6r^2-23r-5=

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Solution for 6r^2-23r-5= equation: 


Simplifying
6r2 + -23r + -5 = 0

Reorder the terms:
-5 + -23r + 6r2 = 0

Solving
-5 + -23r + 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'.
-0.8333333333 + -3.833333333r + r2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + -3.833333333r + 0.8333333333 + r2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + -3.833333333r + r2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + -3.833333333r + r2 = 0 + 0.8333333333
-3.833333333r + r2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
-3.833333333r + r2 = 0.8333333333

The r term is -3.833333333r.  Take half its coefficient (-1.916666667).
Square it (3.673611112) and add it to both sides.

Add '3.673611112' to each side of the equation.
-3.833333333r + 3.673611112 + r2 = 0.8333333333 + 3.673611112

Reorder the terms:
3.673611112 + -3.833333333r + r2 = 0.8333333333 + 3.673611112

Combine like terms: 0.8333333333 + 3.673611112 = 4.5069444453
3.673611112 + -3.833333333r + r2 = 4.5069444453

Factor a perfect square on the left side:
(r + -1.916666667)(r + -1.916666667) = 4.5069444453

Calculate the square root of the right side: 2.122956534

Break this problem into two subproblems by setting 
(r + -1.916666667) equal to 2.122956534 and -2.122956534.


Subproblem 1
r + -1.916666667 = 2.122956534

Simplifying
r + -1.916666667 = 2.122956534

Reorder the terms:
-1.916666667 + r = 2.122956534

Solving
-1.916666667 + r = 2.122956534

Solving for variable 'r'.

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

Add '1.916666667' to each side of the equation.
-1.916666667 + 1.916666667 + r = 2.122956534 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + r = 2.122956534 + 1.916666667
r = 2.122956534 + 1.916666667

Combine like terms: 2.122956534 + 1.916666667 = 4.039623201
r = 4.039623201

Simplifying
r = 4.039623201


Subproblem 2
r + -1.916666667 = -2.122956534

Simplifying
r + -1.916666667 = -2.122956534

Reorder the terms:
-1.916666667 + r = -2.122956534

Solving
-1.916666667 + r = -2.122956534

Solving for variable 'r'.

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

Add '1.916666667' to each side of the equation.
-1.916666667 + 1.916666667 + r = -2.122956534 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + r = -2.122956534 + 1.916666667
r = -2.122956534 + 1.916666667

Combine like terms: -2.122956534 + 1.916666667 = -0.206289867
r = -0.206289867

Simplifying
r = -0.206289867


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {4.039623201, -0.206289867}

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