n^2-16n-31=8

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Solution for n^2-16n-31=8 equation: 


Simplifying
n2 + -16n + -31 = 8

Reorder the terms:
-31 + -16n + n2 = 8

Solving
-31 + -16n + n2 = 8

Solving for variable 'n'.

Reorder the terms:
-31 + -8 + -16n + n2 = 8 + -8

Combine like terms: -31 + -8 = -39
-39 + -16n + n2 = 8 + -8

Combine like terms: 8 + -8 = 0
-39 + -16n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + -16n + 39 + n2 = 0 + 39

Reorder the terms:
-39 + 39 + -16n + n2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + -16n + n2 = 0 + 39
-16n + n2 = 0 + 39

Combine like terms: 0 + 39 = 39
-16n + n2 = 39

The n term is -16n.  Take half its coefficient (-8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
-16n + 64 + n2 = 39 + 64

Reorder the terms:
64 + -16n + n2 = 39 + 64

Combine like terms: 39 + 64 = 103
64 + -16n + n2 = 103

Factor a perfect square on the left side:
(n + -8)(n + -8) = 103

Calculate the square root of the right side: 10.148891565

Break this problem into two subproblems by setting 
(n + -8) equal to 10.148891565 and -10.148891565.


Subproblem 1
n + -8 = 10.148891565

Simplifying
n + -8 = 10.148891565

Reorder the terms:
-8 + n = 10.148891565

Solving
-8 + n = 10.148891565

Solving for variable 'n'.

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

Add '8' to each side of the equation.
-8 + 8 + n = 10.148891565 + 8

Combine like terms: -8 + 8 = 0
0 + n = 10.148891565 + 8
n = 10.148891565 + 8

Combine like terms: 10.148891565 + 8 = 18.148891565
n = 18.148891565

Simplifying
n = 18.148891565


Subproblem 2
n + -8 = -10.148891565

Simplifying
n + -8 = -10.148891565

Reorder the terms:
-8 + n = -10.148891565

Solving
-8 + n = -10.148891565

Solving for variable 'n'.

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

Add '8' to each side of the equation.
-8 + 8 + n = -10.148891565 + 8

Combine like terms: -8 + 8 = 0
0 + n = -10.148891565 + 8
n = -10.148891565 + 8

Combine like terms: -10.148891565 + 8 = -2.148891565
n = -2.148891565

Simplifying
n = -2.148891565


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {18.148891565, -2.148891565}

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