n(n+1)=440

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Solution for n(n+1)=440 equation:


Simplifying
n(n + 1) = 440

Reorder the terms:
n(1 + n) = 440
(1 * n + n * n) = 440
(1n + n2) = 440

Solving
1n + n2 = 440

Solving for variable 'n'.

Reorder the terms:
-440 + 1n + n2 = 440 + -440

Combine like terms: 440 + -440 = 0
-440 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '440' to each side of the equation.
-440 + 1n + 440 + n2 = 0 + 440

Reorder the terms:
-440 + 440 + 1n + n2 = 0 + 440

Combine like terms: -440 + 440 = 0
0 + 1n + n2 = 0 + 440
1n + n2 = 0 + 440

Combine like terms: 0 + 440 = 440
1n + n2 = 440

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 = 440 + 0.25

Reorder the terms:
0.25 + n + n2 = 440 + 0.25

Combine like terms: 440 + 0.25 = 440.25
0.25 + n + n2 = 440.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 440.25

Calculate the square root of the right side: 20.982135258

Break this problem into two subproblems by setting 
(n + 0.5) equal to 20.982135258 and -20.982135258.

Subproblem 1

n + 0.5 = 20.982135258 Simplifying n + 0.5 = 20.982135258 Reorder the terms: 0.5 + n = 20.982135258 Solving 0.5 + n = 20.982135258 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 = 20.982135258 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 20.982135258 + -0.5 n = 20.982135258 + -0.5 Combine like terms: 20.982135258 + -0.5 = 20.482135258 n = 20.482135258 Simplifying n = 20.482135258

Subproblem 2

n + 0.5 = -20.982135258 Simplifying n + 0.5 = -20.982135258 Reorder the terms: 0.5 + n = -20.982135258 Solving 0.5 + n = -20.982135258 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 = -20.982135258 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -20.982135258 + -0.5 n = -20.982135258 + -0.5 Combine like terms: -20.982135258 + -0.5 = -21.482135258 n = -21.482135258 Simplifying n = -21.482135258

Solution

The solution to the problem is based on the solutions from the subproblems. n = {20.482135258, -21.482135258}

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