n(7n+33)+7(n+1)=

Solution for n(7n+33)+7(n+1)= equation:

Simplifying
n(7n + 33) + 7(n + 1) = 0

Reorder the terms:
n(33 + 7n) + 7(n + 1) = 0
(33 * n + 7n * n) + 7(n + 1) = 0
(33n + 7n2) + 7(n + 1) = 0

Reorder the terms:
33n + 7n2 + 7(1 + n) = 0
33n + 7n2 + (1 * 7 + n * 7) = 0
33n + 7n2 + (7 + 7n) = 0

Reorder the terms:
7 + 33n + 7n + 7n2 = 0

Combine like terms: 33n + 7n = 40n
7 + 40n + 7n2 = 0

Solving
7 + 40n + 7n2 = 0

Solving for variable 'n'.

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

Divide each side by '7'.
1 + 5.714285714n + n2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 5.714285714n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + 5.714285714n + n2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 5.714285714n + n2 = 0 + -1
5.714285714n + n2 = 0 + -1

Combine like terms: 0 + -1 = -1
5.714285714n + n2 = -1

The n term is 5.714285714n.  Take half its coefficient (2.857142857).
Square it (8.163265305) and add it to both sides.

Add '8.163265305' to each side of the equation.
5.714285714n + 8.163265305 + n2 = -1 + 8.163265305

Reorder the terms:
8.163265305 + 5.714285714n + n2 = -1 + 8.163265305

Combine like terms: -1 + 8.163265305 = 7.163265305
8.163265305 + 5.714285714n + n2 = 7.163265305

Factor a perfect square on the left side:
(n + 2.857142857)(n + 2.857142857) = 7.163265305

Calculate the square root of the right side: 2.676427713

Break this problem into two subproblems by setting 
(n + 2.857142857) equal to 2.676427713 and -2.676427713.

Subproblem 1
n + 2.857142857 = 2.676427713

Simplifying
n + 2.857142857 = 2.676427713

Reorder the terms:
2.857142857 + n = 2.676427713

Solving
2.857142857 + n = 2.676427713

Solving for variable 'n'.

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

Add '-2.857142857' to each side of the equation.
2.857142857 + -2.857142857 + n = 2.676427713 + -2.857142857

Combine like terms: 2.857142857 + -2.857142857 = 0.000000000
0.000000000 + n = 2.676427713 + -2.857142857
n = 2.676427713 + -2.857142857

Combine like terms: 2.676427713 + -2.857142857 = -0.180715144
n = -0.180715144

Simplifying
n = -0.180715144

Subproblem 2
n + 2.857142857 = -2.676427713

Simplifying
n + 2.857142857 = -2.676427713

Reorder the terms:
2.857142857 + n = -2.676427713

Solving
2.857142857 + n = -2.676427713

Solving for variable 'n'.

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

Add '-2.857142857' to each side of the equation.
2.857142857 + -2.857142857 + n = -2.676427713 + -2.857142857

Combine like terms: 2.857142857 + -2.857142857 = 0.000000000
0.000000000 + n = -2.676427713 + -2.857142857
n = -2.676427713 + -2.857142857

Combine like terms: -2.676427713 + -2.857142857 = -5.53357057
n = -5.53357057

Simplifying
n = -5.53357057

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.180715144, -5.53357057}