n(n+1)=48

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Solution for n(n+1)=48 equation: 


Simplifying
n(n + 1) = 48

Reorder the terms:
n(1 + n) = 48
(1 * n + n * n) = 48
(1n + n2) = 48

Solving
1n + n2 = 48

Solving for variable 'n'.

Reorder the terms:
-48 + 1n + n2 = 48 + -48

Combine like terms: 48 + -48 = 0
-48 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '48' to each side of the equation.
-48 + 1n + 48 + n2 = 0 + 48

Reorder the terms:
-48 + 48 + 1n + n2 = 0 + 48

Combine like terms: -48 + 48 = 0
0 + 1n + n2 = 0 + 48
1n + n2 = 0 + 48

Combine like terms: 0 + 48 = 48
1n + n2 = 48

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 = 48 + 0.25

Reorder the terms:
0.25 + n + n2 = 48 + 0.25

Combine like terms: 48 + 0.25 = 48.25
0.25 + n + n2 = 48.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 48.25

Calculate the square root of the right side: 6.946221995

Break this problem into two subproblems by setting 
(n + 0.5) equal to 6.946221995 and -6.946221995.


Subproblem 1
n + 0.5 = 6.946221995

Simplifying
n + 0.5 = 6.946221995

Reorder the terms:
0.5 + n = 6.946221995

Solving
0.5 + n = 6.946221995

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 = 6.946221995 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 6.946221995 + -0.5
n = 6.946221995 + -0.5

Combine like terms: 6.946221995 + -0.5 = 6.446221995
n = 6.446221995

Simplifying
n = 6.446221995


Subproblem 2
n + 0.5 = -6.946221995

Simplifying
n + 0.5 = -6.946221995

Reorder the terms:
0.5 + n = -6.946221995

Solving
0.5 + n = -6.946221995

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 = -6.946221995 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -6.946221995 + -0.5
n = -6.946221995 + -0.5

Combine like terms: -6.946221995 + -0.5 = -7.446221995
n = -7.446221995

Simplifying
n = -7.446221995


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.446221995, -7.446221995}

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