n^2+16n+51=0

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Solution for n^2+16n+51=0 equation: 


Simplifying
n2 + 16n + 51 = 0

Reorder the terms:
51 + 16n + n2 = 0

Solving
51 + 16n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-51' to each side of the equation.
51 + 16n + -51 + n2 = 0 + -51

Reorder the terms:
51 + -51 + 16n + n2 = 0 + -51

Combine like terms: 51 + -51 = 0
0 + 16n + n2 = 0 + -51
16n + n2 = 0 + -51

Combine like terms: 0 + -51 = -51
16n + n2 = -51

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 = -51 + 64

Reorder the terms:
64 + 16n + n2 = -51 + 64

Combine like terms: -51 + 64 = 13
64 + 16n + n2 = 13

Factor a perfect square on the left side:
(n + 8)(n + 8) = 13

Calculate the square root of the right side: 3.605551275

Break this problem into two subproblems by setting 
(n + 8) equal to 3.605551275 and -3.605551275.


Subproblem 1
n + 8 = 3.605551275

Simplifying
n + 8 = 3.605551275

Reorder the terms:
8 + n = 3.605551275

Solving
8 + n = 3.605551275

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 = 3.605551275 + -8

Combine like terms: 8 + -8 = 0
0 + n = 3.605551275 + -8
n = 3.605551275 + -8

Combine like terms: 3.605551275 + -8 = -4.394448725
n = -4.394448725

Simplifying
n = -4.394448725


Subproblem 2
n + 8 = -3.605551275

Simplifying
n + 8 = -3.605551275

Reorder the terms:
8 + n = -3.605551275

Solving
8 + n = -3.605551275

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 = -3.605551275 + -8

Combine like terms: 8 + -8 = 0
0 + n = -3.605551275 + -8
n = -3.605551275 + -8

Combine like terms: -3.605551275 + -8 = -11.605551275
n = -11.605551275

Simplifying
n = -11.605551275


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-4.394448725, -11.605551275}

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