-0.16666666=n^2+n

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Solution for -0.16666666=n^2+n equation:


Simplifying
-0.16666666 = n2 + n

Reorder the terms:
-0.16666666 = n + n2

Solving
-0.16666666 = n + n2

Solving for variable 'n'.

Reorder the terms:
-0.16666666 + -1n + -1n2 = n + -1n + n2 + -1n2

Combine like terms: n + -1n = 0
-0.16666666 + -1n + -1n2 = 0 + n2 + -1n2
-0.16666666 + -1n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
-0.16666666 + -1n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
0.16666666 + n + n2 = 0

Move the constant term to the right:

Add '-0.16666666' to each side of the equation.
0.16666666 + n + -0.16666666 + n2 = 0 + -0.16666666

Reorder the terms:
0.16666666 + -0.16666666 + n + n2 = 0 + -0.16666666

Combine like terms: 0.16666666 + -0.16666666 = 0.00000000
0.00000000 + n + n2 = 0 + -0.16666666
n + n2 = 0 + -0.16666666

Combine like terms: 0 + -0.16666666 = -0.16666666
n + n2 = -0.16666666

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.
 + 0.25 + n2 = -0.16666666 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = -0.16666666 + 0.25

Combine like terms: -0.16666666 + 0.25 = 0.08333334
1.25 + n2 = 0.08333334

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 0.08333334

Calculate the square root of the right side: 0.288675146

Break this problem into two subproblems by setting 
(n + 0.5) equal to 0.288675146 and -0.288675146.

Subproblem 1

n + 0.5 = 0.288675146 Simplifying n + 0.5 = 0.288675146 Reorder the terms: 0.5 + n = 0.288675146 Solving 0.5 + n = 0.288675146 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 = 0.288675146 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.288675146 + -0.5 n = 0.288675146 + -0.5 Combine like terms: 0.288675146 + -0.5 = -0.211324854 n = -0.211324854 Simplifying n = -0.211324854

Subproblem 2

n + 0.5 = -0.288675146 Simplifying n + 0.5 = -0.288675146 Reorder the terms: 0.5 + n = -0.288675146 Solving 0.5 + n = -0.288675146 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 = -0.288675146 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.288675146 + -0.5 n = -0.288675146 + -0.5 Combine like terms: -0.288675146 + -0.5 = -0.788675146 n = -0.788675146 Simplifying n = -0.788675146

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.211324854, -0.788675146}

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