4n^2+4n+1=75

Solution for 4n^2+4n+1=75 equation:

Simplifying
4n2 + 4n + 1 = 75

Reorder the terms:
1 + 4n + 4n2 = 75

Solving
1 + 4n + 4n2 = 75

Solving for variable 'n'.

Reorder the terms:
1 + -75 + 4n + 4n2 = 75 + -75

Combine like terms: 1 + -75 = -74
-74 + 4n + 4n2 = 75 + -75

Combine like terms: 75 + -75 = 0
-74 + 4n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-37 + 2n + 2n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-37 + 2n + 2n2)' equal to zero and attempt to solve:

Simplifying
-37 + 2n + 2n2 = 0

Solving
-37 + 2n + 2n2 = 0

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

Divide each side by '2'.
-18.5 + n + n2 = 0

Move the constant term to the right:

Add '18.5' to each side of the equation.
-18.5 + n + 18.5 + n2 = 0 + 18.5

Reorder the terms:
-18.5 + 18.5 + n + n2 = 0 + 18.5

Combine like terms: -18.5 + 18.5 = 0.0
0.0 + n + n2 = 0 + 18.5
n + n2 = 0 + 18.5

Combine like terms: 0 + 18.5 = 18.5
n + n2 = 18.5

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + n2 = 18.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 18.5 + 0.25

Combine like terms: 18.5 + 0.25 = 18.75
1.25 + n2 = 18.75

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 18.75

Calculate the square root of the right side: 4.330127019

Break this problem into two subproblems by setting 
(n + 0.5) equal to 4.330127019 and -4.330127019.

Subproblem 1
n + 0.5 = 4.330127019

Simplifying
n + 0.5 = 4.330127019

Reorder the terms:
0.5 + n = 4.330127019

Solving
0.5 + n = 4.330127019

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 4.330127019 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 4.330127019 + -0.5
n = 4.330127019 + -0.5

Combine like terms: 4.330127019 + -0.5 = 3.830127019
n = 3.830127019

Simplifying
n = 3.830127019

Subproblem 2
n + 0.5 = -4.330127019

Simplifying
n + 0.5 = -4.330127019

Reorder the terms:
0.5 + n = -4.330127019

Solving
0.5 + n = -4.330127019

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -4.330127019 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -4.330127019 + -0.5
n = -4.330127019 + -0.5

Combine like terms: -4.330127019 + -0.5 = -4.830127019
n = -4.830127019

Simplifying
n = -4.830127019

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {3.830127019, -4.830127019}
Solution
n = {3.830127019, -4.830127019}