(n)(n+1)=35

Solution for (n)(n+1)=35 equation:

Simplifying
(n)(n + 1) = 35

Reorder the terms:
n(1 + n) = 35
(1 * n + n * n) = 35
(1n + n2) = 35

Solving
1n + n2 = 35

Solving for variable 'n'.

Reorder the terms:
-35 + 1n + n2 = 35 + -35

Combine like terms: 35 + -35 = 0
-35 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '35' to each side of the equation.
-35 + 1n + 35 + n2 = 0 + 35

Reorder the terms:
-35 + 35 + 1n + n2 = 0 + 35

Combine like terms: -35 + 35 = 0
0 + 1n + n2 = 0 + 35
1n + n2 = 0 + 35

Combine like terms: 0 + 35 = 35
1n + n2 = 35

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 = 35 + 0.25

Reorder the terms:
0.25 + n + n2 = 35 + 0.25

Combine like terms: 35 + 0.25 = 35.25
0.25 + n + n2 = 35.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 35.25

Calculate the square root of the right side: 5.937171044

Break this problem into two subproblems by setting 
(n + 0.5) equal to 5.937171044 and -5.937171044.

Subproblem 1
n + 0.5 = 5.937171044

Simplifying
n + 0.5 = 5.937171044

Reorder the terms:
0.5 + n = 5.937171044

Solving
0.5 + n = 5.937171044

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 = 5.937171044 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 5.937171044 + -0.5
n = 5.937171044 + -0.5

Combine like terms: 5.937171044 + -0.5 = 5.437171044
n = 5.437171044

Simplifying
n = 5.437171044

Subproblem 2
n + 0.5 = -5.937171044

Simplifying
n + 0.5 = -5.937171044

Reorder the terms:
0.5 + n = -5.937171044

Solving
0.5 + n = -5.937171044

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 = -5.937171044 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -5.937171044 + -0.5
n = -5.937171044 + -0.5

Combine like terms: -5.937171044 + -0.5 = -6.437171044
n = -6.437171044

Simplifying
n = -6.437171044

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {5.437171044, -6.437171044}