4n^2+2n=8

Solution for 4n^2+2n=8 equation:

Simplifying
4n2 + 2n = 8

Reorder the terms:
2n + 4n2 = 8

Solving
2n + 4n2 = 8

Solving for variable 'n'.

Reorder the terms:
-8 + 2n + 4n2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 2n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-4 + n + 2n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-4 + n + 2n2)' equal to zero and attempt to solve:

Simplifying
-4 + n + 2n2 = 0

Solving
-4 + n + 2n2 = 0

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

Divide each side by '2'.
-2 + 0.5n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.5n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.5n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.5n + n2 = 0 + 2
0.5n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.5n + n2 = 2

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

Add '0.25' to each side of the equation.
0.5n + 0.25 + n2 = 2 + 0.25

Reorder the terms:
0.25 + 0.5n + n2 = 2 + 0.25

Combine like terms: 2 + 0.25 = 2.25
0.25 + 0.5n + n2 = 2.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 2.25

Calculate the square root of the right side: 1.5

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.5 and -1.5.

Subproblem 1
n + 0.5 = 1.5

Simplifying
n + 0.5 = 1.5

Reorder the terms:
0.5 + n = 1.5

Solving
0.5 + n = 1.5

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 1.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 1.5 + -0.5
n = 1.5 + -0.5

Combine like terms: 1.5 + -0.5 = 1
n = 1

Simplifying
n = 1

Subproblem 2
n + 0.5 = -1.5

Simplifying
n + 0.5 = -1.5

Reorder the terms:
0.5 + n = -1.5

Solving
0.5 + n = -1.5

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -1.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -1.5 + -0.5
n = -1.5 + -0.5

Combine like terms: -1.5 + -0.5 = -2
n = -2

Simplifying
n = -2

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1, -2}
Solution
n = {1, -2}