(g+4)3g=1-g

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


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

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

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

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

Solving for variable 'g'.

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

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

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

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

Combine like terms: -1g + g = 0
-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 = 5.0277777793
4.694444446 + 4.333333333g + g2 = 5.0277777793

Factor a perfect square on the left side:
(g + 2.166666667)(g + 2.166666667) = 5.0277777793

Calculate the square root of the right side: 2.242270675

Break this problem into two subproblems by setting 
(g + 2.166666667) equal to 2.242270675 and -2.242270675.

Subproblem 1

g + 2.166666667 = 2.242270675 Simplifying g + 2.166666667 = 2.242270675 Reorder the terms: 2.166666667 + g = 2.242270675 Solving 2.166666667 + g = 2.242270675 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 = 2.242270675 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + g = 2.242270675 + -2.166666667 g = 2.242270675 + -2.166666667 Combine like terms: 2.242270675 + -2.166666667 = 0.075604008 g = 0.075604008 Simplifying g = 0.075604008

Subproblem 2

g + 2.166666667 = -2.242270675 Simplifying g + 2.166666667 = -2.242270675 Reorder the terms: 2.166666667 + g = -2.242270675 Solving 2.166666667 + g = -2.242270675 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 = -2.242270675 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + g = -2.242270675 + -2.166666667 g = -2.242270675 + -2.166666667 Combine like terms: -2.242270675 + -2.166666667 = -4.408937342 g = -4.408937342 Simplifying g = -4.408937342

Solution

The solution to the problem is based on the solutions from the subproblems. g = {0.075604008, -4.408937342}

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