8n(3n+7)=672

Solution for 8n(3n+7)=672 equation:

Simplifying
8n(3n + 7) = 672

Reorder the terms:
8n(7 + 3n) = 672
(7 * 8n + 3n * 8n) = 672
(56n + 24n2) = 672

Solving
56n + 24n2 = 672

Solving for variable 'n'.

Reorder the terms:
-672 + 56n + 24n2 = 672 + -672

Combine like terms: 672 + -672 = 0
-672 + 56n + 24n2 = 0

Factor out the Greatest Common Factor (GCF), '8'.
8(-84 + 7n + 3n2) = 0

Ignore the factor 8.
Subproblem 1
Set the factor '(-84 + 7n + 3n2)' equal to zero and attempt to solve:

Simplifying
-84 + 7n + 3n2 = 0

Solving
-84 + 7n + 3n2 = 0

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

Divide each side by '3'.
-28 + 2.333333333n + n2 = 0

Move the constant term to the right:

Add '28' to each side of the equation.
-28 + 2.333333333n + 28 + n2 = 0 + 28

Reorder the terms:
-28 + 28 + 2.333333333n + n2 = 0 + 28

Combine like terms: -28 + 28 = 0
0 + 2.333333333n + n2 = 0 + 28
2.333333333n + n2 = 0 + 28

Combine like terms: 0 + 28 = 28
2.333333333n + n2 = 28

The n term is 2.333333333n.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333n + 1.361111112 + n2 = 28 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333n + n2 = 28 + 1.361111112

Combine like terms: 28 + 1.361111112 = 29.361111112
1.361111112 + 2.333333333n + n2 = 29.361111112

Factor a perfect square on the left side:
(n + 1.166666667)(n + 1.166666667) = 29.361111112

Calculate the square root of the right side: 5.418589402

Break this problem into two subproblems by setting 
(n + 1.166666667) equal to 5.418589402 and -5.418589402.

Subproblem 1
n + 1.166666667 = 5.418589402

Simplifying
n + 1.166666667 = 5.418589402

Reorder the terms:
1.166666667 + n = 5.418589402

Solving
1.166666667 + n = 5.418589402

Solving for variable 'n'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + n = 5.418589402 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + n = 5.418589402 + -1.166666667
n = 5.418589402 + -1.166666667

Combine like terms: 5.418589402 + -1.166666667 = 4.251922735
n = 4.251922735

Simplifying
n = 4.251922735

Subproblem 2
n + 1.166666667 = -5.418589402

Simplifying
n + 1.166666667 = -5.418589402

Reorder the terms:
1.166666667 + n = -5.418589402

Solving
1.166666667 + n = -5.418589402

Solving for variable 'n'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + n = -5.418589402 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + n = -5.418589402 + -1.166666667
n = -5.418589402 + -1.166666667

Combine like terms: -5.418589402 + -1.166666667 = -6.585256069
n = -6.585256069

Simplifying
n = -6.585256069

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.251922735, -6.585256069}
Solution
n = {4.251922735, -6.585256069}