n^2+8n-51=0

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Solution for n^2+8n-51=0 equation: 


Simplifying
n2 + 8n + -51 = 0

Reorder the terms:
-51 + 8n + n2 = 0

Solving
-51 + 8n + 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 + 8n + 51 + n2 = 0 + 51

Reorder the terms:
-51 + 51 + 8n + n2 = 0 + 51

Combine like terms: -51 + 51 = 0
0 + 8n + n2 = 0 + 51
8n + n2 = 0 + 51

Combine like terms: 0 + 51 = 51
8n + n2 = 51

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 = 51 + 16

Reorder the terms:
16 + 8n + n2 = 51 + 16

Combine like terms: 51 + 16 = 67
16 + 8n + n2 = 67

Factor a perfect square on the left side:
(n + 4)(n + 4) = 67

Calculate the square root of the right side: 8.185352772

Break this problem into two subproblems by setting 
(n + 4) equal to 8.185352772 and -8.185352772.


Subproblem 1
n + 4 = 8.185352772

Simplifying
n + 4 = 8.185352772

Reorder the terms:
4 + n = 8.185352772

Solving
4 + n = 8.185352772

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 = 8.185352772 + -4

Combine like terms: 4 + -4 = 0
0 + n = 8.185352772 + -4
n = 8.185352772 + -4

Combine like terms: 8.185352772 + -4 = 4.185352772
n = 4.185352772

Simplifying
n = 4.185352772


Subproblem 2
n + 4 = -8.185352772

Simplifying
n + 4 = -8.185352772

Reorder the terms:
4 + n = -8.185352772

Solving
4 + n = -8.185352772

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 = -8.185352772 + -4

Combine like terms: 4 + -4 = 0
0 + n = -8.185352772 + -4
n = -8.185352772 + -4

Combine like terms: -8.185352772 + -4 = -12.185352772
n = -12.185352772

Simplifying
n = -12.185352772


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.185352772, -12.185352772}

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