6x(3x+3)=24

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Solution for 6x(3x+3)=24 equation: 


Simplifying
6x(3x + 3) = 24

Reorder the terms:
6x(3 + 3x) = 24
(3 * 6x + 3x * 6x) = 24
(18x + 18x2) = 24

Solving
18x + 18x2 = 24

Solving for variable 'x'.

Reorder the terms:
-24 + 18x + 18x2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + 18x + 18x2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-4 + 3x + 3x2) = 0

Ignore the factor 6.

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

Simplifying
-4 + 3x + 3x2 = 0

Solving
-4 + 3x + 3x2 = 0

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

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

Move the constant term to the right:

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

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

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + x + x2 = 0 + 1.333333333
x + x2 = 0 + 1.333333333

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

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 1.333333333 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 1.333333333 + 0.25

Combine like terms: 1.333333333 + 0.25 = 1.583333333
1.25 + x2 = 1.583333333

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1.583333333

Calculate the square root of the right side: 1.258305739

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.258305739 and -1.258305739.


Subproblem 1
x + 0.5 = 1.258305739

Simplifying
x + 0.5 = 1.258305739

Reorder the terms:
0.5 + x = 1.258305739

Solving
0.5 + x = 1.258305739

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 1.258305739 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 1.258305739 + -0.5
x = 1.258305739 + -0.5

Combine like terms: 1.258305739 + -0.5 = 0.758305739
x = 0.758305739

Simplifying
x = 0.758305739


Subproblem 2
x + 0.5 = -1.258305739

Simplifying
x + 0.5 = -1.258305739

Reorder the terms:
0.5 + x = -1.258305739

Solving
0.5 + x = -1.258305739

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -1.258305739 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -1.258305739 + -0.5
x = -1.258305739 + -0.5

Combine like terms: -1.258305739 + -0.5 = -1.758305739
x = -1.758305739

Simplifying
x = -1.758305739


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.758305739, -1.758305739}
Solution
x = {0.758305739, -1.758305739}

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