n(n+1)=750

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Solution for n(n+1)=750 equation:


Simplifying
n(n + 1) = 750

Reorder the terms:
n(1 + n) = 750
(1 * n + n * n) = 750
(1n + n2) = 750

Solving
1n + n2 = 750

Solving for variable 'n'.

Reorder the terms:
-750 + 1n + n2 = 750 + -750

Combine like terms: 750 + -750 = 0
-750 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '750' to each side of the equation.
-750 + 1n + 750 + n2 = 0 + 750

Reorder the terms:
-750 + 750 + 1n + n2 = 0 + 750

Combine like terms: -750 + 750 = 0
0 + 1n + n2 = 0 + 750
1n + n2 = 0 + 750

Combine like terms: 0 + 750 = 750
1n + n2 = 750

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 = 750 + 0.25

Reorder the terms:
0.25 + n + n2 = 750 + 0.25

Combine like terms: 750 + 0.25 = 750.25
0.25 + n + n2 = 750.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 750.25

Calculate the square root of the right side: 27.39069185

Break this problem into two subproblems by setting 
(n + 0.5) equal to 27.39069185 and -27.39069185.

Subproblem 1

n + 0.5 = 27.39069185 Simplifying n + 0.5 = 27.39069185 Reorder the terms: 0.5 + n = 27.39069185 Solving 0.5 + n = 27.39069185 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 = 27.39069185 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 27.39069185 + -0.5 n = 27.39069185 + -0.5 Combine like terms: 27.39069185 + -0.5 = 26.89069185 n = 26.89069185 Simplifying n = 26.89069185

Subproblem 2

n + 0.5 = -27.39069185 Simplifying n + 0.5 = -27.39069185 Reorder the terms: 0.5 + n = -27.39069185 Solving 0.5 + n = -27.39069185 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 = -27.39069185 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -27.39069185 + -0.5 n = -27.39069185 + -0.5 Combine like terms: -27.39069185 + -0.5 = -27.89069185 n = -27.89069185 Simplifying n = -27.89069185

Solution

The solution to the problem is based on the solutions from the subproblems. n = {26.89069185, -27.89069185}

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