6(x+2)+2=3x(5+x)

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Solution for 6(x+2)+2=3x(5+x) equation: 


Simplifying
6(x + 2) + 2 = 3x(5 + x)

Reorder the terms:
6(2 + x) + 2 = 3x(5 + x)
(2 * 6 + x * 6) + 2 = 3x(5 + x)
(12 + 6x) + 2 = 3x(5 + x)

Reorder the terms:
12 + 2 + 6x = 3x(5 + x)

Combine like terms: 12 + 2 = 14
14 + 6x = 3x(5 + x)
14 + 6x = (5 * 3x + x * 3x)
14 + 6x = (15x + 3x2)

Solving
14 + 6x = 15x + 3x2

Solving for variable 'x'.

Combine like terms: 6x + -15x = -9x
14 + -9x + -3x2 = 15x + 3x2 + -15x + -3x2

Reorder the terms:
14 + -9x + -3x2 = 15x + -15x + 3x2 + -3x2

Combine like terms: 15x + -15x = 0
14 + -9x + -3x2 = 0 + 3x2 + -3x2
14 + -9x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
14 + -9x + -3x2 = 0

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

Divide each side by '-3'.
-4.666666667 + 3x + x2 = 0

Move the constant term to the right:

Add '4.666666667' to each side of the equation.
-4.666666667 + 3x + 4.666666667 + x2 = 0 + 4.666666667

Reorder the terms:
-4.666666667 + 4.666666667 + 3x + x2 = 0 + 4.666666667

Combine like terms: -4.666666667 + 4.666666667 = 0.000000000
0.000000000 + 3x + x2 = 0 + 4.666666667
3x + x2 = 0 + 4.666666667

Combine like terms: 0 + 4.666666667 = 4.666666667
3x + x2 = 4.666666667

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 4.666666667 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 4.666666667 + 2.25

Combine like terms: 4.666666667 + 2.25 = 6.916666667
2.25 + 3x + x2 = 6.916666667

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 6.916666667

Calculate the square root of the right side: 2.62995564

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.62995564 and -2.62995564.


Subproblem 1
x + 1.5 = 2.62995564

Simplifying
x + 1.5 = 2.62995564

Reorder the terms:
1.5 + x = 2.62995564

Solving
1.5 + x = 2.62995564

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 2.62995564 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 2.62995564 + -1.5
x = 2.62995564 + -1.5

Combine like terms: 2.62995564 + -1.5 = 1.12995564
x = 1.12995564

Simplifying
x = 1.12995564


Subproblem 2
x + 1.5 = -2.62995564

Simplifying
x + 1.5 = -2.62995564

Reorder the terms:
1.5 + x = -2.62995564

Solving
1.5 + x = -2.62995564

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -2.62995564 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -2.62995564 + -1.5
x = -2.62995564 + -1.5

Combine like terms: -2.62995564 + -1.5 = -4.12995564
x = -4.12995564

Simplifying
x = -4.12995564


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.12995564, -4.12995564}

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