6r^2+r+1=0

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


Simplifying
6r2 + r + 1 = 0

Reorder the terms:
1 + r + 6r2 = 0

Solving
1 + r + 6r2 = 0

Solving for variable 'r'.

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

Divide each side by '6'.
0.1666666667 + 0.1666666667r + r2 = 0

Move the constant term to the right:

Add '-0.1666666667' to each side of the equation.
0.1666666667 + 0.1666666667r + -0.1666666667 + r2 = 0 + -0.1666666667

Reorder the terms:
0.1666666667 + -0.1666666667 + 0.1666666667r + r2 = 0 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + 0.1666666667r + r2 = 0 + -0.1666666667
0.1666666667r + r2 = 0 + -0.1666666667

Combine like terms: 0 + -0.1666666667 = -0.1666666667
0.1666666667r + r2 = -0.1666666667

The r term is r.  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.
0.1666666667r + 0.25 + r2 = -0.1666666667 + 0.25

Reorder the terms:
0.25 + 0.1666666667r + r2 = -0.1666666667 + 0.25

Combine like terms: -0.1666666667 + 0.25 = 0.0833333333
0.25 + 0.1666666667r + r2 = 0.0833333333

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

Calculate the square root of the right side: 0.288675135

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

Subproblem 1

r + 0.5 = 0.288675135 Simplifying r + 0.5 = 0.288675135 Reorder the terms: 0.5 + r = 0.288675135 Solving 0.5 + r = 0.288675135 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 0.288675135 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 0.288675135 + -0.5 r = 0.288675135 + -0.5 Combine like terms: 0.288675135 + -0.5 = -0.211324865 r = -0.211324865 Simplifying r = -0.211324865

Subproblem 2

r + 0.5 = -0.288675135 Simplifying r + 0.5 = -0.288675135 Reorder the terms: 0.5 + r = -0.288675135 Solving 0.5 + r = -0.288675135 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -0.288675135 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -0.288675135 + -0.5 r = -0.288675135 + -0.5 Combine like terms: -0.288675135 + -0.5 = -0.788675135 r = -0.788675135 Simplifying r = -0.788675135

Solution

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

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