-3g(g+4)=1+g

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Solution for -3g(g+4)=1+g equation: 


Simplifying
-3g(g + 4) = 1 + g

Reorder the terms:
-3g(4 + g) = 1 + g
(4 * -3g + g * -3g) = 1 + g
(-12g + -3g2) = 1 + g

Solving
-12g + -3g2 = 1 + g

Solving for variable 'g'.

Reorder the terms:
-1 + -12g + -1g + -3g2 = 1 + g + -1 + -1g

Combine like terms: -12g + -1g = -13g
-1 + -13g + -3g2 = 1 + g + -1 + -1g

Reorder the terms:
-1 + -13g + -3g2 = 1 + -1 + g + -1g

Combine like terms: 1 + -1 = 0
-1 + -13g + -3g2 = 0 + g + -1g
-1 + -13g + -3g2 = g + -1g

Combine like terms: g + -1g = 0
-1 + -13g + -3g2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 13g + 3g2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(1 + 13g + 3g2)' equal to zero and attempt to solve:

Simplifying
1 + 13g + 3g2 = 0

Solving
1 + 13g + 3g2 = 0

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

Divide each side by '3'.
0.3333333333 + 4.333333333g + g2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 4.333333333g + -0.3333333333 + g2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 4.333333333g + g2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 4.333333333g + g2 = 0 + -0.3333333333
4.333333333g + g2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
4.333333333g + g2 = -0.3333333333

The g term is 4.333333333g.  Take half its coefficient (2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
4.333333333g + 4.694444446 + g2 = -0.3333333333 + 4.694444446

Reorder the terms:
4.694444446 + 4.333333333g + g2 = -0.3333333333 + 4.694444446

Combine like terms: -0.3333333333 + 4.694444446 = 4.3611111127
4.694444446 + 4.333333333g + g2 = 4.3611111127

Factor a perfect square on the left side:
(g + 2.166666667)(g + 2.166666667) = 4.3611111127

Calculate the square root of the right side: 2.088327348

Break this problem into two subproblems by setting 
(g + 2.166666667) equal to 2.088327348 and -2.088327348.


Subproblem 1
g + 2.166666667 = 2.088327348

Simplifying
g + 2.166666667 = 2.088327348

Reorder the terms:
2.166666667 + g = 2.088327348

Solving
2.166666667 + g = 2.088327348

Solving for variable 'g'.

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

Add '-2.166666667' to each side of the equation.
2.166666667 + -2.166666667 + g = 2.088327348 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + g = 2.088327348 + -2.166666667
g = 2.088327348 + -2.166666667

Combine like terms: 2.088327348 + -2.166666667 = -0.078339319
g = -0.078339319

Simplifying
g = -0.078339319


Subproblem 2
g + 2.166666667 = -2.088327348

Simplifying
g + 2.166666667 = -2.088327348

Reorder the terms:
2.166666667 + g = -2.088327348

Solving
2.166666667 + g = -2.088327348

Solving for variable 'g'.

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

Add '-2.166666667' to each side of the equation.
2.166666667 + -2.166666667 + g = -2.088327348 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + g = -2.088327348 + -2.166666667
g = -2.088327348 + -2.166666667

Combine like terms: -2.088327348 + -2.166666667 = -4.254994015
g = -4.254994015

Simplifying
g = -4.254994015


Solution
The solution to the problem is based on the solutions
from the subproblems.
g = {-0.078339319, -4.254994015}
Solution
g = {-0.078339319, -4.254994015}

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