n^2-8n-65=5

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Solution for n^2-8n-65=5 equation: 


Simplifying
n2 + -8n + -65 = 5

Reorder the terms:
-65 + -8n + n2 = 5

Solving
-65 + -8n + n2 = 5

Solving for variable 'n'.

Reorder the terms:
-65 + -5 + -8n + n2 = 5 + -5

Combine like terms: -65 + -5 = -70
-70 + -8n + n2 = 5 + -5

Combine like terms: 5 + -5 = 0
-70 + -8n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '70' to each side of the equation.
-70 + -8n + 70 + n2 = 0 + 70

Reorder the terms:
-70 + 70 + -8n + n2 = 0 + 70

Combine like terms: -70 + 70 = 0
0 + -8n + n2 = 0 + 70
-8n + n2 = 0 + 70

Combine like terms: 0 + 70 = 70
-8n + n2 = 70

The n term is -8n.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8n + 16 + n2 = 70 + 16

Reorder the terms:
16 + -8n + n2 = 70 + 16

Combine like terms: 70 + 16 = 86
16 + -8n + n2 = 86

Factor a perfect square on the left side:
(n + -4)(n + -4) = 86

Calculate the square root of the right side: 9.273618495

Break this problem into two subproblems by setting 
(n + -4) equal to 9.273618495 and -9.273618495.


Subproblem 1
n + -4 = 9.273618495

Simplifying
n + -4 = 9.273618495

Reorder the terms:
-4 + n = 9.273618495

Solving
-4 + n = 9.273618495

Solving for variable 'n'.

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

Add '4' to each side of the equation.
-4 + 4 + n = 9.273618495 + 4

Combine like terms: -4 + 4 = 0
0 + n = 9.273618495 + 4
n = 9.273618495 + 4

Combine like terms: 9.273618495 + 4 = 13.273618495
n = 13.273618495

Simplifying
n = 13.273618495


Subproblem 2
n + -4 = -9.273618495

Simplifying
n + -4 = -9.273618495

Reorder the terms:
-4 + n = -9.273618495

Solving
-4 + n = -9.273618495

Solving for variable 'n'.

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

Add '4' to each side of the equation.
-4 + 4 + n = -9.273618495 + 4

Combine like terms: -4 + 4 = 0
0 + n = -9.273618495 + 4
n = -9.273618495 + 4

Combine like terms: -9.273618495 + 4 = -5.273618495
n = -5.273618495

Simplifying
n = -5.273618495


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {13.273618495, -5.273618495}

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