n^2+8n-6=5n+3

Solution for n^2+8n-6=5n+3 equation:

Simplifying
n2 + 8n + -6 = 5n + 3

Reorder the terms:
-6 + 8n + n2 = 5n + 3

Reorder the terms:
-6 + 8n + n2 = 3 + 5n

Solving
-6 + 8n + n2 = 3 + 5n

Solving for variable 'n'.

Reorder the terms:
-6 + -3 + 8n + -5n + n2 = 3 + 5n + -3 + -5n

Combine like terms: -6 + -3 = -9
-9 + 8n + -5n + n2 = 3 + 5n + -3 + -5n

Combine like terms: 8n + -5n = 3n
-9 + 3n + n2 = 3 + 5n + -3 + -5n

Reorder the terms:
-9 + 3n + n2 = 3 + -3 + 5n + -5n

Combine like terms: 3 + -3 = 0
-9 + 3n + n2 = 0 + 5n + -5n
-9 + 3n + n2 = 5n + -5n

Combine like terms: 5n + -5n = 0
-9 + 3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 3n + 9 + n2 = 0 + 9

Reorder the terms:
-9 + 9 + 3n + n2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 3n + n2 = 0 + 9
3n + n2 = 0 + 9

Combine like terms: 0 + 9 = 9
3n + n2 = 9

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = 9 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 9 + 2.25

Combine like terms: 9 + 2.25 = 11.25
2.25 + 3n + n2 = 11.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(n + 1.5) equal to 3.354101966 and -3.354101966.

Subproblem 1
n + 1.5 = 3.354101966

Simplifying
n + 1.5 = 3.354101966

Reorder the terms:
1.5 + n = 3.354101966

Solving
1.5 + n = 3.354101966

Solving for variable 'n'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + n = 3.354101966 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + n = 3.354101966 + -1.5
n = 3.354101966 + -1.5

Combine like terms: 3.354101966 + -1.5 = 1.854101966
n = 1.854101966

Simplifying
n = 1.854101966

Subproblem 2
n + 1.5 = -3.354101966

Simplifying
n + 1.5 = -3.354101966

Reorder the terms:
1.5 + n = -3.354101966

Solving
1.5 + n = -3.354101966

Solving for variable 'n'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + n = -3.354101966 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + n = -3.354101966 + -1.5
n = -3.354101966 + -1.5

Combine like terms: -3.354101966 + -1.5 = -4.854101966
n = -4.854101966

Simplifying
n = -4.854101966

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.854101966, -4.854101966}