n(n+1)=34

Solution for n(n+1)=34 equation:

Simplifying
n(n + 1) = 34

Reorder the terms:
n(1 + n) = 34
(1 * n + n * n) = 34
(1n + n2) = 34

Solving
1n + n2 = 34

Solving for variable 'n'.

Reorder the terms:
-34 + 1n + n2 = 34 + -34

Combine like terms: 34 + -34 = 0
-34 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '34' to each side of the equation.
-34 + 1n + 34 + n2 = 0 + 34

Reorder the terms:
-34 + 34 + 1n + n2 = 0 + 34

Combine like terms: -34 + 34 = 0
0 + 1n + n2 = 0 + 34
1n + n2 = 0 + 34

Combine like terms: 0 + 34 = 34
1n + n2 = 34

The n term is 1n.  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.
n + 0.25 + n2 = 34 + 0.25

Reorder the terms:
0.25 + n + n2 = 34 + 0.25

Combine like terms: 34 + 0.25 = 34.25
0.25 + n + n2 = 34.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 34.25

Calculate the square root of the right side: 5.852349955

Break this problem into two subproblems by setting 
(n + 0.5) equal to 5.852349955 and -5.852349955.

Subproblem 1
n + 0.5 = 5.852349955

Simplifying
n + 0.5 = 5.852349955

Reorder the terms:
0.5 + n = 5.852349955

Solving
0.5 + n = 5.852349955

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 = 5.852349955 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 5.852349955 + -0.5
n = 5.852349955 + -0.5

Combine like terms: 5.852349955 + -0.5 = 5.352349955
n = 5.352349955

Simplifying
n = 5.352349955

Subproblem 2
n + 0.5 = -5.852349955

Simplifying
n + 0.5 = -5.852349955

Reorder the terms:
0.5 + n = -5.852349955

Solving
0.5 + n = -5.852349955

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 = -5.852349955 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -5.852349955 + -0.5
n = -5.852349955 + -0.5

Combine like terms: -5.852349955 + -0.5 = -6.352349955
n = -6.352349955

Simplifying
n = -6.352349955

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {5.352349955, -6.352349955}