40-(4n+20)n=2

Solution for 40-(4n+20)n=2 equation:

Simplifying
40 + -1(4n + 20) * n = 2

Reorder the terms:
40 + -1(20 + 4n) * n = 2

Reorder the terms for easier multiplication:
40 + -1n(20 + 4n) = 2
40 + (20 * -1n + 4n * -1n) = 2
40 + (-20n + -4n2) = 2

Solving
40 + -20n + -4n2 = 2

Solving for variable 'n'.

Reorder the terms:
40 + -2 + -20n + -4n2 = 2 + -2

Combine like terms: 40 + -2 = 38
38 + -20n + -4n2 = 2 + -2

Combine like terms: 2 + -2 = 0
38 + -20n + -4n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(19 + -10n + -2n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(19 + -10n + -2n2)' equal to zero and attempt to solve:

Simplifying
19 + -10n + -2n2 = 0

Solving
19 + -10n + -2n2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-9.5 + 5n + n2 = 0

Move the constant term to the right:

Add '9.5' to each side of the equation.
-9.5 + 5n + 9.5 + n2 = 0 + 9.5

Reorder the terms:
-9.5 + 9.5 + 5n + n2 = 0 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + 5n + n2 = 0 + 9.5
5n + n2 = 0 + 9.5

Combine like terms: 0 + 9.5 = 9.5
5n + n2 = 9.5

The n term is 5n.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5n + 6.25 + n2 = 9.5 + 6.25

Reorder the terms:
6.25 + 5n + n2 = 9.5 + 6.25

Combine like terms: 9.5 + 6.25 = 15.75
6.25 + 5n + n2 = 15.75

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 15.75

Calculate the square root of the right side: 3.968626967

Break this problem into two subproblems by setting 
(n + 2.5) equal to 3.968626967 and -3.968626967.

Subproblem 1
n + 2.5 = 3.968626967

Simplifying
n + 2.5 = 3.968626967

Reorder the terms:
2.5 + n = 3.968626967

Solving
2.5 + n = 3.968626967

Solving for variable 'n'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + n = 3.968626967 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + n = 3.968626967 + -2.5
n = 3.968626967 + -2.5

Combine like terms: 3.968626967 + -2.5 = 1.468626967
n = 1.468626967

Simplifying
n = 1.468626967

Subproblem 2
n + 2.5 = -3.968626967

Simplifying
n + 2.5 = -3.968626967

Reorder the terms:
2.5 + n = -3.968626967

Solving
2.5 + n = -3.968626967

Solving for variable 'n'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + n = -3.968626967 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + n = -3.968626967 + -2.5
n = -3.968626967 + -2.5

Combine like terms: -3.968626967 + -2.5 = -6.468626967
n = -6.468626967

Simplifying
n = -6.468626967

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.468626967, -6.468626967}
Solution
n = {1.468626967, -6.468626967}