5n(4n+7)=1

Solution for 5n(4n+7)=1 equation:

Simplifying
5n(4n + 7) = 1

Reorder the terms:
5n(7 + 4n) = 1
(7 * 5n + 4n * 5n) = 1
(35n + 20n2) = 1

Solving
35n + 20n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + 35n + 20n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 35n + 20n2 = 0

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

Divide each side by '20'.
-0.05 + 1.75n + n2 = 0.0

Move the constant term to the right:

Add '0.05' to each side of the equation.
-0.05 + 1.75n + 0.05 + n2 = 0.0 + 0.05

Reorder the terms:
-0.05 + 0.05 + 1.75n + n2 = 0.0 + 0.05

Combine like terms: -0.05 + 0.05 = 0.00
0.00 + 1.75n + n2 = 0.0 + 0.05
1.75n + n2 = 0.0 + 0.05

Combine like terms: 0.0 + 0.05 = 0.05
1.75n + n2 = 0.05

The n term is 1.75n.  Take half its coefficient (0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
1.75n + 0.765625 + n2 = 0.05 + 0.765625

Reorder the terms:
0.765625 + 1.75n + n2 = 0.05 + 0.765625

Combine like terms: 0.05 + 0.765625 = 0.815625
0.765625 + 1.75n + n2 = 0.815625

Factor a perfect square on the left side:
(n + 0.875)(n + 0.875) = 0.815625

Calculate the square root of the right side: 0.903119593

Break this problem into two subproblems by setting 
(n + 0.875) equal to 0.903119593 and -0.903119593.

Subproblem 1
n + 0.875 = 0.903119593

Simplifying
n + 0.875 = 0.903119593

Reorder the terms:
0.875 + n = 0.903119593

Solving
0.875 + n = 0.903119593

Solving for variable 'n'.

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

Add '-0.875' to each side of the equation.
0.875 + -0.875 + n = 0.903119593 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + n = 0.903119593 + -0.875
n = 0.903119593 + -0.875

Combine like terms: 0.903119593 + -0.875 = 0.028119593
n = 0.028119593

Simplifying
n = 0.028119593

Subproblem 2
n + 0.875 = -0.903119593

Simplifying
n + 0.875 = -0.903119593

Reorder the terms:
0.875 + n = -0.903119593

Solving
0.875 + n = -0.903119593

Solving for variable 'n'.

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

Add '-0.875' to each side of the equation.
0.875 + -0.875 + n = -0.903119593 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + n = -0.903119593 + -0.875
n = -0.903119593 + -0.875

Combine like terms: -0.903119593 + -0.875 = -1.778119593
n = -1.778119593

Simplifying
n = -1.778119593

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.028119593, -1.778119593}