8n^2+69n+10=0

Solution for 8n^2+69n+10=0 equation:

Simplifying
8n2 + 69n + 10 = 0

Reorder the terms:
10 + 69n + 8n2 = 0

Solving
10 + 69n + 8n2 = 0

Solving for variable 'n'.

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

Divide each side by '8'.
1.25 + 8.625n + n2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + 8.625n + -1.25 + n2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + 8.625n + n2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + 8.625n + n2 = 0 + -1.25
8.625n + n2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
8.625n + n2 = -1.25

The n term is 8.625n.  Take half its coefficient (4.3125).
Square it (18.59765625) and add it to both sides.

Add '18.59765625' to each side of the equation.
8.625n + 18.59765625 + n2 = -1.25 + 18.59765625

Reorder the terms:
18.59765625 + 8.625n + n2 = -1.25 + 18.59765625

Combine like terms: -1.25 + 18.59765625 = 17.34765625
18.59765625 + 8.625n + n2 = 17.34765625

Factor a perfect square on the left side:
(n + 4.3125)(n + 4.3125) = 17.34765625

Calculate the square root of the right side: 4.16505177

Break this problem into two subproblems by setting 
(n + 4.3125) equal to 4.16505177 and -4.16505177.

Subproblem 1
n + 4.3125 = 4.16505177

Simplifying
n + 4.3125 = 4.16505177

Reorder the terms:
4.3125 + n = 4.16505177

Solving
4.3125 + n = 4.16505177

Solving for variable 'n'.

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

Add '-4.3125' to each side of the equation.
4.3125 + -4.3125 + n = 4.16505177 + -4.3125

Combine like terms: 4.3125 + -4.3125 = 0.0000
0.0000 + n = 4.16505177 + -4.3125
n = 4.16505177 + -4.3125

Combine like terms: 4.16505177 + -4.3125 = -0.14744823
n = -0.14744823

Simplifying
n = -0.14744823

Subproblem 2
n + 4.3125 = -4.16505177

Simplifying
n + 4.3125 = -4.16505177

Reorder the terms:
4.3125 + n = -4.16505177

Solving
4.3125 + n = -4.16505177

Solving for variable 'n'.

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

Add '-4.3125' to each side of the equation.
4.3125 + -4.3125 + n = -4.16505177 + -4.3125

Combine like terms: 4.3125 + -4.3125 = 0.0000
0.0000 + n = -4.16505177 + -4.3125
n = -4.16505177 + -4.3125

Combine like terms: -4.16505177 + -4.3125 = -8.47755177
n = -8.47755177

Simplifying
n = -8.47755177

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.14744823, -8.47755177}