4[1 + (-8 + 4x)] = 6x(x + 9)

Combine like terms: 1 + -8 = -7
9 + 4[-7 + 4x] = 6x(x + 9)
9 + [-7 * 4 + 4x * 4] = 6x(x + 9)
9 + [-28 + 16x] = 6x(x + 9)

Combine like terms: 9 + -28 = -19
-19 + 16x = 6x(x + 9)

Reorder the terms:
-19 + 16x = 6x(9 + x)
-19 + 16x = (9 * 6x + x * 6x)
-19 + 16x = (54x + 6x2)

Solving
-19 + 16x = 54x + 6x2

Solving for variable 'x'.

Combine like terms: 16x + -54x = -38x
-19 + -38x + -6x2 = 54x + 6x2 + -54x + -6x2

Reorder the terms:
-19 + -38x + -6x2 = 54x + -54x + 6x2 + -6x2

Combine like terms: 54x + -54x = 0
-19 + -38x + -6x2 = 0 + 6x2 + -6x2
-19 + -38x + -6x2 = 6x2 + -6x2

Combine like terms: 6x2 + -6x2 = 0
-19 + -38x + -6x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(19 + 38x + 6x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(19 + 38x + 6x2)' equal to zero and attempt to solve:

Simplifying
19 + 38x + 6x2 = 0

Solving
19 + 38x + 6x2 = 0

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

Divide each side by '6'.
3.166666667 + 6.333333333x + x2 = 0

Move the constant term to the right:

Add '-3.166666667' to each side of the equation.
3.166666667 + 6.333333333x + -3.166666667 + x2 = 0 + -3.166666667

Reorder the terms:
3.166666667 + -3.166666667 + 6.333333333x + x2 = 0 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + 6.333333333x + x2 = 0 + -3.166666667
6.333333333x + x2 = 0 + -3.166666667

Combine like terms: 0 + -3.166666667 = -3.166666667
6.333333333x + x2 = -3.166666667

The x term is 6.333333333x.  Take half its coefficient (3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
6.333333333x + 10.02777778 + x2 = -3.166666667 + 10.02777778

Reorder the terms:
10.02777778 + 6.333333333x + x2 = -3.166666667 + 10.02777778

Combine like terms: -3.166666667 + 10.02777778 = 6.861111113
10.02777778 + 6.333333333x + x2 = 6.861111113

Factor a perfect square on the left side:
(x + 3.166666667)(x + 3.166666667) = 6.861111113

Calculate the square root of the right side: 2.619372275

Break this problem into two subproblems by setting 
(x + 3.166666667) equal to 2.619372275 and -2.619372275.


Subproblem 1
x + 3.166666667 = 2.619372275

Simplifying
x + 3.166666667 = 2.619372275

Reorder the terms:
3.166666667 + x = 2.619372275

Solving
3.166666667 + x = 2.619372275

Solving for variable 'x'.

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

Add '-3.166666667' to each side of the equation.
3.166666667 + -3.166666667 + x = 2.619372275 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + x = 2.619372275 + -3.166666667
x = 2.619372275 + -3.166666667

Combine like terms: 2.619372275 + -3.166666667 = -0.547294392
x = -0.547294392

Simplifying
x = -0.547294392


Subproblem 2
x + 3.166666667 = -2.619372275

Simplifying
x + 3.166666667 = -2.619372275

Reorder the terms:
3.166666667 + x = -2.619372275

Solving
3.166666667 + x = -2.619372275

Solving for variable 'x'.

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

Add '-3.166666667' to each side of the equation.
3.166666667 + -3.166666667 + x = -2.619372275 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + x = -2.619372275 + -3.166666667
x = -2.619372275 + -3.166666667

Combine like terms: -2.619372275 + -3.166666667 = -5.786038942
x = -5.786038942

Simplifying
x = -5.786038942


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.547294392, -5.786038942}
Solution
x = {-0.547294392, -5.786038942}

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