6r^2-7r-1=0

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Solution for 6r^2-7r-1=0 equation: 


Simplifying
6r2 + -7r + -1 = 0

Reorder the terms:
-1 + -7r + 6r2 = 0

Solving
-1 + -7r + 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 + -1.166666667r + r2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + -1.166666667r + 0.1666666667 + r2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + -1.166666667r + r2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + -1.166666667r + r2 = 0 + 0.1666666667
-1.166666667r + r2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
-1.166666667r + r2 = 0.1666666667

The r term is -1.166666667r.  Take half its coefficient (-0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
-1.166666667r + 0.3402777780 + r2 = 0.1666666667 + 0.3402777780

Reorder the terms:
0.3402777780 + -1.166666667r + r2 = 0.1666666667 + 0.3402777780

Combine like terms: 0.1666666667 + 0.3402777780 = 0.5069444447
0.3402777780 + -1.166666667r + r2 = 0.5069444447

Factor a perfect square on the left side:
(r + -0.5833333335)(r + -0.5833333335) = 0.5069444447

Calculate the square root of the right side: 0.712000312

Break this problem into two subproblems by setting 
(r + -0.5833333335) equal to 0.712000312 and -0.712000312.


Subproblem 1
r + -0.5833333335 = 0.712000312

Simplifying
r + -0.5833333335 = 0.712000312

Reorder the terms:
-0.5833333335 + r = 0.712000312

Solving
-0.5833333335 + r = 0.712000312

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '0.5833333335' to each side of the equation.
-0.5833333335 + 0.5833333335 + r = 0.712000312 + 0.5833333335

Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000
0.0000000000 + r = 0.712000312 + 0.5833333335
r = 0.712000312 + 0.5833333335

Combine like terms: 0.712000312 + 0.5833333335 = 1.2953336455
r = 1.2953336455

Simplifying
r = 1.2953336455


Subproblem 2
r + -0.5833333335 = -0.712000312

Simplifying
r + -0.5833333335 = -0.712000312

Reorder the terms:
-0.5833333335 + r = -0.712000312

Solving
-0.5833333335 + r = -0.712000312

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '0.5833333335' to each side of the equation.
-0.5833333335 + 0.5833333335 + r = -0.712000312 + 0.5833333335

Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000
0.0000000000 + r = -0.712000312 + 0.5833333335
r = -0.712000312 + 0.5833333335

Combine like terms: -0.712000312 + 0.5833333335 = -0.1286669785
r = -0.1286669785

Simplifying
r = -0.1286669785


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1.2953336455, -0.1286669785}

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