n(n+1)=57

Solution for n(n+1)=57 equation:

Simplifying
n(n + 1) = 57

Reorder the terms:
n(1 + n) = 57
(1 * n + n * n) = 57
(1n + n2) = 57

Solving
1n + n2 = 57

Solving for variable 'n'.

Reorder the terms:
-57 + 1n + n2 = 57 + -57

Combine like terms: 57 + -57 = 0
-57 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '57' to each side of the equation.
-57 + 1n + 57 + n2 = 0 + 57

Reorder the terms:
-57 + 57 + 1n + n2 = 0 + 57

Combine like terms: -57 + 57 = 0
0 + 1n + n2 = 0 + 57
1n + n2 = 0 + 57

Combine like terms: 0 + 57 = 57
1n + n2 = 57

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 = 57 + 0.25

Reorder the terms:
0.25 + n + n2 = 57 + 0.25

Combine like terms: 57 + 0.25 = 57.25
0.25 + n + n2 = 57.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 57.25

Calculate the square root of the right side: 7.566372975

Break this problem into two subproblems by setting 
(n + 0.5) equal to 7.566372975 and -7.566372975.

Subproblem 1
n + 0.5 = 7.566372975

Simplifying
n + 0.5 = 7.566372975

Reorder the terms:
0.5 + n = 7.566372975

Solving
0.5 + n = 7.566372975

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 = 7.566372975 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 7.566372975 + -0.5
n = 7.566372975 + -0.5

Combine like terms: 7.566372975 + -0.5 = 7.066372975
n = 7.066372975

Simplifying
n = 7.066372975

Subproblem 2
n + 0.5 = -7.566372975

Simplifying
n + 0.5 = -7.566372975

Reorder the terms:
0.5 + n = -7.566372975

Solving
0.5 + n = -7.566372975

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 = -7.566372975 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -7.566372975 + -0.5
n = -7.566372975 + -0.5

Combine like terms: -7.566372975 + -0.5 = -8.066372975
n = -8.066372975

Simplifying
n = -8.066372975

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {7.066372975, -8.066372975}