n(n+1)=4012

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Solution for n(n+1)=4012 equation:


Simplifying
n(n + 1) = 4012

Reorder the terms:
n(1 + n) = 4012
(1 * n + n * n) = 4012
(1n + n2) = 4012

Solving
1n + n2 = 4012

Solving for variable 'n'.

Reorder the terms:
-4012 + 1n + n2 = 4012 + -4012

Combine like terms: 4012 + -4012 = 0
-4012 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4012' to each side of the equation.
-4012 + 1n + 4012 + n2 = 0 + 4012

Reorder the terms:
-4012 + 4012 + 1n + n2 = 0 + 4012

Combine like terms: -4012 + 4012 = 0
0 + 1n + n2 = 0 + 4012
1n + n2 = 0 + 4012

Combine like terms: 0 + 4012 = 4012
1n + n2 = 4012

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 = 4012 + 0.25

Reorder the terms:
0.25 + n + n2 = 4012 + 0.25

Combine like terms: 4012 + 0.25 = 4012.25
0.25 + n + n2 = 4012.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 4012.25

Calculate the square root of the right side: 63.342323923

Break this problem into two subproblems by setting 
(n + 0.5) equal to 63.342323923 and -63.342323923.

Subproblem 1

n + 0.5 = 63.342323923 Simplifying n + 0.5 = 63.342323923 Reorder the terms: 0.5 + n = 63.342323923 Solving 0.5 + n = 63.342323923 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 = 63.342323923 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 63.342323923 + -0.5 n = 63.342323923 + -0.5 Combine like terms: 63.342323923 + -0.5 = 62.842323923 n = 62.842323923 Simplifying n = 62.842323923

Subproblem 2

n + 0.5 = -63.342323923 Simplifying n + 0.5 = -63.342323923 Reorder the terms: 0.5 + n = -63.342323923 Solving 0.5 + n = -63.342323923 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 = -63.342323923 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -63.342323923 + -0.5 n = -63.342323923 + -0.5 Combine like terms: -63.342323923 + -0.5 = -63.842323923 n = -63.842323923 Simplifying n = -63.842323923

Solution

The solution to the problem is based on the solutions from the subproblems. n = {62.842323923, -63.842323923}

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