n(n+1)=10

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Solution for n(n+1)=10 equation: 


Simplifying
n(n + 1) = 10

Reorder the terms:
n(1 + n) = 10
(1 * n + n * n) = 10
(1n + n2) = 10

Solving
1n + n2 = 10

Solving for variable 'n'.

Reorder the terms:
-10 + 1n + n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 1n + 10 + n2 = 0 + 10

Reorder the terms:
-10 + 10 + 1n + n2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 1n + n2 = 0 + 10
1n + n2 = 0 + 10

Combine like terms: 0 + 10 = 10
1n + n2 = 10

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 = 10 + 0.25

Reorder the terms:
0.25 + n + n2 = 10 + 0.25

Combine like terms: 10 + 0.25 = 10.25
0.25 + n + n2 = 10.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(n + 0.5) equal to 3.201562119 and -3.201562119.


Subproblem 1
n + 0.5 = 3.201562119

Simplifying
n + 0.5 = 3.201562119

Reorder the terms:
0.5 + n = 3.201562119

Solving
0.5 + n = 3.201562119

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 = 3.201562119 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 3.201562119 + -0.5
n = 3.201562119 + -0.5

Combine like terms: 3.201562119 + -0.5 = 2.701562119
n = 2.701562119

Simplifying
n = 2.701562119


Subproblem 2
n + 0.5 = -3.201562119

Simplifying
n + 0.5 = -3.201562119

Reorder the terms:
0.5 + n = -3.201562119

Solving
0.5 + n = -3.201562119

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 = -3.201562119 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -3.201562119 + -0.5
n = -3.201562119 + -0.5

Combine like terms: -3.201562119 + -0.5 = -3.701562119
n = -3.701562119

Simplifying
n = -3.701562119


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.701562119, -3.701562119}

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