n(n+2)=12.5316

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Solution for n(n+2)=12.5316 equation:


Simplifying
n(n + 2) = 12.5316

Reorder the terms:
n(2 + n) = 12.5316
(2 * n + n * n) = 12.5316
(2n + n2) = 12.5316

Solving
2n + n2 = 12.5316

Solving for variable 'n'.

Reorder the terms:
-12.5316 + 2n + n2 = 12.5316 + -12.5316

Combine like terms: 12.5316 + -12.5316 = 0.0000
-12.5316 + 2n + n2 = 0.0000

Begin completing the square.

Move the constant term to the right:

Add '12.5316' to each side of the equation.
-12.5316 + 2n + 12.5316 + n2 = 0.0000 + 12.5316

Reorder the terms:
-12.5316 + 12.5316 + 2n + n2 = 0.0000 + 12.5316

Combine like terms: -12.5316 + 12.5316 = 0.0000
0.0000 + 2n + n2 = 0.0000 + 12.5316
2n + n2 = 0.0000 + 12.5316

Combine like terms: 0.0000 + 12.5316 = 12.5316
2n + n2 = 12.5316

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 12.5316 + 1

Reorder the terms:
1 + 2n + n2 = 12.5316 + 1

Combine like terms: 12.5316 + 1 = 13.5316
1 + 2n + n2 = 13.5316

Factor a perfect square on the left side:
(n + 1)(n + 1) = 13.5316

Calculate the square root of the right side: 3.678532316

Break this problem into two subproblems by setting 
(n + 1) equal to 3.678532316 and -3.678532316.

Subproblem 1

n + 1 = 3.678532316 Simplifying n + 1 = 3.678532316 Reorder the terms: 1 + n = 3.678532316 Solving 1 + n = 3.678532316 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 3.678532316 + -1 Combine like terms: 1 + -1 = 0 0 + n = 3.678532316 + -1 n = 3.678532316 + -1 Combine like terms: 3.678532316 + -1 = 2.678532316 n = 2.678532316 Simplifying n = 2.678532316

Subproblem 2

n + 1 = -3.678532316 Simplifying n + 1 = -3.678532316 Reorder the terms: 1 + n = -3.678532316 Solving 1 + n = -3.678532316 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -3.678532316 + -1 Combine like terms: 1 + -1 = 0 0 + n = -3.678532316 + -1 n = -3.678532316 + -1 Combine like terms: -3.678532316 + -1 = -4.678532316 n = -4.678532316 Simplifying n = -4.678532316

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.678532316, -4.678532316}

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