n^2+n=360

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Solution for n^2+n=360 equation:


Simplifying
n2 + n = 360

Reorder the terms:
n + n2 = 360

Solving
n + n2 = 360

Solving for variable 'n'.

Reorder the terms:
-360 + n + n2 = 360 + -360

Combine like terms: 360 + -360 = 0
-360 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '360' to each side of the equation.
-360 + n + 360 + n2 = 0 + 360

Reorder the terms:
-360 + 360 + n + n2 = 0 + 360

Combine like terms: -360 + 360 = 0
0 + n + n2 = 0 + 360
n + n2 = 0 + 360

Combine like terms: 0 + 360 = 360
n + n2 = 360

The n term is n.  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 = 360 + 0.25

Reorder the terms:
0.25 + n + n2 = 360 + 0.25

Combine like terms: 360 + 0.25 = 360.25
0.25 + n + n2 = 360.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 360.25

Calculate the square root of the right side: 18.980252896

Break this problem into two subproblems by setting 
(n + 0.5) equal to 18.980252896 and -18.980252896.

Subproblem 1

n + 0.5 = 18.980252896 Simplifying n + 0.5 = 18.980252896 Reorder the terms: 0.5 + n = 18.980252896 Solving 0.5 + n = 18.980252896 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 = 18.980252896 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 18.980252896 + -0.5 n = 18.980252896 + -0.5 Combine like terms: 18.980252896 + -0.5 = 18.480252896 n = 18.480252896 Simplifying n = 18.480252896

Subproblem 2

n + 0.5 = -18.980252896 Simplifying n + 0.5 = -18.980252896 Reorder the terms: 0.5 + n = -18.980252896 Solving 0.5 + n = -18.980252896 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 = -18.980252896 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -18.980252896 + -0.5 n = -18.980252896 + -0.5 Combine like terms: -18.980252896 + -0.5 = -19.480252896 n = -19.480252896 Simplifying n = -19.480252896

Solution

The solution to the problem is based on the solutions from the subproblems. n = {18.480252896, -19.480252896}

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