6n^2+3n=5

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

Simplifying
6n2 + 3n = 5

Reorder the terms:
3n + 6n2 = 5

Solving
3n + 6n2 = 5

Solving for variable 'n'.

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

Combine like terms: 5 + -5 = 0
-5 + 3n + 6n2 = 0

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

Divide each side by '6'.
-0.8333333333 + 0.5n + n2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + 0.5n + 0.8333333333 + n2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + 0.5n + n2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + 0.5n + n2 = 0 + 0.8333333333
0.5n + n2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
0.5n + n2 = 0.8333333333

The n term is 0.5n.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5n + 0.0625 + n2 = 0.8333333333 + 0.0625

Reorder the terms:
0.0625 + 0.5n + n2 = 0.8333333333 + 0.0625

Combine like terms: 0.8333333333 + 0.0625 = 0.8958333333
0.0625 + 0.5n + n2 = 0.8958333333

Factor a perfect square on the left side:
(n + 0.25)(n + 0.25) = 0.8958333333

Calculate the square root of the right side: 0.946484724

Break this problem into two subproblems by setting 
(n + 0.25) equal to 0.946484724 and -0.946484724.

Subproblem 1
n + 0.25 = 0.946484724

Simplifying
n + 0.25 = 0.946484724

Reorder the terms:
0.25 + n = 0.946484724

Solving
0.25 + n = 0.946484724

Solving for variable 'n'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + n = 0.946484724 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + n = 0.946484724 + -0.25
n = 0.946484724 + -0.25

Combine like terms: 0.946484724 + -0.25 = 0.696484724
n = 0.696484724

Simplifying
n = 0.696484724

Subproblem 2
n + 0.25 = -0.946484724

Simplifying
n + 0.25 = -0.946484724

Reorder the terms:
0.25 + n = -0.946484724

Solving
0.25 + n = -0.946484724

Solving for variable 'n'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + n = -0.946484724 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + n = -0.946484724 + -0.25
n = -0.946484724 + -0.25

Combine like terms: -0.946484724 + -0.25 = -1.196484724
n = -1.196484724

Simplifying
n = -1.196484724

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.696484724, -1.196484724}