4n^2+88n+22=0

Solution for 4n^2+88n+22=0 equation:

Simplifying
4n2 + 88n + 22 = 0

Reorder the terms:
22 + 88n + 4n2 = 0

Solving
22 + 88n + 4n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(11 + 44n + 2n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(11 + 44n + 2n2)' equal to zero and attempt to solve:

Simplifying
11 + 44n + 2n2 = 0

Solving
11 + 44n + 2n2 = 0

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

Divide each side by '2'.
5.5 + 22n + n2 = 0

Move the constant term to the right:

Add '-5.5' to each side of the equation.
5.5 + 22n + -5.5 + n2 = 0 + -5.5

Reorder the terms:
5.5 + -5.5 + 22n + n2 = 0 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + 22n + n2 = 0 + -5.5
22n + n2 = 0 + -5.5

Combine like terms: 0 + -5.5 = -5.5
22n + n2 = -5.5

The n term is 22n.  Take half its coefficient (11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
22n + 121 + n2 = -5.5 + 121

Reorder the terms:
121 + 22n + n2 = -5.5 + 121

Combine like terms: -5.5 + 121 = 115.5
121 + 22n + n2 = 115.5

Factor a perfect square on the left side:
(n + 11)(n + 11) = 115.5

Calculate the square root of the right side: 10.74709263

Break this problem into two subproblems by setting 
(n + 11) equal to 10.74709263 and -10.74709263.

Subproblem 1
n + 11 = 10.74709263

Simplifying
n + 11 = 10.74709263

Reorder the terms:
11 + n = 10.74709263

Solving
11 + n = 10.74709263

Solving for variable 'n'.

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

Add '-11' to each side of the equation.
11 + -11 + n = 10.74709263 + -11

Combine like terms: 11 + -11 = 0
0 + n = 10.74709263 + -11
n = 10.74709263 + -11

Combine like terms: 10.74709263 + -11 = -0.25290737
n = -0.25290737

Simplifying
n = -0.25290737

Subproblem 2
n + 11 = -10.74709263

Simplifying
n + 11 = -10.74709263

Reorder the terms:
11 + n = -10.74709263

Solving
11 + n = -10.74709263

Solving for variable 'n'.

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

Add '-11' to each side of the equation.
11 + -11 + n = -10.74709263 + -11

Combine like terms: 11 + -11 = 0
0 + n = -10.74709263 + -11
n = -10.74709263 + -11

Combine like terms: -10.74709263 + -11 = -21.74709263
n = -21.74709263

Simplifying
n = -21.74709263

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.25290737, -21.74709263}
Solution
n = {-0.25290737, -21.74709263}