n^2+14n-31=0

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Solution for n^2+14n-31=0 equation: 


Simplifying
n2 + 14n + -31 = 0

Reorder the terms:
-31 + 14n + n2 = 0

Solving
-31 + 14n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + 14n + 31 + n2 = 0 + 31

Reorder the terms:
-31 + 31 + 14n + n2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + 14n + n2 = 0 + 31
14n + n2 = 0 + 31

Combine like terms: 0 + 31 = 31
14n + n2 = 31

The n term is 14n.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14n + 49 + n2 = 31 + 49

Reorder the terms:
49 + 14n + n2 = 31 + 49

Combine like terms: 31 + 49 = 80
49 + 14n + n2 = 80

Factor a perfect square on the left side:
(n + 7)(n + 7) = 80

Calculate the square root of the right side: 8.94427191

Break this problem into two subproblems by setting 
(n + 7) equal to 8.94427191 and -8.94427191.


Subproblem 1
n + 7 = 8.94427191

Simplifying
n + 7 = 8.94427191

Reorder the terms:
7 + n = 8.94427191

Solving
7 + n = 8.94427191

Solving for variable 'n'.

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

Add '-7' to each side of the equation.
7 + -7 + n = 8.94427191 + -7

Combine like terms: 7 + -7 = 0
0 + n = 8.94427191 + -7
n = 8.94427191 + -7

Combine like terms: 8.94427191 + -7 = 1.94427191
n = 1.94427191

Simplifying
n = 1.94427191


Subproblem 2
n + 7 = -8.94427191

Simplifying
n + 7 = -8.94427191

Reorder the terms:
7 + n = -8.94427191

Solving
7 + n = -8.94427191

Solving for variable 'n'.

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

Add '-7' to each side of the equation.
7 + -7 + n = -8.94427191 + -7

Combine like terms: 7 + -7 = 0
0 + n = -8.94427191 + -7
n = -8.94427191 + -7

Combine like terms: -8.94427191 + -7 = -15.94427191
n = -15.94427191

Simplifying
n = -15.94427191


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.94427191, -15.94427191}

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