8n(2n+5)=19

Solution for 8n(2n+5)=19 equation:

Simplifying
8n(2n + 5) = 19

Reorder the terms:
8n(5 + 2n) = 19
(5 * 8n + 2n * 8n) = 19
(40n + 16n2) = 19

Solving
40n + 16n2 = 19

Solving for variable 'n'.

Reorder the terms:
-19 + 40n + 16n2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + 40n + 16n2 = 0

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

Divide each side by '16'.
-1.1875 + 2.5n + n2 = 0

Move the constant term to the right:

Add '1.1875' to each side of the equation.
-1.1875 + 2.5n + 1.1875 + n2 = 0 + 1.1875

Reorder the terms:
-1.1875 + 1.1875 + 2.5n + n2 = 0 + 1.1875

Combine like terms: -1.1875 + 1.1875 = 0.0000
0.0000 + 2.5n + n2 = 0 + 1.1875
2.5n + n2 = 0 + 1.1875

Combine like terms: 0 + 1.1875 = 1.1875
2.5n + n2 = 1.1875

The n term is 2.5n.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5n + 1.5625 + n2 = 1.1875 + 1.5625

Reorder the terms:
1.5625 + 2.5n + n2 = 1.1875 + 1.5625

Combine like terms: 1.1875 + 1.5625 = 2.75
1.5625 + 2.5n + n2 = 2.75

Factor a perfect square on the left side:
(n + 1.25)(n + 1.25) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(n + 1.25) equal to 1.658312395 and -1.658312395.

Subproblem 1
n + 1.25 = 1.658312395

Simplifying
n + 1.25 = 1.658312395

Reorder the terms:
1.25 + n = 1.658312395

Solving
1.25 + n = 1.658312395

Solving for variable 'n'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + n = 1.658312395 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + n = 1.658312395 + -1.25
n = 1.658312395 + -1.25

Combine like terms: 1.658312395 + -1.25 = 0.408312395
n = 0.408312395

Simplifying
n = 0.408312395

Subproblem 2
n + 1.25 = -1.658312395

Simplifying
n + 1.25 = -1.658312395

Reorder the terms:
1.25 + n = -1.658312395

Solving
1.25 + n = -1.658312395

Solving for variable 'n'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + n = -1.658312395 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + n = -1.658312395 + -1.25
n = -1.658312395 + -1.25

Combine like terms: -1.658312395 + -1.25 = -2.908312395
n = -2.908312395

Simplifying
n = -2.908312395

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.408312395, -2.908312395}