-8n(n+7)=8n-40

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Solution for -8n(n+7)=8n-40 equation:


Simplifying
-8n(n + 7) = 8n + -40

Reorder the terms:
-8n(7 + n) = 8n + -40
(7 * -8n + n * -8n) = 8n + -40
(-56n + -8n2) = 8n + -40

Reorder the terms:
-56n + -8n2 = -40 + 8n

Solving
-56n + -8n2 = -40 + 8n

Solving for variable 'n'.

Reorder the terms:
40 + -56n + -8n + -8n2 = -40 + 8n + 40 + -8n

Combine like terms: -56n + -8n = -64n
40 + -64n + -8n2 = -40 + 8n + 40 + -8n

Reorder the terms:
40 + -64n + -8n2 = -40 + 40 + 8n + -8n

Combine like terms: -40 + 40 = 0
40 + -64n + -8n2 = 0 + 8n + -8n
40 + -64n + -8n2 = 8n + -8n

Combine like terms: 8n + -8n = 0
40 + -64n + -8n2 = 0

Factor out the Greatest Common Factor (GCF), '8'.
8(5 + -8n + -1n2) = 0

Ignore the factor 8.

Subproblem 1
Set the factor '(5 + -8n + -1n2)' equal to zero and attempt to solve:

Simplifying
5 + -8n + -1n2 = 0

Solving
5 + -8n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-5 + 8n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 8n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + 8n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 8n + n2 = 0 + 5
8n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
8n + n2 = 5

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 = 5 + 16

Reorder the terms:
16 + 8n + n2 = 5 + 16

Combine like terms: 5 + 16 = 21
16 + 8n + n2 = 21

Factor a perfect square on the left side:
(n + 4)(n + 4) = 21

Calculate the square root of the right side: 4.582575695

Break this problem into two subproblems by setting 
(n + 4) equal to 4.582575695 and -4.582575695.


Subproblem 1
n + 4 = 4.582575695

Simplifying
n + 4 = 4.582575695

Reorder the terms:
4 + n = 4.582575695

Solving
4 + n = 4.582575695

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 = 4.582575695 + -4

Combine like terms: 4 + -4 = 0
0 + n = 4.582575695 + -4
n = 4.582575695 + -4

Combine like terms: 4.582575695 + -4 = 0.582575695
n = 0.582575695

Simplifying
n = 0.582575695


Subproblem 2
n + 4 = -4.582575695

Simplifying
n + 4 = -4.582575695

Reorder the terms:
4 + n = -4.582575695

Solving
4 + n = -4.582575695

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 = -4.582575695 + -4

Combine like terms: 4 + -4 = 0
0 + n = -4.582575695 + -4
n = -4.582575695 + -4

Combine like terms: -4.582575695 + -4 = -8.582575695
n = -8.582575695

Simplifying
n = -8.582575695


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.582575695, -8.582575695}
Solution
n = {0.582575695, -8.582575695}

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