4n^2+80n+40=0

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Solution for 4n^2+80n+40=0 equation: 


Simplifying
4n2 + 80n + 40 = 0

Reorder the terms:
40 + 80n + 4n2 = 0

Solving
40 + 80n + 4n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '4'.
4(10 + 20n + n2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(10 + 20n + n2)' equal to zero and attempt to solve:

Simplifying
10 + 20n + n2 = 0

Solving
10 + 20n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-10' to each side of the equation.
10 + 20n + -10 + n2 = 0 + -10

Reorder the terms:
10 + -10 + 20n + n2 = 0 + -10

Combine like terms: 10 + -10 = 0
0 + 20n + n2 = 0 + -10
20n + n2 = 0 + -10

Combine like terms: 0 + -10 = -10
20n + n2 = -10

The n term is 20n.  Take half its coefficient (10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
20n + 100 + n2 = -10 + 100

Reorder the terms:
100 + 20n + n2 = -10 + 100

Combine like terms: -10 + 100 = 90
100 + 20n + n2 = 90

Factor a perfect square on the left side:
(n + 10)(n + 10) = 90

Calculate the square root of the right side: 9.486832981

Break this problem into two subproblems by setting 
(n + 10) equal to 9.486832981 and -9.486832981.


Subproblem 1
n + 10 = 9.486832981

Simplifying
n + 10 = 9.486832981

Reorder the terms:
10 + n = 9.486832981

Solving
10 + n = 9.486832981

Solving for variable 'n'.

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

Add '-10' to each side of the equation.
10 + -10 + n = 9.486832981 + -10

Combine like terms: 10 + -10 = 0
0 + n = 9.486832981 + -10
n = 9.486832981 + -10

Combine like terms: 9.486832981 + -10 = -0.513167019
n = -0.513167019

Simplifying
n = -0.513167019


Subproblem 2
n + 10 = -9.486832981

Simplifying
n + 10 = -9.486832981

Reorder the terms:
10 + n = -9.486832981

Solving
10 + n = -9.486832981

Solving for variable 'n'.

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

Add '-10' to each side of the equation.
10 + -10 + n = -9.486832981 + -10

Combine like terms: 10 + -10 = 0
0 + n = -9.486832981 + -10
n = -9.486832981 + -10

Combine like terms: -9.486832981 + -10 = -19.486832981
n = -19.486832981

Simplifying
n = -19.486832981


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.513167019, -19.486832981}
Solution
n = {-0.513167019, -19.486832981}

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