2+4+6+8+2n=n(n+1)

Solution for 2+4+6+8+2n=n(n+1) equation:

Simplifying
2 + 4 + 6 + 8 + 2n = n(n + 1)

Combine like terms: 2 + 4 = 6
6 + 6 + 8 + 2n = n(n + 1)

Combine like terms: 6 + 6 = 12
12 + 8 + 2n = n(n + 1)

Combine like terms: 12 + 8 = 20
20 + 2n = n(n + 1)

Reorder the terms:
20 + 2n = n(1 + n)
20 + 2n = (1 * n + n * n)
20 + 2n = (1n + n2)

Solving
20 + 2n = 1n + n2

Solving for variable 'n'.

Combine like terms: 2n + -1n = 1n
20 + 1n + -1n2 = 1n + n2 + -1n + -1n2

Reorder the terms:
20 + 1n + -1n2 = 1n + -1n + n2 + -1n2

Combine like terms: 1n + -1n = 0
20 + 1n + -1n2 = 0 + n2 + -1n2
20 + 1n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
20 + 1n + -1n2 = 0

Factor a trinomial.
(5 + -1n)(4 + n) = 0

Subproblem 1
Set the factor '(5 + -1n)' equal to zero and attempt to solve:

Simplifying
5 + -1n = 0

Solving
5 + -1n = 0

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

Add '-5' to each side of the equation.
5 + -5 + -1n = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -1n = 0 + -5
-1n = 0 + -5

Combine like terms: 0 + -5 = -5
-1n = -5

Divide each side by '-1'.
n = 5

Simplifying
n = 5
Subproblem 2
Set the factor '(4 + n)' equal to zero and attempt to solve:

Simplifying
4 + n = 0

Solving
4 + n = 0

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

Add '-4' to each side of the equation.
4 + -4 + n = 0 + -4

Combine like terms: 4 + -4 = 0
0 + n = 0 + -4
n = 0 + -4

Combine like terms: 0 + -4 = -4
n = -4

Simplifying
n = -4
Solution
n = {5, -4}