n(n+2)=220

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Solution for n(n+2)=220 equation:


Simplifying
n(n + 2) = 220

Reorder the terms:
n(2 + n) = 220
(2 * n + n * n) = 220
(2n + n2) = 220

Solving
2n + n2 = 220

Solving for variable 'n'.

Reorder the terms:
-220 + 2n + n2 = 220 + -220

Combine like terms: 220 + -220 = 0
-220 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '220' to each side of the equation.
-220 + 2n + 220 + n2 = 0 + 220

Reorder the terms:
-220 + 220 + 2n + n2 = 0 + 220

Combine like terms: -220 + 220 = 0
0 + 2n + n2 = 0 + 220
2n + n2 = 0 + 220

Combine like terms: 0 + 220 = 220
2n + n2 = 220

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 = 220 + 1

Reorder the terms:
1 + 2n + n2 = 220 + 1

Combine like terms: 220 + 1 = 221
1 + 2n + n2 = 221

Factor a perfect square on the left side:
(n + 1)(n + 1) = 221

Calculate the square root of the right side: 14.866068747

Break this problem into two subproblems by setting 
(n + 1) equal to 14.866068747 and -14.866068747.

Subproblem 1

n + 1 = 14.866068747 Simplifying n + 1 = 14.866068747 Reorder the terms: 1 + n = 14.866068747 Solving 1 + n = 14.866068747 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 = 14.866068747 + -1 Combine like terms: 1 + -1 = 0 0 + n = 14.866068747 + -1 n = 14.866068747 + -1 Combine like terms: 14.866068747 + -1 = 13.866068747 n = 13.866068747 Simplifying n = 13.866068747

Subproblem 2

n + 1 = -14.866068747 Simplifying n + 1 = -14.866068747 Reorder the terms: 1 + n = -14.866068747 Solving 1 + n = -14.866068747 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 = -14.866068747 + -1 Combine like terms: 1 + -1 = 0 0 + n = -14.866068747 + -1 n = -14.866068747 + -1 Combine like terms: -14.866068747 + -1 = -15.866068747 n = -15.866068747 Simplifying n = -15.866068747

Solution

The solution to the problem is based on the solutions from the subproblems. n = {13.866068747, -15.866068747}

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