15+5g(g+5)=415

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Solution for 15+5g(g+5)=415 equation: 


Simplifying
15 + 5g(g + 5) = 415

Reorder the terms:
15 + 5g(5 + g) = 415
15 + (5 * 5g + g * 5g) = 415
15 + (25g + 5g2) = 415

Solving
15 + 25g + 5g2 = 415

Solving for variable 'g'.

Reorder the terms:
15 + -415 + 25g + 5g2 = 415 + -415

Combine like terms: 15 + -415 = -400
-400 + 25g + 5g2 = 415 + -415

Combine like terms: 415 + -415 = 0
-400 + 25g + 5g2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-80 + 5g + g2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-80 + 5g + g2)' equal to zero and attempt to solve:

Simplifying
-80 + 5g + g2 = 0

Solving
-80 + 5g + g2 = 0

Begin completing the square.

Move the constant term to the right:

Add '80' to each side of the equation.
-80 + 5g + 80 + g2 = 0 + 80

Reorder the terms:
-80 + 80 + 5g + g2 = 0 + 80

Combine like terms: -80 + 80 = 0
0 + 5g + g2 = 0 + 80
5g + g2 = 0 + 80

Combine like terms: 0 + 80 = 80
5g + g2 = 80

The g term is 5g.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5g + 6.25 + g2 = 80 + 6.25

Reorder the terms:
6.25 + 5g + g2 = 80 + 6.25

Combine like terms: 80 + 6.25 = 86.25
6.25 + 5g + g2 = 86.25

Factor a perfect square on the left side:
(g + 2.5)(g + 2.5) = 86.25

Calculate the square root of the right side: 9.287087811

Break this problem into two subproblems by setting 
(g + 2.5) equal to 9.287087811 and -9.287087811.


Subproblem 1
g + 2.5 = 9.287087811

Simplifying
g + 2.5 = 9.287087811

Reorder the terms:
2.5 + g = 9.287087811

Solving
2.5 + g = 9.287087811

Solving for variable 'g'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + g = 9.287087811 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + g = 9.287087811 + -2.5
g = 9.287087811 + -2.5

Combine like terms: 9.287087811 + -2.5 = 6.787087811
g = 6.787087811

Simplifying
g = 6.787087811


Subproblem 2
g + 2.5 = -9.287087811

Simplifying
g + 2.5 = -9.287087811

Reorder the terms:
2.5 + g = -9.287087811

Solving
2.5 + g = -9.287087811

Solving for variable 'g'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + g = -9.287087811 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + g = -9.287087811 + -2.5
g = -9.287087811 + -2.5

Combine like terms: -9.287087811 + -2.5 = -11.787087811
g = -11.787087811

Simplifying
g = -11.787087811


Solution
The solution to the problem is based on the solutions
from the subproblems.
g = {6.787087811, -11.787087811}
Solution
g = {6.787087811, -11.787087811}

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