n^2+8n-21=0

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Solution for n^2+8n-21=0 equation: 


Simplifying
n2 + 8n + -21 = 0

Reorder the terms:
-21 + 8n + n2 = 0

Solving
-21 + 8n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 8n + 21 + n2 = 0 + 21

Reorder the terms:
-21 + 21 + 8n + n2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 8n + n2 = 0 + 21
8n + n2 = 0 + 21

Combine like terms: 0 + 21 = 21
8n + n2 = 21

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 = 21 + 16

Reorder the terms:
16 + 8n + n2 = 21 + 16

Combine like terms: 21 + 16 = 37
16 + 8n + n2 = 37

Factor a perfect square on the left side:
(n + 4)(n + 4) = 37

Calculate the square root of the right side: 6.08276253

Break this problem into two subproblems by setting 
(n + 4) equal to 6.08276253 and -6.08276253.


Subproblem 1
n + 4 = 6.08276253

Simplifying
n + 4 = 6.08276253

Reorder the terms:
4 + n = 6.08276253

Solving
4 + n = 6.08276253

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 = 6.08276253 + -4

Combine like terms: 4 + -4 = 0
0 + n = 6.08276253 + -4
n = 6.08276253 + -4

Combine like terms: 6.08276253 + -4 = 2.08276253
n = 2.08276253

Simplifying
n = 2.08276253


Subproblem 2
n + 4 = -6.08276253

Simplifying
n + 4 = -6.08276253

Reorder the terms:
4 + n = -6.08276253

Solving
4 + n = -6.08276253

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 = -6.08276253 + -4

Combine like terms: 4 + -4 = 0
0 + n = -6.08276253 + -4
n = -6.08276253 + -4

Combine like terms: -6.08276253 + -4 = -10.08276253
n = -10.08276253

Simplifying
n = -10.08276253


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.08276253, -10.08276253}

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