n^2+8+15n=0

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Solution for n^2+8+15n=0 equation: 


Simplifying
n2 + 8 + 15n = 0

Reorder the terms:
8 + 15n + n2 = 0

Solving
8 + 15n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 15n + -8 + n2 = 0 + -8

Reorder the terms:
8 + -8 + 15n + n2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 15n + n2 = 0 + -8
15n + n2 = 0 + -8

Combine like terms: 0 + -8 = -8
15n + n2 = -8

The n term is 15n.  Take half its coefficient (7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
15n + 56.25 + n2 = -8 + 56.25

Reorder the terms:
56.25 + 15n + n2 = -8 + 56.25

Combine like terms: -8 + 56.25 = 48.25
56.25 + 15n + n2 = 48.25

Factor a perfect square on the left side:
(n + 7.5)(n + 7.5) = 48.25

Calculate the square root of the right side: 6.946221995

Break this problem into two subproblems by setting 
(n + 7.5) equal to 6.946221995 and -6.946221995.


Subproblem 1
n + 7.5 = 6.946221995

Simplifying
n + 7.5 = 6.946221995

Reorder the terms:
7.5 + n = 6.946221995

Solving
7.5 + n = 6.946221995

Solving for variable 'n'.

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

Add '-7.5' to each side of the equation.
7.5 + -7.5 + n = 6.946221995 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + n = 6.946221995 + -7.5
n = 6.946221995 + -7.5

Combine like terms: 6.946221995 + -7.5 = -0.553778005
n = -0.553778005

Simplifying
n = -0.553778005


Subproblem 2
n + 7.5 = -6.946221995

Simplifying
n + 7.5 = -6.946221995

Reorder the terms:
7.5 + n = -6.946221995

Solving
7.5 + n = -6.946221995

Solving for variable 'n'.

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

Add '-7.5' to each side of the equation.
7.5 + -7.5 + n = -6.946221995 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + n = -6.946221995 + -7.5
n = -6.946221995 + -7.5

Combine like terms: -6.946221995 + -7.5 = -14.446221995
n = -14.446221995

Simplifying
n = -14.446221995


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.553778005, -14.446221995}

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