3n(n+5)+2=26

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Solution for 3n(n+5)+2=26 equation: 


Simplifying
3n(n + 5) + 2 = 26

Reorder the terms:
3n(5 + n) + 2 = 26
(5 * 3n + n * 3n) + 2 = 26
(15n + 3n2) + 2 = 26

Reorder the terms:
2 + 15n + 3n2 = 26

Solving
2 + 15n + 3n2 = 26

Solving for variable 'n'.

Reorder the terms:
2 + -26 + 15n + 3n2 = 26 + -26

Combine like terms: 2 + -26 = -24
-24 + 15n + 3n2 = 26 + -26

Combine like terms: 26 + -26 = 0
-24 + 15n + 3n2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-8 + 5n + n2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-8 + 5n + n2)' equal to zero and attempt to solve:

Simplifying
-8 + 5n + n2 = 0

Solving
-8 + 5n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 5n + 8 + n2 = 0 + 8

Reorder the terms:
-8 + 8 + 5n + n2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 5n + n2 = 0 + 8
5n + n2 = 0 + 8

Combine like terms: 0 + 8 = 8
5n + n2 = 8

The n term is 5n.  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.
5n + 6.25 + n2 = 8 + 6.25

Reorder the terms:
6.25 + 5n + n2 = 8 + 6.25

Combine like terms: 8 + 6.25 = 14.25
6.25 + 5n + n2 = 14.25

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(n + 2.5) equal to 3.774917218 and -3.774917218.


Subproblem 1
n + 2.5 = 3.774917218

Simplifying
n + 2.5 = 3.774917218

Reorder the terms:
2.5 + n = 3.774917218

Solving
2.5 + n = 3.774917218

Solving for variable 'n'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + n = 3.774917218 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + n = 3.774917218 + -2.5
n = 3.774917218 + -2.5

Combine like terms: 3.774917218 + -2.5 = 1.274917218
n = 1.274917218

Simplifying
n = 1.274917218


Subproblem 2
n + 2.5 = -3.774917218

Simplifying
n + 2.5 = -3.774917218

Reorder the terms:
2.5 + n = -3.774917218

Solving
2.5 + n = -3.774917218

Solving for variable 'n'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + n = -3.774917218 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + n = -3.774917218 + -2.5
n = -3.774917218 + -2.5

Combine like terms: -3.774917218 + -2.5 = -6.274917218
n = -6.274917218

Simplifying
n = -6.274917218


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.274917218, -6.274917218}
Solution
n = {1.274917218, -6.274917218}

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