6n^2+6n+1=0

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Solution for 6n^2+6n+1=0 equation:


Simplifying
6n2 + 6n + 1 = 0

Reorder the terms:
1 + 6n + 6n2 = 0

Solving
1 + 6n + 6n2 = 0

Solving for variable 'n'.

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

Divide each side by '6'.
0.1666666667 + n + n2 = 0

Move the constant term to the right:

Add '-0.1666666667' to each side of the equation.
0.1666666667 + n + -0.1666666667 + n2 = 0 + -0.1666666667

Reorder the terms:
0.1666666667 + -0.1666666667 + n + n2 = 0 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + n + n2 = 0 + -0.1666666667
n + n2 = 0 + -0.1666666667

Combine like terms: 0 + -0.1666666667 = -0.1666666667
n + n2 = -0.1666666667

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 = -0.1666666667 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = -0.1666666667 + 0.25

Combine like terms: -0.1666666667 + 0.25 = 0.0833333333
1.25 + n2 = 0.0833333333

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 0.0833333333

Calculate the square root of the right side: 0.288675135

Break this problem into two subproblems by setting 
(n + 0.5) equal to 0.288675135 and -0.288675135.

Subproblem 1

n + 0.5 = 0.288675135 Simplifying n + 0.5 = 0.288675135 Reorder the terms: 0.5 + n = 0.288675135 Solving 0.5 + n = 0.288675135 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 = 0.288675135 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.288675135 + -0.5 n = 0.288675135 + -0.5 Combine like terms: 0.288675135 + -0.5 = -0.211324865 n = -0.211324865 Simplifying n = -0.211324865

Subproblem 2

n + 0.5 = -0.288675135 Simplifying n + 0.5 = -0.288675135 Reorder the terms: 0.5 + n = -0.288675135 Solving 0.5 + n = -0.288675135 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 = -0.288675135 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.288675135 + -0.5 n = -0.288675135 + -0.5 Combine like terms: -0.288675135 + -0.5 = -0.788675135 n = -0.788675135 Simplifying n = -0.788675135

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.211324865, -0.788675135}

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