3x(4+3x)=9

Solution for 3x(4+3x)=9 equation:

Simplifying
3x(4 + 3x) = 9
(4 * 3x + 3x * 3x) = 9
(12x + 9x2) = 9

Solving
12x + 9x2 = 9

Solving for variable 'x'.

Reorder the terms:
-9 + 12x + 9x2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 12x + 9x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-3 + 4x + 3x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-3 + 4x + 3x2)' equal to zero and attempt to solve:

Simplifying
-3 + 4x + 3x2 = 0

Solving
-3 + 4x + 3x2 = 0

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

Divide each side by '3'.
-1 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.333333333x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.333333333x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.333333333x + x2 = 0 + 1
1.333333333x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.333333333x + x2 = 1

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 1 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 1 + 0.4444444442

Combine like terms: 1 + 0.4444444442 = 1.4444444442
0.4444444442 + 1.333333333x + x2 = 1.4444444442

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 1.4444444442

Calculate the square root of the right side: 1.201850425

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 1.201850425 and -1.201850425.

Subproblem 1
x + 0.6666666665 = 1.201850425

Simplifying
x + 0.6666666665 = 1.201850425

Reorder the terms:
0.6666666665 + x = 1.201850425

Solving
0.6666666665 + x = 1.201850425

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = 1.201850425 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = 1.201850425 + -0.6666666665
x = 1.201850425 + -0.6666666665

Combine like terms: 1.201850425 + -0.6666666665 = 0.5351837585
x = 0.5351837585

Simplifying
x = 0.5351837585

Subproblem 2
x + 0.6666666665 = -1.201850425

Simplifying
x + 0.6666666665 = -1.201850425

Reorder the terms:
0.6666666665 + x = -1.201850425

Solving
0.6666666665 + x = -1.201850425

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = -1.201850425 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = -1.201850425 + -0.6666666665
x = -1.201850425 + -0.6666666665

Combine like terms: -1.201850425 + -0.6666666665 = -1.8685170915
x = -1.8685170915

Simplifying
x = -1.8685170915

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.5351837585, -1.8685170915}
Solution
x = {0.5351837585, -1.8685170915}