n^2-2n-2=38.5

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Solution for n^2-2n-2=38.5 equation: 


Simplifying
n2 + -2n + -2 = 38.5

Reorder the terms:
-2 + -2n + n2 = 38.5

Solving
-2 + -2n + n2 = 38.5

Solving for variable 'n'.

Reorder the terms:
-2 + -38.5 + -2n + n2 = 38.5 + -38.5

Combine like terms: -2 + -38.5 = -40.5
-40.5 + -2n + n2 = 38.5 + -38.5

Combine like terms: 38.5 + -38.5 = 0.0
-40.5 + -2n + n2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '40.5' to each side of the equation.
-40.5 + -2n + 40.5 + n2 = 0.0 + 40.5

Reorder the terms:
-40.5 + 40.5 + -2n + n2 = 0.0 + 40.5

Combine like terms: -40.5 + 40.5 = 0.0
0.0 + -2n + n2 = 0.0 + 40.5
-2n + n2 = 0.0 + 40.5

Combine like terms: 0.0 + 40.5 = 40.5
-2n + n2 = 40.5

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 40.5 + 1

Reorder the terms:
1 + -2n + n2 = 40.5 + 1

Combine like terms: 40.5 + 1 = 41.5
1 + -2n + n2 = 41.5

Factor a perfect square on the left side:
(n + -1)(n + -1) = 41.5

Calculate the square root of the right side: 6.442049363

Break this problem into two subproblems by setting 
(n + -1) equal to 6.442049363 and -6.442049363.


Subproblem 1
n + -1 = 6.442049363

Simplifying
n + -1 = 6.442049363

Reorder the terms:
-1 + n = 6.442049363

Solving
-1 + n = 6.442049363

Solving for variable 'n'.

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

Add '1' to each side of the equation.
-1 + 1 + n = 6.442049363 + 1

Combine like terms: -1 + 1 = 0
0 + n = 6.442049363 + 1
n = 6.442049363 + 1

Combine like terms: 6.442049363 + 1 = 7.442049363
n = 7.442049363

Simplifying
n = 7.442049363


Subproblem 2
n + -1 = -6.442049363

Simplifying
n + -1 = -6.442049363

Reorder the terms:
-1 + n = -6.442049363

Solving
-1 + n = -6.442049363

Solving for variable 'n'.

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

Add '1' to each side of the equation.
-1 + 1 + n = -6.442049363 + 1

Combine like terms: -1 + 1 = 0
0 + n = -6.442049363 + 1
n = -6.442049363 + 1

Combine like terms: -6.442049363 + 1 = -5.442049363
n = -5.442049363

Simplifying
n = -5.442049363


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {7.442049363, -5.442049363}

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