n(n+1)=126

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Solution for n(n+1)=126 equation:


Simplifying
n(n + 1) = 126

Reorder the terms:
n(1 + n) = 126
(1 * n + n * n) = 126
(1n + n2) = 126

Solving
1n + n2 = 126

Solving for variable 'n'.

Reorder the terms:
-126 + 1n + n2 = 126 + -126

Combine like terms: 126 + -126 = 0
-126 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '126' to each side of the equation.
-126 + 1n + 126 + n2 = 0 + 126

Reorder the terms:
-126 + 126 + 1n + n2 = 0 + 126

Combine like terms: -126 + 126 = 0
0 + 1n + n2 = 0 + 126
1n + n2 = 0 + 126

Combine like terms: 0 + 126 = 126
1n + n2 = 126

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 = 126 + 0.25

Reorder the terms:
0.25 + n + n2 = 126 + 0.25

Combine like terms: 126 + 0.25 = 126.25
0.25 + n + n2 = 126.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 126.25

Calculate the square root of the right side: 11.236102527

Break this problem into two subproblems by setting 
(n + 0.5) equal to 11.236102527 and -11.236102527.

Subproblem 1

n + 0.5 = 11.236102527 Simplifying n + 0.5 = 11.236102527 Reorder the terms: 0.5 + n = 11.236102527 Solving 0.5 + n = 11.236102527 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 = 11.236102527 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 11.236102527 + -0.5 n = 11.236102527 + -0.5 Combine like terms: 11.236102527 + -0.5 = 10.736102527 n = 10.736102527 Simplifying n = 10.736102527

Subproblem 2

n + 0.5 = -11.236102527 Simplifying n + 0.5 = -11.236102527 Reorder the terms: 0.5 + n = -11.236102527 Solving 0.5 + n = -11.236102527 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 = -11.236102527 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -11.236102527 + -0.5 n = -11.236102527 + -0.5 Combine like terms: -11.236102527 + -0.5 = -11.736102527 n = -11.736102527 Simplifying n = -11.736102527

Solution

The solution to the problem is based on the solutions from the subproblems. n = {10.736102527, -11.736102527}

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