4x(2+3x)=54

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Solution for 4x(2+3x)=54 equation: 


Simplifying
4x(2 + 3x) = 54
(2 * 4x + 3x * 4x) = 54
(8x + 12x2) = 54

Solving
8x + 12x2 = 54

Solving for variable 'x'.

Reorder the terms:
-54 + 8x + 12x2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + 8x + 12x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-27 + 4x + 6x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-27 + 4x + 6x2)' equal to zero and attempt to solve:

Simplifying
-27 + 4x + 6x2 = 0

Solving
-27 + 4x + 6x2 = 0

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

Divide each side by '6'.
-4.5 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 0.6666666667x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 0.6666666667x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 0.6666666667x + x2 = 0 + 4.5
0.6666666667x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
0.6666666667x + x2 = 4.5

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 4.5 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 4.5 + 0.1111111112

Combine like terms: 4.5 + 0.1111111112 = 4.6111111112
0.1111111112 + 0.6666666667x + x2 = 4.6111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 4.6111111112

Calculate the square root of the right side: 2.147349788

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 2.147349788 and -2.147349788.


Subproblem 1
x + 0.3333333334 = 2.147349788

Simplifying
x + 0.3333333334 = 2.147349788

Reorder the terms:
0.3333333334 + x = 2.147349788

Solving
0.3333333334 + x = 2.147349788

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = 2.147349788 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = 2.147349788 + -0.3333333334
x = 2.147349788 + -0.3333333334

Combine like terms: 2.147349788 + -0.3333333334 = 1.8140164546
x = 1.8140164546

Simplifying
x = 1.8140164546


Subproblem 2
x + 0.3333333334 = -2.147349788

Simplifying
x + 0.3333333334 = -2.147349788

Reorder the terms:
0.3333333334 + x = -2.147349788

Solving
0.3333333334 + x = -2.147349788

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = -2.147349788 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = -2.147349788 + -0.3333333334
x = -2.147349788 + -0.3333333334

Combine like terms: -2.147349788 + -0.3333333334 = -2.4806831214
x = -2.4806831214

Simplifying
x = -2.4806831214


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.8140164546, -2.4806831214}
Solution
x = {1.8140164546, -2.4806831214}

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